Pacific Journal of Mathematics Vol. 249 (2011), No. 1

Paciﬁc Journal of Mathematics

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Vyjayanthi Chari Darren Long Sorin Popa Department of Mathematics Department of Mathematics Department of Mathematics University of California University of California University of California Riverside, CA 92521-0135 Santa Barbara, CA 93106-3080 Los Angeles, CA 90095-1555 [email protected] [email protected] [email protected]

Robert Finn Jiang-Hua Lu Jie Qing Department of Mathematics Department of Mathematics Department of Mathematics Stanford University The University of Hong Kong University of California Stanford, CA 94305-2125 Pokfulam Rd., Hong Kong Santa Cruz, CA 95064 ﬁ[email protected] [email protected] [email protected]

Kefeng Liu Alexander Merkurjev Jonathan Rogawski Department of Mathematics Department of Mathematics Department of Mathematics University of California University of California University of California Los Angeles, CA 90095-1555 Los Angeles, CA 90095-1555 Los Angeles, CA 90095-1555 [email protected] [email protected] [email protected]

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METABELIAN SL(n, ރ) REPRESENTATIONS OF KNOT GROUPS, II: FIXED POINTS

HANS U.BODENAND STEFAN FRIEDL

Given a knot K in an integral homology sphere 6 with exterior NK , there is a natural action of the cyclic group ޚ/n on the space of SL(n, ރ) representations of the knot group π1(NK ), which induces an action on the SL(n, ރ) character variety. We identify the ﬁxed points of this action in terms of characters of metabelian representations, and we apply this in order to show α that the twisted Alexander polynomial 1K,1(t) associated to an irreducible metabelian SL(n, ރ) representation α is actually a polynomial in tn.

1. Introduction

The study of metabelian representations and metabelian quotients of knot groups goes back to the pioneering work of Neuwirth [1965], de Rham [1967], Burde [1967], and Fox [1970]; see also [Burde and Zieschang 2003, Section 14]. The theory was further developed by many authors, including Hartley [1979; 1983], Livingston [1995], Letsche [2000], Lin [2001], Nagasato [2007], and Jebali [2008]. In [Boden and Friedl 2008], we proved a classiﬁcation theorem for irreducible metabelian representations and in this paper we continue our study of metabelian representations of knot groups. Throughout this paper, when we say that K is a knot, we will always understand that K is an oriented, simple closed curve in an integral homology 3-sphere 6. We 3 write NK = 6 \ τ(K ), where τ(K ) denotes an open tubular neighborhood of K . Given a topological space M, let Rn(M) be the space of SL(n, ރ) representations of π1(M), and let Xn(M) be the associated character variety. We use ξα to denote the character of the representation α : π1(M) → SL(n, ރ). We will often make use of the important fact that two irreducible representations determine the same character if and only if they are conjugate; see [Lubotzky and Magid 1985, Corollary 1.33]. Now suppose K is a knot. The group ޚ/n has an action on the representation k 2πik/n variety Rn(NK ), given by twisting by the n-th roots of unity ω = e ∈ U(1).

MSC2000: primary 57M25; secondary 20C15. Keywords: metabelian representation, knot group, character variety, group action, ﬁxed point.

1 2 HANSU.BODEN AND STEFAN FRIEDL

(This is a special case of the more general twisting operation described in [Lubotzky and Magid 1985, Chapter 5].) More precisely, we write ޚ/n = hσ | σ n = 1i and ε(g) set (σ ·α)(g) = ω α(g) for each g ∈ π1(NK ), where ε : π1(NK ) → H1(NK ) = ޚ is determined by the given orientation of the knot. This constructs an action of ޚ/n on Rn(NK ) which, in turn, descends to an action on the character variety Xn(NK ). Our main result identiﬁes the ﬁxed points ∗ of ޚ/n in Xn(NK ) — the irreducible characters — as those associated to metabelian representations.

Theorem 1. The character ξα of an irreducible representation α : π1(NK ) → SL(n, ރ) is fixed under the ޚ/n action if and only if α is metabelian. In proving this result, we will actually characterize the entire fixed point set ޚ/n Xn(NK ) in terms of characters ξα of the metabelian representations α = α(n,χ) described in Section 2.3 (see Theorem 4). When n = 2, it turns out that every metabelian SL(2, ރ) representation is dihedral. For this case, Theorem 1 was first proved by F. Nagasato and Y. Yamaguchi [2008, Proposition 4.8]. As an application of Theorem 1, we prove a result about the twisted Alexander polynomials associated to metabelian representations. This result was first shown by C. Herald, P. Kirk and C. Livingston [2010] using completely different methods. Our approach is elementary and natural, and is explained in Section 3.2, where we apply it to give an answer to a question raised by Hirasawa and Murasugi [2009].

2. The classiﬁcation of metabelian representations of knot groups

We recall some results from [Boden and Friedl 2008] regarding the classiﬁcation of metabelian representations of knot groups.

2.1. Preliminaries. Given a group π, we write π (n) for the n-th term of the derived series of π. These subgroups are deﬁned inductively by setting π (0) = π and π (i+1) = [π (i), π (i)]. The group π is called metabelian if π (2) = {e}. Suppose V is a ﬁnite-dimensional vector space over ރ. A representation % : π → Aut(V ) is called metabelian if % factors through π/π (2). The representation % is called reducible if there exists a proper subspace U ⊂ V invariant under %(γ ) for all γ ∈ π. Otherwise, % is called irreducible or simple. If % is the direct sum of simple representations, then % is called semisimple. Two representations %1 : π → Aut(V ) and %2 : π → Aut(W) are called isomor- −1 phic if there exists an isomorphism ϕ : V → W such that ϕ ◦ %1(g) ◦ ϕ = %2(g) for all g ∈ π.

2.2. Metabelian quotients of knot groups. Let K ⊂ 63 be a knot in an integral homology 3-sphere. Denote by NeK the inﬁnite cyclic cover of NK corresponding METABELIAN SL(n, ރ) REPRESENTATIONSOFKNOTGROUPS,II 3

∼ (1) to the abelianization π1(NK ) → H1(NK ) = ޚ. Thus, π1(NeK ) = π1(NK ) and ±1 ∼ (1) (2) H1 NK ; ޚ[t ] = H1(NeK ) = π1(NK ) /π1(NK ) . The ޚ[t±1]-module structure is given on the right hand side by tn · g := µ−n g µn, where µ is a meridian of K . For a knot K , we set π := π1(NK ) and consider the short exact sequence 1 → π (1)/π (2) → π/π (2) → π/π (1) → 1.

(1) ∼ Since π/π = H1(NK ) = ޚ, this sequence splits, and we get isomorphisms (2) ∼ (1) (1) (2) ∼ (1) (2) ∼ ±1 π/π = π/π n π /π = ޚ n π /π = ޚ n H1(NK ; ޚ[t ]), g 7→ (µε(g), µ−ε(g)g) 7→ (ε(g), µ−ε(g)g), where the semidirect products are taken with respect to the ޚ actions deﬁned by (1) (2) n ±1 letting n ∈ ޚ act on π /π by conjugation by µ , and on H1(NK ; ޚ[t ]) by multiplication by tn.

2.3. Irreducible metabelian SL(n, ރ) representations of knot groups. Let K be ±1 a knot and write H = H1(NK ; ޚ[t ]). The discussion of the previous section shows that irreducible metabelian SL(n, ރ) representations of π1(NK ) correspond precisely to the irreducible SL(n, ރ) representations of ޚ n H. Let χ : H → ރ∗ be a character that factors through H/(tn − 1), and take z ∈ S1 with zn = (−1)n+1. It follows from [Boden and Friedl 2008, Section 3] that, for ( j, h) ∈ ޚ n H, j 0 0 ··· z χ(h) 0 ··· 0   ···   ···  z 0 0  0 χ(th) 0  α(χ,z)( j, h) =     ......   . . .. .  . . . .  . . . .  0 ··· z 0 0 0 ··· χ(tn−1h) deﬁnes an SL(n, ރ) representation whose isomorphism type does not depend on the choice of z. In our notation we will not normally distinguish between metabelian representations of π1(NK ) and representations of ޚ n H. We say that a character χ : H → ރ∗ has order n if it factors through H/(tn −1) but not through H/(t` −1) for any ` < n. Given a character χ : H → ރ∗, let ti χ be the character deﬁned by (ti χ)(h) = χ(ti h). Any character χ : H → ރ∗ that factors through H/(tn − 1) must have order k for some divisor k of n. The next statement is a combination of [Boden and Friedl 2008, Lemma 2.2 and Theorem 3.3]. Theorem 2. Suppose χ : H → ރ∗ is a character that factors through H/(tn − 1).

(i) α(n,χ) : ޚ n H → SL(n, ރ) is irreducible if and only if the character χ has order n. 4 HANSU.BODEN AND STEFAN FRIEDL

0 ∗ (ii) Given two characters χ, χ : H → ރ of order n, the representations α(n,χ) k 0 and α(n,χ 0) are conjugate if and only if χ = t χ for some k. (iii) For any irreducible representation α : ޚn H → SL(n, ރ), there is a character ∗ χ : H → ރ of order n such that α is conjugate to α(n,χ).

3. Main results

3.1. Metabelian characters as ﬁxed points. Set ω = e2πi/n and recall the action n of the cyclic group ޚ/n = hσ | σ = 1i on representations α : π1(NK ) → SL(n, ރ) ε(g) obtained by setting (σ ·α)(g) = ω α(g) for all g ∈ π1(NK ), where ε : π1(NK ) → H1(NK ) = ޚ.

Lemma 3. Suppose α : π1(NK ) → SL(n, ރ) is a representation whose associated character ξα ∈ Xn(NK ) is a ﬁxed point of the ޚ/n action. Up to conjugation, 0 0 ··· z z 0 ··· 0 (1) α(µ) =   ......  . . . . 0 ··· z 0 for some (in fact, any) z ∈ U(1) such that zn = (−1)n+1. Proof. Let c(t) = det(α(µ) − t I ) denote the characteristic polynomial of α(µ), which we can write as

n n n−1 c(t) = (−1) t + cn−1t + · · · + c1t + 1.

Note that c(t) is determined by the character ξα ∈ Xn(NK ) and so, assuming ξα is a ﬁxed point of the ޚ/n action, we conclude that α(µ) and ωkα(µ) have the same characteristic polynomials for all k. In particular, c(t) = det(α(µ) − t I ) = det(ω−1α(µ) − t I ) = det(ω−1α(µ) − (ω−1ω)t I ) = det(ω−1 I ) det(α(µ) − ωt I ) = det(α(µ) − tωI ) = c(ωt).

k However, ω 6= 1 unless n divides k, and this implies 0 = cn−1 = cn−2 = · · · = c1 and c(t) = (−1)ntn +1. In particular, the matrix α(µ) and the matrix appearing in Equation (1) have the same set of n distinct eigenvalues. This implies that the two matrices are conjugate. To prove Theorem 1, we establish the following more general result:

Theorem 4. The ﬁxed point set of the ޚ/n action on Xn(NK ) consists of characters ξα of the metabelian representations α = α(n,χ) described in Section 2.3. In other ޚ/n ±1 ∗ words, Xn(NK ) = {ξα | α = α(n,χ) for χ : H1(NK ; ޚ[t ]) → ރ }. METABELIAN SL(n, ރ) REPRESENTATIONSOFKNOTGROUPS,II 5

Theorem 1 can be viewed as the special case of Theorem 4 when α is irreducible. (Recall that irreducible representations are conjugate if and only if they deﬁne the same character.) Note that not every reducible metabelian representation is of the form α(n,χ).

Proof. We ﬁrst show that if α : π1(NK ) → SL(n, ރ) is given as α = α(n,χ), then σ · α is conjugate to α. This of course implies that ξα = ξσ·α. Assume then that α = α(n,χ). We have

0 0 ··· z z 0 ··· 0 α(µ) =   , ......  . . . . 0 ··· z 0

n n+1 where z satisfies z =(−1) . Also α(g) is diagonal for all g ∈[π1(NK ), π1(NK )]. From the definition of σ · α, we see that  0 0 ··· ωz ωz 0 ··· 0  (σ · α)(µ) = ω α(µ) =    ......   . . . .  0 ··· ωz 0 and that (σ · α)(g) = α(g) for all g ∈ [π1(NK ), π1(NK )]. It follows easily from Theorem 2(ii) that σ · α and α(n,χ) are conjugate; however, it is easy to see this directly too. Simply take 1 0   ω  P =   ,  ..   .  0 ωn−1 and compute that σ · α = P αP−1 as claimed. ޚ/n We now show the other implication, namely, that each point ξ ∈ Xn(NK ) in the fixed point set can be represented as the character ξ = ξα of a metabelian ±1 ∗ representation α = α(n,χ), where χ : H1(NK ; ޚ[t ]) → ރ is a character that ±1 n factors through H1(NK ; ޚ[t ])/(t − 1) and hence has order k for some k that divides n. (Note that Theorem 2(i) tells us that α(n,χ) is irreducible if and only if χ has order n.) From the general results on representation spaces and character varieties (see [Lubotzky and Magid 1985]), it follows that every point in the character variety Xn(NK ) can be represented as ξα for some semisimple representation α : π1(NK ) → SL(n, ރ). Further, two semisimple representations α1 and α2 determine 6 HANSU.BODEN AND STEFAN FRIEDL the same character if and only if α1 is conjugate to α2. (This is evident from the fact that the orbits of the semisimple representations under conjugation are closed.) ޚ/n Given ξ ∈ Xn(NK ) , we can therefore suppose that ξ =ξα for some semisimple representation α. Clearly σ ·α is also semisimple, and since ξα = ξσ·α, we conclude from the previous argument that α and σ · α are conjugate representations. This means that there exists a matrix A ∈ SL(n, ރ) such that Aα A−1 = σ · α. In other words, for all g ∈ π1(NK ), we have (2) Aα(g)A−1 = ωε(g)α(g). Lemma 3 implies that α(µ) is conjugate to the matrix in Equation (1). It is convenient to conjugate α so that α(µ) is diagonal, meaning that z 0   ωz  α(µ) =   ,  ..   .  0 ωn−1z where z satisfies zn = (−1)n+1. We now apply (2) to the meridian to conclude that Aα(µ) = ω α(µ) A, which implies that A = (ai j ) satisfies ai j = 0 unless j = i + 1 mod (n). Thus, we see that   0 λ1 0 ··· 0  0 0 λ2 ··· 0    =  ......  A  . . . . .     0 0 ··· 0 λn−1 λn 0 ··· 0 0 n+1 for some λ1, . . . , λn satisfying λ1 ··· λn = (−1) . It is completely straightforward to see that the characteristic polynomial of A is det(A − t I ) = (−1)n(tn − (−1)n+1). From this, we conclude that A has as its eigenvalues the n distinct n-th roots of (−1)n+1. In particular, the subset of matrices in SL(n, ރ) that commute with A is ∼ ∗ n−1 just a copy of the unique maximal torus TA = (ރ ) containing A. For any g ∈ [π1(NK ), π1(NK )], we have α(g) = (σ ·α)(g). Thus it follows that −1 Aα(g)A = α(g), and this implies that α(g) ∈ TA for all g ∈ [π1(NK ), π1(NK )]. This shows that the restriction of α to the commutator subgroup [π1(NK ), π1(NK )] is abelian. We conclude from this that α is indeed metabelian. Notice that this, and an application of Theorem 2(iii), completes the proof in case α is irreducible. METABELIAN SL(n, ރ) REPRESENTATIONSOFKNOTGROUPS,II 7

In the general case, it follows from the discussion in Section 2.2 that α factors ±1 ±1 through ޚ n H1(NK ; ޚ[t ]). Let H = H1(NK ; ޚ[t ]). Given a character χ : ∗ H → ރ , we deﬁne the associated weight space Vχ by setting

n Vχ = {v ∈ ރ | χ(h) · v = α(h)v for all h ∈ H}.

Recall that A · α(h) · A−1 = α(h) for any h ∈ H. It is straightforward to show that A restricts to an automorphism of Vχ . Since H is abelian, there exists at least ∗ i one character χ : H → ރ such that Vχ is nontrivial. For any i, denote by t χ the character given by (ti χ)(h) = χ(ti h) for h ∈ H. Note that A has n distinct eigenvalues and therefore is diagonalizable. Since A restricts to an automorphism of Vχ , there is an eigenvector v of A that lies in Vχ . Let λ be the corresponding eigenvalue. By the proof of [Boden and Friedl 2008, Theorem 2.3], the map α(µ) induces an isomorphism Vχ → Vtχ . We now calculate

A · α(µ)v = (Aα(µ)A−1) · Av = ω α(µ) · λv = λω · α(µ)v; that is, α(µ)v ∈ Vtχ is an eigenvector of A with eigenvalue ωλ. i Iterating this argument, we see that α(µ) v lies in Vti χ and is an eigenvector of A with eigenvalue ωi λ. Since ω is a primitive n-th root of unity, the eigenvalues λ, ωλ, . . . , ωn−1λ are all distinct, and this implies that the corresponding eigenvectors v, α(µ)v, . . . , α(µ)n−1v form a basis for ރn. Let m be the order of χ; that is, m is the minimal number such that χ = tmχ. n From the previous argument, we see that ރ is generated by Vχ , Vtχ ,..., Vtm χ . Since the characters χ, tχ, . . . , tmχ are pairwise distinct, it follows that ރn is given as the direct sum Vχ ⊕ Vtχ ⊕ · · · ⊕ Vtm−1χ . m We write k = dimރ(Vχ ) and note that n = km. We note further that α(µ) has eigenvalues

(3) {zm, zme2πi/k,..., zme2πi(k−1)/k}, and each eigenvalue has multiplicity m. Clearly α(µ)m restricts to an automorphism of Vti χ for i = 0,..., m −1, and equally clearly we see that the restrictions all give conjugate representations. This implies that the restriction of α(µ)m to Vχ has eigenvalues in the set (3) above, each occurring with multiplicity 1. In m particular, we can ﬁnd a basis {v1, . . . , vk} for Vχ in which the matrix of α(µ) has the form  0 0 ··· zm zm 0 ··· 0  α(µm) =   .  ......   . . . .  0 ··· zm 0 8 HANSU.BODEN AND STEFAN FRIEDL

It is now straightforward to verify that, with respect to the ordered basis    z−1 ··· z−(m−1) m−1   v1, α(µ)v1, , α(µ) v1,     −1 −(m−1) m−1   v2, z α(µ)v2, ··· , z α(µ) v2,  . . . ,  . . .       −1 −(m−1) m−1   vk, z α(µ)vk, ··· , z α(µ) vk 

α is given by α(n, χ). 3.2. Application to twisted Alexander polynomials. As an application, we prove a the following result regarding twisted Alexander polynomials of knots correspond- α ing to metabelian representations. Denote by 1K,i (t) the i-th twisted Alexander polynomial for a given representation α : π1(NK ) → SL(n, ރ), as presented in [Friedl and Vidussi 2009].

Proposition 5. Let α be a metabelian representation of the form α = α(n,χ) : π1(NK ) → SL(n, ރ). Then 1 − tn if χ is trivial, 1α (t) = K,0 1 otherwise. α n Further, the twisted Alexander polynomial 1K,1(t) is actually a polynomial in t . Remark 6. In their paper, C. Herald, P. Kirk, and C. Livingston prove the same result using an entirely different approach; compare with [Herald et al. 2010, page 10] We also point out that Proposition 5 gives a positive answer to [Hirasawa and Murasugi 2009, Conjecture A]. Proof. The proof of the first statement is not difficult. It is immediate when χ is trivial, and it follows by a direct calculation when χ is nontrivial. We turn to the proof of the second statement. For θ ∈ U(1) and any representation β : π1(NK ) → GL(n, ރ), define the θ-twist of β to be the represen- ε(g) tation that sends g ∈ π1(NK ) to θ β(g), where ε : π1(NK ) → ޚ is determined by the orientation of K . We denote the newly obtained representation by 2πik/n βθ : π1(NK ) → GL(n, ރ). Note that in case α : π1(NK ) → SL(n, ރ) and θ = e is an n-th root of unity, αθ is again an SL(n, ރ) representation. The proof of the proposition relies on the formula

βθ = β (4) 1K,1(t) 1K,1(θt). This formula is well known, and follows directly from the deﬁnition of the twisted Alexander polynomial. Equation (4) combines with Theorem 1 to complete the 2πi/n proof, as we now explain. Take ω = e . If α = α(n,χ) is metabelian, Theorem 1 METABELIAN SL(n, ރ) REPRESENTATIONSOFKNOTGROUPS,II 9 shows that its conjugacy class is ﬁxed under the ޚ/n action. In particular, since α and αω are conjugate, Equation (4) shows that

α = αω = α 1K,1(t) 1K,1(t) 1K,1(ωt). α = P i k = k Expanding 1K,1(t) ai t and using the fact that t (ωt) if and only if k is a multiple of n, this shows that ak = 0 unless k is a multiple of n.

Acknowledgments

We would like to thank Steven Boyer, Christopher Herald, Michael Heusener, Paul Kirk, Charles Livingston, Andrew Nicas, and Adam Sikora for generously sharing their knowledge, wisdom, and insight. We would also like to thank Fumikazu Nagasato and Yoshikazu Yamaguchi for communicating the results of their paper to us.

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Received September 20, 2009. Revised February 9, 2010.

HANS U.BODEN DEPARTMENT OF MATHEMATICS AND STATISTICS MCMASTER UNIVERSITY 1280 MAIN STREET WEST HAMILTON,ONTARIO L8S 4K1 CANADA [email protected] http://www.math.mcmaster.ca/boden

STEFAN FRIEDL MATHEMATICS INSTITUTE ZEEMAN BUILDING UNIVERSITYOF WARWICK COVENTRY CV4 7AL UNITED KINGDOM [email protected] http://www.warwick.ac.uk/~masgaw PACIFIC JOURNAL OF MATHEMATICS Vol. 249, No. 1, 2011

LEWIS–ZAGIER CORRESPONDENCE FOR HIGHER-ORDER FORMS

ANTON DEITMAR

The Lewis–Zagier correspondence attaches period functions to Maaß wave forms. We extend this correspondence to wave forms of higher order, which are higher-order invariants of the Fuchsian group in question. The key ingredient, an identiﬁcation of higher-order invariants with ordinary invariants of unipotent twists, makes it possible to apply standard methods of automorphic forms to higher-order forms.

Introduction

The Lewis–Zagier correspondence — [Lewis 1997; Lewis and Zagier 1997; 2001]; see also [Bruggeman 1997] — is a bijection between the space of Maaß wave forms of a fixed Laplace-eigenvalue λ and the space of real-analytic functions on the line, satisfying a functional equation that involves the eigenvalue. The latter functions are called period functions. In [Deitmar and Hilgert 2007], this correspondence was extended to subgroups 0 of finite index in the full modular group 0(1). One can assume 0 to be normal in 0(1). The central idea of the latter paper is to consider the action of the finite group 0(1)/ 0 and, in this way, to consider Maaß forms for 0 as vector-valued Maaß forms for 0(1). This technique can be applied to higher-order forms as well [Chinta et al. 2002; Deitmar 2009; Deitmar and Diamantis 2009; Diamantis and Sreekantan 2006; Diamantis et al. 2006; Diamantis and O’Sullivan 2008; Diamantis and Sim 2008], turning the somewhat unfamiliar notion of a higher-order invariant into the notion of a classical invariant of a twist by unipotent representation. In the case of Eisenstein series, this viewpoint has already been used in [Jorgenson and O’Sullivan 2008]. The general framework of higher-order invariants and unipotent twists is described in Section 1. This way of viewing higher-order forms has the advantage that it allows techniques of classical automorphic forms to be applied in the context of higher-order forms. The example of the trace formula will be treated in forthcoming work. This paper only applies this technique to extend the Lewis–Zagier correspondence to higher-order forms.

MSC2000: 11F12, 11F67. Keywords: higher-order invariants, automorphic forms.

11 12 ANTON DEITMAR

We define the corresponding spaces of automorphic forms of higher order in Section 2. Various authors have defined holomorphic forms of higher order. Maaß forms are more subtle, as it is not immediately clear how to establish the L2 structure on higher-order invariants. Deitmar and Diamantis [2009] resorted to the obvious L2 structure for the quotient spaces of consecutive higher-order forms; however, this is unsatisfactory because one wishes to view L2 higher-order forms as higher-order invariants themselves. In this paper this flaw is remedied: we give a space of locally square-integrable functions on the universal cover of the Borel–Serre compactification, whose higher-order invariants give the sought-for L2 invariants. We use a common definition of higher-order forms that insists on full invariance under parabolic elements. In the language of Section 1, that means that we take the subgroup P to be the subgroup generated by all parabolic elements. It is an open question whether the results here can be extended to the case P = {1}, that is, to full higher-order invariants. In this case, Fourier expansions have to be replaced with Fourier–Taylor expansions, and thus it is unclear how the correspondence should be defined.

1. Higher-order invariants and unipotent representations

We describe higher-order forms by using invariants in unipotent representations. Let 0 be a group and let W be a ރ[0]-module. We take the ﬁeld ރ of complex numbers as the base ring. Most of the general theory works over any ring, but our applications are over ރ. Let I0 be the augmentation ideal in ރ[0]; that is, the kernel of the augmentation homomorphism X X A : ރ[0] → ރ, cγ γ 7→ cγ . γ γ

The ideal I0 is a vector space with basis {γ − 1}γ ∈0\{1}. We will need two simple properties of the augmentation ideal that, for the reader’s convenience, we will prove in the next lemma. A set S of generators of the group 0 is called symmetric when s ∈ S if and only if s−1 ∈ S. It is easy to P see that ރ[0] = ރ ⊕ I0 and that I0 = s∈S ރ[0](s − 1) for any set of generators S of 0. We also ﬁx a normal subgroup P of 0. We let IP denote the augmentation ˜ ˜ ideal of P, and IP = ރ[0] IP . As P is normal, IP is a two-sided ideal of ރ[0]. For q ˜ 0 0 any integer q ≥ 0, we set Jq = I0 + IP . The set of 0-invariants W = H (0, W) in W can be described as the set of all w ∈ W with I0w = 0. For q = 1, 2,... , we deﬁne the set of invariants of type P and order q to be

0 Hq,P (0, W) = {w ∈ W : Jq w = 0}. LEWIS–ZAGIER CORRESPONDENCE FOR HIGHER-ORDER FORMS 13

0 = 0 Then Hq,P Hq,P (0, W) is a submodule of W, and we have a natural filtration 0 0 0 0 ⊂ H1,P ⊂ H2,P ⊂ · · · ⊂ Hq,P ⊂ · · · . 0 ⊂ 0 0 0 Since I0 Hq,P Hq−1,P , the group 0 acts trivially on Hq,P /Hq−1,P . A representation (η, Vη) of 0 on a complex vector space Vη is called a unipotent length-q representation if Vη has a 0-stable filtration 0 ⊂ Vη,1 ⊂ · · · ⊂ Vη,q = Vη such that 0 acts trivially on each quotient Vη,k/Vη,k−1, where k = 1,..., q and Vη,0 = 0. Let (η, Vη) be a unipotent length-q representation. Assume that it is P-trivial, that is, its restriction to the subgroup P is the trivial representation. There is a natural map 8η : Hom0(Vη, W) ⊗ Vη → W given by α ⊗ v 7→ α(v). [ ] 0 Lemma 1.1. Let W be a ރ 0 -module. The submodule Hq,P (0, W) constitutes a P-trivial, unipotent length-q representation of 0. If the group 0 is finitely gener- 0 ated, then the space Hq,P (0, W) is the sum of all images 8η, when η runs over the set of all P-trivial, unipotent length-q representations that are finite-dimensional over ރ. Proof. The first assertion is clear. Assume now that 0 is finitely generated. 0 The space Hq,P (0, W) needn’t be finite-dimensional. We use induction on q to 0 show that for each w ∈ Hq,par(0, W) the complex vector space ރ[0]v is finite- dimensional. For q = 1, we have ރ[0]w = ރw, and the claim follows. Next, let ∈ 0 w Hq+1,P (0, W) and let S be a finite set of generators of 0. Then X ރ[0]w = ރw + I0w = ރw + ރ[0](s − 1)w. s∈S − ∈ 0 Since (s 1)w Hq,P (0, W), the claim follows from the induction hypothesis. From now on, assume that 0 is finitely generated. The philosophy pursued in the rest of the paper is this:

Once you know Hom0(Vη, W) for every P-trivial ﬁnite-dimensional uni- 0 potent length-q representation, you know the space Hq,P (0, W). 0 So, instead of investigating Hq,P (0, W), one should rather look at ∼ ∗ 0 Hom0(Vη, W) = (Vη ⊗ W) , which is often easier to handle. In fact, it is enough to restrict to a generic set of η. As an example of this approach, consider the case q = 2. For each group homomorphism χ : 0/P → (ރ, +), one gets a P-trivial unipotent length-q representation 2 ηχ on ރ given by 1 χ(γ ) η (γ ) = . χ 0 1 14 ANTON DEITMAR

We introduce the notation

0 0 0 0 0 0 H q,P = H q,P (0, W) = Hq,P (0, W)/Hq−1,P (0, W) = Hq,P /Hq−1,P .

0 Proposition 1.2. The space H2,P (0, W) is the sum of all images 8ηχ when χ ranges over Hom(0/P, ރ) \{0}. For any two χ 6= χ 0, one has ∩ = 0 Im(8ηχ ) Im(8ηχ0 ) H (0, W). In other words, 0 = M 0 H 1 Im(8ηχ )/H . χ Proof. We the order-lowering operator

0 0 ∼ 0 3 : H q,P → Hom(0/P, H q−1,P ) = Hom(0/P, ރ) ⊗ H q−1,P , where the last isomorphism exists because 0 is ﬁnitely generated. This operator is deﬁned by 3(w)(γ ) = (γ − 1)w. One sees that it is indeed a homomorphism in γ by using the fact that (γ τ − 1) ≡ (γ − 1) + (τ − 1) mod I 2 for any two γ, τ ∈ 0. The map 3 is clearly injective. ∈ ∩ 6= 0 ∈ ⊗ 0 ∩ 0 ⊗ 0 Pick now w Im(8ηχ ) Im(8ηχ0 ) for χ χ . Then 3(w) χ H χ H , 6= 0 ∈ 0 and the latter space is zero since χ χ . For surjectivity, pick w H1 . Then 3(w) = Pn χ ⊗ w with w ∈ H 0, and so w ∈ Pn Im(ϕ ). i=1 i i i i=1 ηχi

2. Higher-order forms

We now remedy the aforementioned shortcoming of [Deitmar and Diamantis 2009]. We also give a guide on how to set up higher-order L2 invariants in more general cases (like general lattices in locally compact groups) when there is no gadget such as the Borel–Serre compactification around. In that case, Lemma 2.1 tells you how to define the L2 structure once you have chosen a fundamental domain for the group action. Let G denote the group PSL2(ޒ) = SL2(ޒ)/{±1} and K its maximal compact subgroup PSO(2) = SO(2)/{±1}. Let 0(1) = PSL2(ޚ) be the full modular group and 0 ⊂ 0(1) be a normal subgroup of finite index that is torsion-free. For every cusp c of 0, fix σc ∈ 0(1) such that σc ∞ = c and 1 N ޚ σ −10 σ = ± c . c c c 0 1

The number Nc ∈ ގ is uniquely determined and is called the width of the cusp c. Let ވ = {z ∈ ރ : Im(z) > 0} be the upper half plane, and let ᏻ(ވ) be the set of holomorphic functions on ވ. We fix a weight k ∈ 2ޚ and define a (right) action LEWIS–ZAGIER CORRESPONDENCE FOR HIGHER-ORDER FORMS 15 of G on functions f on ވ by az + b a b f | γ (z) = (cz + d)−k f for γ = . k cz + d c d M ∈ We define the space ᏻ0,k(ވ) to be the set of all f ᏻ(ވ) such that for every cusp c of 0, the function f |kσc is bounded on the domain {Im(z) > 1} by a constant times Im(z)A for some A > 0. S ∈ Further, we consider the space ᏻ0,k(ވ) of all f ᏻ(ވ) such that for every cusp c of 0, the function f |k σc is bounded on the domain {Im(z) > 1} by a constant times e−A Im(z) for some A > 0. These two spaces are preserved not only by 0, but also by the action of the full modular group 0(1). The normal subgroup P of 0 will be the subgroup 0par generated by all parabolic 0 0 elements. We write Hq,par for Hq,P , and we consider the space 0 M Mk,q (0) = Hq,par(0, ᏻ0,k(ވ)) of modular functions of weight k and order q, as well as the corresponding space

0 S Sk,q (0) = Hq,par(0, ᏻ0,k(ވ)) of cusp forms. Then, every f ∈ Mk,q (0) possesses a Fourier expansion at each cusp c of the form ∞ X 2πi(n/Nc)z f |k σc(z) = ac,n e . n=0 A function f ∈ Mk,q (0) belongs to the subset Sk,q (0) if and only if ac,0 = 0 for every cusp c of 0. Since the group 0 is normal in 0(1), the latter acts on the finite-dimensional spaces Mk,q (0) and Sk,q (0). These therefore give examples of finite-dimensional representations of 0(1) that become unipotent length-q when restricted to 0. By a Maaß wave form for the group 0 and parameter ν ∈ ރ, we mean a function u ∈ L2(0\ވ) that is twice continuously differentiable and satisfies = 1 − 2 1u ( 4 ν )u. By the regularity of solutions of elliptic differential equations, this condition implies that u is real analytic. Let ᏹν = ᏹν(0) be the space of all Maaß wave forms for 0. Sometimes in the definition of Maaß forms, instead of the L2 condition, a weaker condition on the growth at the cusps is imposed. Next, we define Maaß wave forms of higher order. First, we need the higher- order version of the Hilbert space L2(0\ވ). For this, recall the construction of the Borel–Serre [1973] compactification 0\ވ of 0\ވ. One constructs a space 16 ANTON DEITMAR

ވ0 ⊃ ވ by attaching to ވ a real line at each cusp c of 0, and then one equips this set with a suitable topology such that 0 acts properly discontinuously. The quotient 0\ވ0 is the Borel–Serre compactification. The space ވ0 is constructed so that, for a given (closed) fundamental domain D ⊂ ވ of 0\ވ with finitely many geodesic sides, the closure D in ވ0 is a fundamental domain for 0\ވ0. By the discontinuity of the group action, this has the consequence that for every compact set K ⊂ ވ0, there exists a finite set F ⊂ 0 S such that K ⊂ F D = γ ∈F γ D. Now we extend the hyperbolic measure to ވ0 so that the boundary ∂ވ0 = ވ0 \ވ 2 2 is a nullset. Let Lloc(ވ0) be the space of local L functions on ވ0. Then, 0 acts 2 on Lloc(ވ0). Since 0 acts discontinuously with compact quotient on ވ0, one has 2 0 2 Lloc(ވ0) = L (0\ވ). 2 \ ∈ 2 = We define Lq (0 ވ) as the space of all f Lloc(ވ0) such that Jq f 0; in other words, 2 0 2 Lq (0\ވ) = Hq,par(0, Lloc(ވ0)). 2 \ = 2 \ Then, L1(0 ވ) L (0 ވ) is a Hilbert space in a natural way. 2 We want to endow the spaces Lq (0\ވ) with Hilbert space structures when q ≥2 as well. For this, we introduce the space Fq of all measurable functions f : ވ → ރ 2 such that Jq f = 0 modulo nullfunctions. Then, Lq (0\ވ) is a subset of Fq . Lemma 2.1. Let S ⊂ 0 be a finite set of generators, assumed to be symmetric and to contain the unit element. Let D ⊂ ވ be a closed fundamental domain of 0 with finitely many geodesic sides. Any f ∈ Fq is uniquely determined by its restriction to

q−1 [ S D = s1 ... sq−1 D.

s1,...,sq−1∈S Further, one has 2 q−1 2 Fq ∩ L (S D) = Lq (0\ވ), where on both sides we mean the restriction to Sq D, which is unambiguous by 2 the ﬁrst assertion. In this way the space Lq (0\ވ) is a closed subspace of the 2 q−1 2 Hilbert space L (S D). The induced Hilbert-space topology on Lq (0\ވ) is independent of the choices of S and D, although the inner product is not. The 2 action of the group 0(1) on Lq (0\ވ) is continuous, but not unitary unless q = 1. 2 q−1 Proof. We have to show that any f ∈ Lq = Lq (0\ވ) that vanishes on S D is zero. We use induction on q. The case q = 1 is clear. Take q ≥ 2 and write Lq = Lq /Lq−1. Consider the order-lowering operator ∼ 3 : Lq → Hom(0, Lq−1) = Hom(0, ރ) ⊗ Lq−1, LEWIS–ZAGIER CORRESPONDENCE FOR HIGHER-ORDER FORMS 17 given by 3( f )(γ ) = (γ − 1) f. q−1 The kernel of 3 is Lq−1. Assume f (S D) = 0. Then, for every s ∈ S, we have (s − 1) f (Sq−2 D) = 0, and hence by the induction hypothesis we conclude that (s − 1) f = 0. However, since S generates 0, this means that 3( f ) = 0 and so f ∈ Lq−1, so again by the induction hypothesis, we get f = 0. We next show that

2 q−1 2 q−1+ j Fq ∩ L (S D) = Fq ∩ L (S D) for every j ≥ 0.

The inclusion “⊃” is clear. We show the other inclusion by induction on q and j. For q = 1 or j = 0, there is no problem. So, assume the claim proved for q. Let 2 q+ j f ∈ Fq+1 ∩ L (S D) and s ∈ S. Then f (sz) = f (z)+ f (sz)− f (z), the function 2 q+ j 2 q+ j−1 f (z) is in L (S D), and the function f (sz) − f (z) is in Fq ∩ L (S D) = L2(Sq+ j D) by the induction hypothesis. It follows that f ∈ L2(s−1 Sq+ j D), and since this holds for every s, we get f ∈ L2(Sq+ j+1 D) as claimed. We now come to 2 q−1 2 Fq ∩ L (S D) = Lq (0\ވ). 2 q−1 Let f ∈ Fq ∩ L (S D). For every compact subset K of ވ0, there exists j ≥ 0 such that K ⊂ Sq−1+ j D. Therefore f is in L2(K ) for every compact subset K 2 q = of H0. Since the latter space is locally compact, f is in Lloc(ވ0). Since I0 f 0, 2 2 q−1 we get f ∈ Lq (0\ވ). For the other inclusion, let f ∈ Lq (0\ވ). Since S D is 2 q−1 relatively compact in ވ0, it follows that f ∈ L (S D) as claimed. We next prove the independence of the topology of S. Let S0 be another set of generators. There exists l ∈ ގ such that S0 ⊂ Sl . Hence, it suffices to show that for 2 2 q−1 every j ≥ 0, the topology from the inclusion Lq (0\ވ) ⊂ L (S D) coincides 2 2 q+ j with the topology from the inclusion Lq (0\ވ) ⊂ L (S D). If a sequence tends to zero in the latter topology, it clearly tends to zero in the first as well. The other way around is proved by an induction on j similar to the one above. In particular, the continuity of the 0(1)-action follows. Finally, we prove the independence of D. Let D0 be another closed fundamental domain with finitely many geodesic sides. Then there exists l ∈ ގ such that 0 l D ⊂ S D, and the claim follows along the same lines as before.

We deﬁne the space ᏹν,q = ᏹν,q (0) of Maaß wave forms of order q to be the 2 space of all u ∈ Lq (0\ވ) that are twice continuously differentiable and satisfy = 1 − 2 1u ( 4 ν )u.

Fix a ﬁnite-dimensional representation (η, Vη) of 0(1) that is 0par-trivial and that becomes a unipotent length-q representation on restriction to 0. 18 ANTON DEITMAR

0(1) ˜ ˜ We set ᏹν,q,η equal to (Vη ⊗ ᏹν,q ) . Likewise, we define ᏹν,q (0) = ᏹν,q as the space of all u ∈ Fq (0) that are twice continuously differentiable and satisfy = 1 − 2 ˜ = ⊗ ˜ 0(1) 1u ( 4 ν )u, and we set ᏹν,q,η (Vη ᏹν,q ) . 0 = 1 − 2 Lemma 2.2. If Ᏸν is the space of all distributions u on ވ with 1u ( 4 ν )u, then ˜ ˜ 0(1) 0 0(1) ᏹν,q,η = (Vη ⊗ ᏹν,q ) = (Vη ⊗ Ᏸν) and 0(1) 0 2 0(1) ᏹν,q,η = (Vη ⊗ ᏹν,q ) = (Vη ⊗ (Ᏸν ∩ Lloc(ވ0))) . Proof. The inclusion “⊂” is obvious in both cases. We show “⊃”. In the first case, the space on the left can be described as the space of all smooth functions : → = 1 − 2 = = u ވ Vη satisfying 1u ( 4 ν )u, as well as Jq+1u 0 and u(γ z) η(γ )u(z) 0 0(1) for every γ ∈ 0(1). Let u ∈ (Vη ⊗ Ᏸν) . As u satisfies an elliptic differential = 1 − 2 equation with smooth coefficients, u is a smooth function with 1u ( 4 ν )u. The condition u(γ z) = η(γ )u(z) is obvious. Finally, the condition Jq+1u = 0 follows from the fact that η|0 satisfies η(Jq+1) = 0 since it is 0par-trivial and unipotent of length q. Hence, the first claim is proven. The second is similar.

As in the holomorphic case, every Maaß form f ∈ ᏹν,q (0) has a Fourier expansion ∞ X 2πi(n/Nc)x f (σcz) = ac,n(y) e n=0 at each cusp c, with smooth functions ac,n(y). We deﬁne the space ᏿ν,q of Maaß cusp forms to be the space of all f ∈ ᏹν,q 0(1) with ac,0(y) = 0 for every cusp c. We also set ᏿ν,q,η = (Vη ⊗ ᏿ν,q ) . 0(1) Note that since η is unipotent of length q on 0, we have ᏿ν,q,η = (Vη ⊗᏿ν,q0 ) for every q0 ≥ q.

3. Setting up the transform

It is the aim of this note to extend the Lewis correspondence [Deitmar and Hilgert 2007; Lewis 1997; Lewis and Zagier 1997; Lewis and Zagier 2001] to the case of higher-order forms. We will first explain our approach in the case of cusp forms. Throughout, (η, Vη) will be a finite-dimensional representation of 0(1) that becomes 0par-trivial and unipotent of length q when restricted to 0. For the canonical generators of 0(1), we fix the notation

0 1 1 1 S = ± and T = ± . −1 0 0 1

The, S2 = 1 = (ST )3 and T is of inﬁnite order. Note that since 0(1)/ 0 is a ﬁnite group, there exists N ∈ ގ such that T N ∈ 0; let N be minimal with this property. LEWIS–ZAGIER CORRESPONDENCE FOR HIGHER-ORDER FORMS 19

N N Then, N = N∞ is the width of the cusp ∞ of 0. We have η(T ) = η(T ) = 1 since η is trivial on the parabolic elements of 0. Let 9ν,η be the space of all holomorphic functions ψ : ރ \ (−∞, 0] → Vη satisfying the Lewis equation z (1) η(T ) ψ(z) = ψ(z + 1) + (z + 1)−2ν−1 η(ST −1) ψ , z+1 and the asymptotic formula

+ − − −1 (2) 0 = e πiν lim ψ(z) + z 2ν 1 η(S)ψ Im(z)→∞ z − − − −1 + e πiν lim ψ(z) + z 2ν 1 η(S)ψ , Im(z)→−∞ z where both limits are assumed to exist. Let A denote the subgroup of G consisting of diagonal matrices, and let N be the subgroup of upper-triangular matrices with ±1 on the diagonal. As a manifold, the group G is a direct product G = ANK . For ν ∈ ރ and a = ± diag(t, t−1) ∈ A with t > 0, let aν = t2ν. (We insert the factor 2 for compatibility reasons.)

Let (πν, Vπν ) denote the principal series representation of G with parameter ν. : → The representation space Vπν is the Hilbert space of all functions ϕ G ރ with = ν+1/2 ∈ ∈ ∈ R | |2 ∞ ϕ(anx) a ϕ(x) for a A, n N, x G, and K ϕ(k) dk < modulo nullfunctions. The representation is πν(x)ϕ(y) = ϕ(yx). There is a special vector = ν+1/2 ϕ0 in Vπν given by ϕ0(ank) a . This vector is called the basic spherical function with parameter ν. For a continuous G-representation (π, Vπ ) on a topological vector space Vπ , ω let π denote the subrepresentation on the space of analytic vectors; that is, Vπ ω consists of all vectors v in Vπ such that for every continuous linear map α : Vπ → ރ, the map g 7→ α(π(g)v) is real analytic on G. This space comes with a natural −ω topology. Let π be its topological dual. In the case of π = πν, it is known that ω −ω πν and πν are in perfect duality; that is, they are each other’s topological duals. −ω The vectors in πν are called hyperfunction vectors of the representation πν. As a crucial tool, we will use the space −ω −ω 0(1) 0 −ω Aν,η = (πν ⊗ η) = H (0(1), πν ⊗ η) and call it the space of η-automorphic hyperfunctions. −ω For an automorphic hyperfunction α ∈ Aν,η , we consider the function u : G → Vη given by u(g) = hπ−ν(g) ϕ0, αi . ω −ω Here, h · , · i is the canonical pairing π−ν × π−ν ⊗ η → Vη. Then, u is right K - ˜ invariant, and hence can be viewed as a function on ވ. As such, it lies in ᏹν,η since α is 0-equivariant, and the Casimir operator on G (which induces 1) is scalar on πν 20 ANTON DEITMAR

1 − 2 : 7→ with eigenvalue 4 ν . The transform P α u is called the Poisson transform. It follows from [Schlichtkrull 1984, Theorem 5.4.3] that the Poisson transform −ω ˜ P : Aν,η → ᏹν,η 6∈ 1 + is an isomorphism for ν 2 ޚ. −ω For α ∈ Aν,η , set − − −1 ψ (z) = f (z) − z 2ν 1 η(S) f , α α α z 7→ + 2 ν+1/2 | with fα such that the function z (1 z ) fα(z) represents the restriction α ޒ. −ω Then, the Bruggeman transform B : α 7→ ψα maps Aν,η to 9ν,η. It is a bijection ∈ 1 + if ν / 2 ޚ, as can be seen in a manner similar to [Deitmar and Hilgert 2007, Proposition 2.2]. 6∈ 1 + : → When ν 2 ޚ, we ﬁnally deﬁne the Lewis transform as the map L ᏹν,η 9ν,η given by L = B ◦ P−1. Theorem 3.1 (Lewis transform; see [Lewis and Zagier 2001, Theorem 1.1]). When 6∈ 1 + < − 1 ν 2 ޚ and (ν) > 2 , the Lewis transform is a bijective linear map from the o space of Maaß cusp forms ᏿ν,q,η to the space 9ν,η of period functions. Proof. The proof runs, with small obvious changes, along the lines of the corresponding result [Deitmar and Hilgert 2007, Theorem 3.3].

References

[Borel and Serre 1973] A. Borel and J.-P. Serre, “Corners and arithmetic groups”, Comment. Math. Helv. 48 (1973), 436–491. MR 52 #8337 Zbl 0274.22011 [Bruggeman 1997] R. W. Bruggeman, “Automorphic forms, hyperfunction cohomology, and period functions”, J. Reine Angew. Math. 492 (1997), 1–39. MR 98i:11035 Zbl 0914.11023 [Chinta et al. 2002] G. Chinta, N. Diamantis, and C. O’Sullivan, “Second order modular forms”, Acta Arith. 103:3 (2002), 209–223. MR 2003b:11037 Zbl 1020.11025 [Deitmar 2009] A. Deitmar, “Higher order group cohomology and the Eichler–Shimura map”, J. Reine Angew. Math. 629 (2009), 221–235. MR 2527419 Zbl 05551467 [Deitmar and Diamantis 2009] A. Deitmar and N. Diamantis, “Automorphic forms of higher order”, J. Lond. Math. Soc. (2) 80:1 (2009), 18–34. MR 2010m:11055 Zbl 05588911 [Deitmar and Hilgert 2007] A. Deitmar and J. Hilgert, “A Lewis correspondence for submodular groups”, Forum Math. 19:6 (2007), 1075–1099. MR 2008j:11058 Zbl 05237815 [Diamantis and O’Sullivan 2008] N. Diamantis and C. O’Sullivan, “The dimensions of spaces of holomorphic second-order automorphic forms and their cohomology”, Trans. Amer. Math. Soc. 360:11 (2008), 5629–5666. MR 2009h:11067 Zbl 1158.11021 [Diamantis and Sim 2008] N. Diamantis and D. Sim, “The classiﬁcation of higher-order cusp forms”, J. Reine Angew. Math. 622 (2008), 121–153. MR 2010a:11067 Zbl 05344066 [Diamantis and Sreekantan 2006] N. Diamantis and R. Sreekantan, “Iterated integrals and higher order automorphic forms”, Commentarii Math. Helv. 81:2 (2006), 481–494. MR 2007b:11050 Zbl 1156.11019 LEWIS–ZAGIER CORRESPONDENCE FOR HIGHER-ORDER FORMS 21

[Diamantis et al. 2006] N. Diamantis, M. Knopp, G. Mason, and C. O’Sullivan, “L-functions of second-order cusp forms”, Ramanujan J. 12:3 (2006), 327–347. MR 2008c:11073 Zbl 05141552 [Jorgenson and O’Sullivan 2008] J. Jorgenson and C. O’Sullivan, “Unipotent vector bundles and higher-order non-holomorphic Eisenstein series”, J. Théor. Nombres Bordeaux 20:1 (2008), 131– 163. MR 2010g:11089 Zbl 05543194 [Lewis 1997] J. B. Lewis, “Spaces of holomorphic functions equivalent to the even Maass cusp forms”, Invent. Math. 127:2 (1997), 271–306. MR 98f:11036 Zbl 0922.11043 [Lewis and Zagier 1997] J. Lewis and D. Zagier, “Period functions and the Selberg zeta function for the modular group”, pp. 83–97 in The mathematical beauty of physics (Saclay, 1996), edited by J. M. Drouffe and J. B. Zuber, Adv. Ser. Math. Phys. 24, World Sci. Publ., River Edge, NJ, 1997. MR 99c:11108 Zbl 1058.11544 [Lewis and Zagier 2001] J. Lewis and D. Zagier, “Period functions for Maass wave forms. I”, Ann. of Math. (2) 153:1 (2001), 191–258. MR 2003d:11068 Zbl 1061.11021 [Schlichtkrull 1984] H. Schlichtkrull, Hyperfunctions and harmonic analysis on symmetric spaces, Progress in Mathematics 49, Birkhäuser, Boston, MA, 1984. MR 86g:22021 Zbl 0555.43002

Received September 29, 2009. Revised April 19, 2010.

ANTON DEITMAR UNIVERSITÄT TÜBINGEN MATHEMATISCHES INSTITUT AUFDER MORGENSTELLE 10 D-72076 TÜBINGEN GERMANY [email protected]

PACIFIC JOURNAL OF MATHEMATICS Vol. 249, No. 1, 2011

TOPOLOGY OF POSITIVELY CURVED 8-DIMENSIONAL MANIFOLDS WITH SYMMETRY

ANAND DESSAI

We show that a simply connected 8-dimensional manifold M of positive sectional curvature and symmetry rank ≥ 2 resembles a rank-one symmetric space in several ways. For example, the Euler characteristic of M is equal to the Euler characteristic of S8, ވP2 or ރP4. If M is rationally elliptic, then M is rationally isomorphic to a rank-one symmetric space. For torsion- free manifolds, we derive a much stronger classiﬁcation. We also study the bordism type of 8-dimensional manifolds of positive sectional curvature and symmetry rank ≥ 2. As an illustration, we apply our results to various families of 8-manifolds.

1. Introduction

We study the topology of positively curved 8-dimensional manifolds with symmetry rank ≥ 2. A Riemannian manifold M is said to have positive curvature if the sectional curvature of all its tangent planes is positive. The symmetry rank of M is deﬁned as the rank of its isometry group. Throughout this paper, all manifolds are assumed to be closed, that is, compact without boundary. At present only a few manifolds are known to admit a Riemannian metric of positive curvature. Besides the examples of Eschenburg and Bazaikin (which are biquotients of dimension 6, 7 or 13), all other simply connected positively curved examples1 are homogeneous, that is, they admit a metric of positive curvature with transitive isometry group (the latter were classiﬁed by Berger, Wallach, Aloff and Berard´ Bergery). In dimension greater than 24 all known examples are symmetric of rank one.

MSC2000: 53C20, 57R19, 57S25. Keywords: positive curvature, torus actions, Euler characteristic, classiﬁcation of 8-manifolds. Research partially supported by the German Research Foundation (DFG-Schwerpunktprogramm 1154 “Globale Differentialgeometrie”) and the Swiss National Science Foundation (grant number 200021-117701). 1Recently, Petersen and Wilhelm [2008], Grove, Verdiani and Ziller [2008], and Dearricott all announced the discovery of new 7-dimensional examples with positive curvature.

23 24 ANAND DESSAI

Classifications of various strength have been obtained for positively curved manifolds with large symmetry; see the survey [Wilking 2007, Section 4]. Among the measures of “largeness”, we shall focus on the symmetry rank. Grove and Searle [1994] showed that the symmetry rank of a positively curved simply connected n-dimensional manifold M is ≤ [(n + 1)/2], and that equality occurs if and only if M is diffeomorphic to a sphere or a complex projective space. Wilking [2003] proved that, if the symmetry rank of M is ≥ n/4+1 and if n ≥ 10, then either M is homeomorphic to a sphere or a quaternionic projective space, or M is homotopically equivalent to a complex projective space (it follows from [Dessai and Wilking 2004] that “homotopically equivalent” can be strengthened to “tangentially equivalent” in this classification). Building on [Wilking 2003], Fang and Rong [2005] showed that, if the symmetry rank is no less than [(n −1)/2] and if n ≥ 8, then M is homeomorphic to a sphere, a quaternionic projective space, or a complex projective space. In dimension eight, the rank-one symmetric spaces S8, ވP2 and ރP4 are the only known simply connected positively curved examples. In this dimension, the aforementioned work of Grove and Searle and of Fang and Rong says that a positively curved simply connected manifold M is diffeomorphic to S8 or ރP4 if the symmetry rank of M is ≥ 4, and homeomorphic to S8, ވP2 or ރP4 if the symmetry rank is ≥ 3. The main purpose of this paper is to give some information on the topology of positively curved 8-dimensional manifolds with symmetry rank ≥ 2. Our first result concerns the Euler characteristic: Theorem 1.1. Let M be a simply connected 8-dimensional manifold. If M admits a metric of positive curvature and symmetry rank ≥ 2, then the Euler characteristic of M is equal to the Euler characteristic of S8, ވP2 or ރP4; that is, χ(M) = 2, 3 or 5. This information on the Euler characteristic leads to a rather strong classification if one assumes in addition that the manifold is rationally elliptic. Recall that a closed simply connected manifold M is rationally elliptic if its rational homotopy π∗(M) ⊗ ޑ is of finite rank. A conjecture attributed to Bott asserts that any nonnegatively curved manifold is rationally elliptic; see [Grove and Halperin 1982, page 172]. Theorem 1.2. Let M be a simply connected positively curved 8-dimensional manifold of symmetry rank ≥ 2. (1) If M is rationally elliptic, M has the rational cohomology ring and rational homotopy type of a rank-one symmetric space, that is, of S8, ވP2 or ރP4. (2) If M is rationally elliptic and H ∗(M; ޚ) is torsion-free, M is homeomorphic to S8, diffeomorphic to ވP2, or tangentially equivalent to ރP4. TOPOLOGY OF POSITIVELY CURVED 8-MANIFOLDS WITH SYMMETRY 25

If one drops the assumption on rational ellipticity and weakens the assumption on the symmetry rank, one can still prove a bound on the Euler characteristic: Theorem 1.3. Suppoer M is a simply connected positively curved manifold of even dimension ≤ 8. Assume S1 acts smoothly on M. If some σ ∈ S1 acts isometrically and nontrivially on M, then χ(M) ≥ 2. This fits well with the Hopf conjecture on the positivity of the Euler characteristic of even-dimensional positively curved manifolds. To put our results in perspective, we briefly recall what is known about positively curved manifolds in low dimensions. Next to surfaces, manifolds of positive curvature are only classified in dimension 3; see [Hamilton 1982] and [Perelman 2002; 2003b; 2003a]. In higher dimensions, the only known obstructions to positive curvature for simply connected manifolds are given by the Betti number theorem of [Gromov 1981] and obstructions to positive scalar curvature (for example, the α- invariant of Lichnerowicz and Hitchin, and the obstructions in dimension 4 coming from Seiberg–Witten theory). In particular, the Hopf problem that asks whether S2 × S2 admits a metric of positive curvature is still open. The study of low-dimensional positively curved manifolds with positive symmetry rank began with Hsiang and Kleiner [1989] on 4-dimensional manifolds. Their main result says that the Euler characteristic of a simply connected positively curved 4-dimensional manifold M with positive symmetry rank is ≤ 3. Using [Freedman 1982], they conclude that M is homeomorphic2 to S4 or ރP2. Rong [2002] showed that a simply connected positively curved 5-dimensional manifold with symmetry rank 2 is diffeomorphic to S5. In dimension 6 and 7 there are examples with symmetry rank 2 and 3, respectively, that are not homotopically equivalent to a rank-one symmetric space [Aloff and Wallach 1975; Eschenburg 1982]. This indicates that, in these dimensions, a classification below the maximal symmetry rank is more complicated. Theorems 1.1 and 1.3 imply that, for many nonnegatively curved manifolds, any metric of positive curvature must be quite nonsymmetric. For example, if M is of even dimension ≤ 8 and has Euler characteristic < 2 (for example, if M is a product of two simply connected odd-dimensional spheres, or M is a simply connected Lie group), then it follows from Theorem 1.3 that, for any positively curved metric g on M, the only isometry of (M, g) sitting in a compact connected Lie subgroup of the diffeomorphism group is the identity. As a further illustration of our results, we consider the following classes of manifolds:

Product manifolds. Let M = N1 × N2 be a simply connected product manifold of dimension 8 (with dim Ni > 0). It is straightforward to see that the Euler characteristic of M is 6= 2, 3 or 5. By Theorem 1.1, M does not admit a metric of

2According to the recent preprint [Kim 2008], M is diffeomorphic to S4 or ރP2. 26 ANAND DESSAI positive curvature with symmetry rank ≥ 2. In particular, the product of two simply 4 4 connected nonnegatively curved manifolds N1 and N2 (for example, S × S ) does not admit a metric of positive curvature and symmetry rank ≥ 2. It is interesting to compare this with [Hsiang and Kleiner 1989], which implies that S2 × S2 does not admit a metric of positive curvature and symmetry rank ≥ 1. Connected sum of rank-one symmetric spaces. Cheeger [1973] has shown that ރP4 ] ±ރP4, ރP4 ] ±ވP2 and ވP2 ] ±ވP2 admit a metric of nonnegative curvature. The Euler characteristic of these manifolds is 8, 6 and 4, respectively. By Theorem 1.1, none of them admits a metric of positive curvature and symmetry rank ≥ 2. Cohomogeneity-one manifolds. Grove and Ziller [2000] constructed invariant metrics of nonnegative curvature on cohomogeneity-one manifolds with codimension- two singular orbits. Using this construction, they exhibited metrics of nonnegative curvature on certain infinite families of simply connected manifolds that fiber over S4, ރP2, S2 × S2 or ރP2 ] ±ރP2 [Grove and Ziller 2000; 2008]; see also the survey [Ziller 2007]. In dimension 8, the Euler characteristic of all these manifolds turns out to be 6= 2, 3, 5. Again, by Theorem 1.1, none of them admits a metric of positive curvature and symmetry rank ≥ 2. Biquotients. Another interesting class of manifolds known to admit metrics of nonnegative curvature are biquotients. A biquotient of a compact Lie group G is the quotient of a homogeneous space G/H by a free action of a subgroup K of G, where the K-action is induced from the left G-action on G/H. Note that any homogeneous space can be described as a biquotient by taking one of the factors to be trivial. If G is equipped with a biinvariant metric, the biquotient M = K \G/H inherits a metric of nonnegative curvature, a consequence of O’Neill’s formula for Riemannian submersions. As pointed out by Eschenburg [1992], a manifold M is a biquotient if and only if M is the quotient of a compact Lie group G by a free action of a compact Lie group L, where the action of L on G is given by a homomorphism L → G × G together with the two-sided action of G × G on G := · · −1 given by (g1, g2)(g) g1 g g2 . The topology of biquotients has been investigated in [Eschenburg 1992; Singhof 1993; Kapovitch 2002; Kapovitch and Ziller 2004; Totaro 2002]. Kapovitch and Ziller [2004] have classified biquotients with singly generated rational cohomology. Combining their classification with the first part of Theorem 1.2 gives this: Corollary 1.4. A simply connected 8-dimensional biquotient of positive curvature 8 4 2 and symmetry rank ≥ 2 is diffeomorphic to S , ރP , ވP or G2/SO(4). In view of Theorems 1.1 and 1.3, the examples above contain plenty of examples of simply connected nonnegatively curved manifolds with positive Ricci curvature TOPOLOGY OF POSITIVELY CURVED 8-MANIFOLDS WITH SYMMETRY 27 for which the metric cannot be deformed to a metric of positive curvature via a symmetry-preserving process such as the Ricci flow. The paper is structured as follows: In Section 2, we recall basic geometric and topological properties of positively curved manifolds with symmetry. In Section 3, we prove the statements on the Euler characteristic. In Section 4, we prove our classification theorem 1.2 for rationally elliptic manifolds and the corollary 1.4 for biquotients. In the final Section 5, we study the bordism type of positively curved 8-dimensional manifolds with symmetry.

2. Tools from geometry and topology

On the topological side, the proofs rely on arguments from equivariant index theory (Theorem 2.4) and the cohomological structure of fixed point sets of smooth actions on cohomology spheres and cohomology projective spaces (Theorem 2.5). On the geometric side, the proofs rely on the work of Hsiang and Kleiner on positively curved 4-dimensional manifolds with symmetry (Theorem 2.3), the fixed point theorems of Berger, Synge and Weinstein for isometries [Kobayashi 1972, Chapter II, Corollary 5.7; Synge 1936; Weinstein 1968], and the following two properties of totally geodesic submanifolds, which we state for further reference: Intersection theorem [Frankel 1961]. Let M be a connected positively curved manifold of dimension n, and let N1 and N2 be totally geodesic submanifolds of dimension n1 and n2, respectively. If n1 + n2 ≥ n, then N1 and N2 intersect. Here, the dimension of a manifold is defined to be the maximal number occurring as the dimension of a connected component of it. Similarly, the codimension of a submanifold N of a connected manifold M is defined to be the minimal number occurring as the codimension of a connected component of N in M. Building on the intersection theorem, Frankel [1966, page 71] observed that the inclusion of a connected totally geodesic submanifold N is 1-connected if the codimension of N in M is at most half of the dimension of M. Using a Morse theory argument, Wilking proved the following far-reaching generalization: Connectivity theorem [Wilking 2003]. Let M be a connected positively curved manifold, and let N1 and N2 be connected totally geodesic submanifolds of codimension k1 and k2, respectively.

(1) Then the inclusion Ni ,→ M is (n−2ki +1)-connected.

(2) If k1 + k2 ≤ n and k1 ≤ k2, then the intersection N1 ∩ N2 is a totally geodesic submanifold, and the inclusion of N1 ∩ N2 in N2 is (n−(k1 +k2))-connected. The connectivity theorem leads to strong restrictions on the topology of positively curved manifolds with large symmetry. Wilking [2003] used this property to classify positively curved manifolds of dimension n ≥ 10 (respectively, n ≥ 6000) 28 ANAND DESSAI with symmetry rank ≥ n/4+1 (respectively, ≥ n/6+1). For further reference, we point out the following rather elementary consequences: Corollary 2.1. Let M be a simply connected positively curved manifold of even dimension n = 2m ≥ 6. (1) If M admits a totally geodesic connected submanifold N of codimension 2, then N is simply connected. Moreover, the integral cohomology of M and N is concentrated in even degrees and satisﬁes H 2i (M; ޚ) =∼ H 2 j (N; ޚ) for all 0 < 2i < n and 0 < 2 j < n − 2.

(2) If M admits two different totally geodesic connected submanifolds N1 and N2 of codimension 2, then M is homeomorphic to Sn or homotopy equivalent to ރPm. Proof. Part (1) follows directly from the connectivity theorem; see [Wilking 2003]. For the convenience of the reader, we recall the argument. We begin with a more general discussion: i Let M be an oriented n-dimensional connected manifold, and N,→M an oriented connected submanifold of codimension k. Let u ∈ H k(M; ޚ) be the Poincare´ dual of the fundamental class of N in M. Then, the cup product with u is given by the composition

∗ ∼ ∼ i i i = i∗ = i+k H (M; ޚ) −→ H (N; ޚ) −→ Hn−k−i (N; ޚ) −→ Hn−k−i (M; ޚ) −→ H (M; ޚ) of maps, where the second and fourth map are the Poincare´ isomorphism maps of N and M, respectively; see for example [Milnor and Stasheff 1974, page 137]. Now, assume that the inclusion i : N ,→ M is (n−k −l)-connected. Then it is straightforward to check that the homomorphism

∪u : H i (M; ޚ) → H i+k(M; ޚ), x 7→ x ∪ u, is surjective for l ≤ i < n − k − l, and injective for l < i ≤ n − k − l (for all this, see [Wilking 2003, Lemma 2.2]). In the first part of the Corollary 2.1, we have k = 2. By the connectivity theorem, the inclusion N ,→ M is (n−3)-connected, that is, l = 1. Hence, the map ∪u is surjective for 1 ≤ i < n − 3, and injective for 1 < i ≤ n − 3. Since M is simply connected and N ,→ M is at least 3-connected, the first part follows. Next, assume N1 and N2 are different totally geodesic connected submanifolds of codimension 2. The second part of the connectivity theorem implies that N := N1 ∩ N2 ,→ N1 is a totally geodesic submanifold, and that the inclusion N := N1 ∩ N2 ,→ N1 is at least 2-connected. 0 2 If the codimension of N in N1 is 2, the map H (N1; ޚ) → H (N1; ޚ) (given by multiplication with the Poincare´ dual of N in N1 for some fixed orientation of N) is TOPOLOGY OF POSITIVELY CURVED 8-MANIFOLDS WITH SYMMETRY 29 surjective by the connectivity theorem. Hence, b2(M) = b2(N1) ≤ 1. Using the first part of the Corollary 2.1, we conclude that M is an integral-cohomology sphere if m b2(M) = 0, or an integral-cohomology ރP if b2(M) = 1. If b2(M) = 0, then M n is actually homeomorphic to S by the work of [Smale 1961]; if b2(M) = 1, then M is homotopy equivalent to ރPm since M is simply connected. Next, assume the codimension of N in N1 is 1. Using the connectivity theorem, we see that the inclusion N ,→ N1 is (n−3)-connected. Arguing along the lines above, it follows that M is homeomorphic to Sn. This completes the proof of the second part. Remark 2.2. For 8-dimensional manifolds, one can show that, under the assumptions of Corollary 2.1(2), M is homeomorphic to S8 or homeomorphic to ރP4. This follows from the Sullivan’s classification of homotopy complex projective spaces [1996]; see the argument in [Fang and Rong 2005, page 85].

Another important geometric ingredient in our proofs is the classiﬁcation (due to Hsiang and Kleiner) up to homeomorphism of positively curved 4-dimensional manifolds with positive symmetry rank.

Theorem 2.3 [Hsiang and Kleiner 1989]. If M is a positively curved simply connected 4-dimensional manifold with positive symmetry rank, the Euler characteristic of M is 2 or 3, and hence M is homeomorphic to S4 or ރP2 by Freedman’s work.

In particular, S2 × S2 does not admit a metric of positive curvature and positive symmetry rank. Note that Theorem 1.1 gives analogous restrictions for the Euler characteristic in dimension 8. Among the topological tools used in the proofs are the classical Lefschetz ﬁxed point formula for the Euler characteristic, the rigidity of the signature on oriented manifolds with S1-action, and its applications to involutions [Hirzebruch 1968], as well as the Aˆ-vanishing theorem of Atiyah and Hirzebruch [1970] for S1-actions on Spin-manifolds.

Theorem 2.4. Let M be an oriented manifold with a smooth nontrivial S1-action, and σ ∈ S1 be the element of order 2. If M S1 and Mσ denote the ﬁxed point manifolds with respect to the S1-action and the σ -action, respectively, then

(1) χ(M) = χ(M S1 ); 1 (2) the equivariant signature signS1 (M) is constant as a character of S ; (3) sign(M) = sign(M S1 ), where the orientation of each component of the ﬁxed point manifold M S1 is chosen to be compatible with the complex structure of its normal bundle (induced by the S1-action) and the orientation of M; and 30 ANAND DESSAI

(4) the signature of M is equal to the signature of a transversal self-intersection Mσ ◦ Mσ . If , in addition, M is a Spin-manifold, then (5) the A-genusˆ vanishes and (6) the connected components of Mσ are either all of codimension ≡ 0 mod 4 (even case), or are all of codimension ≡ 2 mod 4 (odd case). Proof. Part (1) is just a version of the classical Lefschetz fixed point theorem; see for example [Kobayashi 1972, Theorem 5.5]. We give here a simple argument: For any prime p, choose a triangulation of M that is equivariant with respect to the action of ޚ/pޚ ⊂ S1 on M. Then a counting argument shows that χ(M) ≡ χ(Mޚ/pޚ) mod p. For p large enough, this implies that χ(M) = χ(M S1 ) (note that the proof also applies to nonorientable manifolds). Part (2) follows directly from the homotopy invariance of cohomology, or from the Lefschetz fixed point formula of Atiyah, Bott, Segal and Singer (see [Atiyah and Singer 1968, Theorem 6.12, page 582]), as explained for example in [Bott and Taubes 1989, page 142]. For part (3), we consider as a function in λ ∈ ރ the S1-equivariant signature 1 ∼ −1 signS1 (M) ∈ R(S ) = ޚ[λ, λ ], and compute the limit λ → ∞ using the Lefschetz fixed point formula; details can be found in [Hirzebruch et al. 1992, page 68]. Part (4) is a result of Hirzebruch; see [Hirzebruch 1968; Atiyah and Singer 1968, Proposition 6.15, page 583]. Hirzebruch shows that the signature of a transversal σ σ self-intersection M ◦ M is the equivariant signature signS1 (M) evaluated at σ . The claim follows from the rigidity of the signature; see part (2). Part (5) is the celebrated Aˆ-vanishing theorem of Atiyah and Hirzebruch [1970]. Using the Lefschetz fixed point formula, they show that the S1-equivariant Aˆ-genus extends to a holomorphic function on ރ that vanishes at infinity. By a classical result of Liouville, this function has to vanish identically. For a proof part (6), see [Atiyah and Bott 1968, Proposition 8.46, page 487]. We also point out certain properties of smooth actions on cohomology spheres and cohomology projective spaces: Theorem 2.5. (1) If M is a ޚ/2ޚ-cohomology sphere with a ޚ/2ޚ-action, the ޚ/2ޚ-fixed point manifold is again a ޚ/2ޚ-cohomology sphere, or is empty. (2) If M is an integral-cohomology sphere with an S1-action, the S1-fixed point manifold is again an integral-cohomology sphere, or is empty. (3) If M is a ޚ/2ޚ-cohomology complex projective space with a ޚ/2ޚ-action such that the ޚ/2ޚ-action extends to an S1-action, then each component of the ޚ/2ޚ-fixed point manifold is again a ޚ/2ޚ-cohomology complex projective space. TOPOLOGY OF POSITIVELY CURVED 8-MANIFOLDS WITH SYMMETRY 31

(4) The action of an involution on a ޚ/2ޚ-cohomology ރP2 cannot have only isolated fixed points. Proof. The first two statements are well-known applications of Smith theory; see [Bredon 1972, Chapter III, Theorems 5.1 and 10.2]. The last two statements follow directly from the general theory on fixed point sets of actions on projective spaces; see [Bredon 1972, Chapter VII, Theorems 3.1 and 3.3].

3. Euler characteristic

Here we prove the statements on the Euler characteristic given in Section 1. Theorem 1.3. Let M be a simply connected positively curved manifold of even dimension ≤ 8. Assume S1 acts smoothly on M. If some σ ∈ S1 acts isometrically and nontrivially on M, then χ(M) ≥ 2. Proof. Since σ ∈ S1 acts nontrivially, the dimension of M is positive; that is, dim M = 2, 4, 6 or 8. In dimension ≤ 4, the theorem is true for purely topological reasons (Poincare´ duality). So, assume the dimension of M is 6 or 8. Note that Mσ is nonempty [Weinstein 1968] and the connected components of the fixed point manifold Mσ are totally geodesic submanifolds. Each is of even codimension, since σ preserves orientation. If Mσ contains a connected component of codimension 2, then, as pointed out in Corollary 2.1, the connectivity theorem implies that all odd Betti numbers of M vanish. Hence, χ(M) ≥ 2 by Poincare´ duality. So, assume codim Mσ > 2. Note that any connected component F ⊂ Mσ is an S1-invariant totally geodesic submanifold of even dimension ≤ 4. If F is not a point, then F inherits positive curvature from M. In this case, F or a twofold cover of F is simply connected by [Synge 1936]. Hence the Euler characteristic of any connected component of Mσ is positive. From the Lefschetz fixed point formula for the Euler characteristic (see Theorem 2.4(1)), we get 1 1 X χ(M) = χ(M S ) = χ(Mσ )S = χ(Mσ ) = χ(F) ≥ 1. F⊂Mσ Here, equality holds if and only if Mσ is connected and χ(Mσ ) = 1. If so, Mσ must have the ޚ/2ޚ-cohomology of a point, of ޒP2, or of ޒP4. Note that the connected components of M S1 = (Mσ )S1 are orientable submanifolds of even dimension ≤ 4. This implies that the case χ(M) = 1 can only happen if M S1 is a point; see [Bredon 1972, Chapter VII, Theorem 3.1]. However, a smooth S1-action on a closed orientable manifold M cannot have exactly one fixed point. To show this, consider the Lefschetz fixed point formula 1 [Atiyah and Singer 1968] for the S -equivariant signature signS1 (M). The local 1 contribution for signS1 (M) at an isolated S -fixed point extends to a meromorphic 32 ANAND DESSAI function on ރ, which has at least one pole on the unit circle; see for example [Bott 1 and Taubes 1989, page 142]. Since signS1 (M), being a character of S , has no poles on the unit circle, the S1-action cannot have exactly one fixed point (more generally, this is true for any diffeomorphism of order pl , with p an odd prime, as shown by [Atiyah and Bott 1968, Theorem 7.1]). Hence, χ(M) ≥ 2. We remark that the proof simplifies drastically if M has positive symmetry rank; see for example [Puttmann¨ and Searle 2002, Theorem 2]. Note that, by the result above, any metric of positive curvature on S3× S3, S2× S3× S3, S3× S5 or SU(3) must be very nonsymmetric. In the remaining part of this section, we restrict to positively curved simply connected 8-dimensional manifolds with symmetry rank ≥ 2, and prove the statement on the Euler characteristic given in Theorem 1.1. Let T be a two-dimensional torus that acts isometrically and effectively on M, ∼ let T2 = ޚ/2ޚ × ޚ/2ޚ denote the 2-torus in T, and let σ ∈ T be a nontrivial involution, that is, σ ∈ T2 with σ 6= id. By the fixed point theorem of [Weinstein 1968], the fixed point manifold Mσ is nonempty. Each connected component F of Mσ is a totally geodesic T-invariant submanifold of M. Since σ preserves orientation, F is of even codimension. By Berger’s fixed point theorem (see [Kobayashi 1972, Chapter II, Corollary 5.7]), the torus T acts with fixed point on F. Lemma 3.1. F is orientable. If dim F 6= 6, then F is homeomorphic to S4, ރP2, S2 or a point. Proof. If dim F = 6, then F is simply connected by the connectivity theorem, and hence orientable. Next, suppose that dim F = 4 and T acts trivially on F. In this case, we can choose an S1 subgroup of T such that M S1 contains a 6-dimensional connected component Y , and F is a T -fixed point component of Y . In this case, by [Grove and Searle 1994, Theorem 1.2] M, Y and F are diffeomorphic to spheres or complex projective spaces, Suppose that dim F = 4 and T acts nontrivially on F. In this case, we can find a subgroup S1 ⊂ T and a connected component Y of M S1 such that F ∩ Y has positive dimension. If dim Y = 6, then M is diffeomorphic to S8 or ރP4; see [Grove and Searle 1994, Theorem 1.2], and F is a ޚ/2ޚ-cohomology sphere or a ޚ/2ޚ-cohomology complex projective space; see Theorem 2.5. Since the universal cover of F is homeomorphic to S4 or ރP2 (see Theorem 2.3), we conclude that F is simply connected and homeomorphic to S4 or ރP2. If dim Y = 4, then Y is homeomorphic to S4 or ރP2 by Theorem 2.3. It follows from Theorem 2.5 that any connected component of Y ∩ F is a ޚ/2ޚ-cohomology sphere or a ޚ/2ޚ- cohomology complex projective space. Since F ∩ Y has positive dimension, we TOPOLOGY OF POSITIVELY CURVED 8-MANIFOLDS WITH SYMMETRY 33 have χ(F)≥2. Applying Theorem 2.3 again, we conclude that F is homeomorphic to S4 or ރP2. Finally, assume that dim F = 2. We choose an S1 subgroup of T that fixes F pointwise. Let Y be the connected component of M S1 that contains F. If Y is 2-dimensional, then F = Y is orientable of positive curvature and hence diffeomorphic to S2. If Y is 6-dimensional, then M is diffeomorphic to S8 or ރP4 [Grove and Searle 1994, Theorem 1.2], which implies that F =∼ S2. If Y is 4-dimensional, then we may assume that T acts nontrivially on Y (otherwise, we can replace Y by a 6-dimensional connected component for some other S1 subgroup). By Theorem 2.3, Y is homeomorphic to S4 or ރP2. Using Theorem 2.5, 2 we conclude that F is diffeomorphic to S . In the proof of Theorem 1.1, we will use the concept of the “type” of an involution at a fixed point component of the T2-action. This concept is defined as follows: T For each nontrivial involution σ ∈ T2 and each connected component X of M 2 , we consider the dimension of the connected component F of Mσ containing X. For fixed X, this gives an unordered triple of dimensions, which we call the type of X. Note that F is orientable by Lemma 3.1. Since the action of T is orientation-preserving on F, any connected component X of M T2 is a totally geodesic T-invariant submanifold of even dimension. By Berger’s fixed point theorem (see [Kobayashi 1972, Chapter II, Corollary 5.7]), X T is nonempty. We will use the following information on type, which can be easily verified by considering the isotropy representation at a T-fixed point in X.

Lemma 3.2. (1) The type of X is (6, 6, 4), (6, 4, 2), (6, 2, 0), (4, 4, 4), (4, 4, 0) or (4, 2, 2).

(2) X is an isolated T2-ﬁxed point if and only if X is of type (6, 2, 0), (4, 4, 0) or (4, 2, 2).

Example 3.3. Let M be the quaternionic plane

2 ވP = {[q0, q1, q2] | qi ∈ ވ,(q0, q1, q2) 6= 0}, where [q0, q1, q2] denotes the orbit of (q0, q1, q2) with respect to the diagonal action of nonzero quaternions on ވ3 from the right. Consider the action of T = S1 × S1 = {(λ, µ) | λ, µ ∈ S1} ⊂ ރ × ރ on M via √ √ √ (λ, µ)([q0, q1, q2]) := [λ · µ · q0, µ · q1, µ · q2]. √ Note that although the square root · is only well-defined up to sign, the action is independent of this choice. Let σ1 and σ2 be the involutions in the first and second 1 S factor of T, and let σ3 := σ1 ·σ2 denote the third nontrivial involution. Then, we 34 ANAND DESSAI have the following fixed point manifolds:

σ1 ∼ 4 M = {[1, 0, 0]} ∪ {[0, q1, q2] | qi ∈ ވ,(q1, q2) 6= 0} = pt ∪ S ,

σ2 ∼ 2 M = {[q0, q1, q2] | qi ∈ ރ,(q0, q1, q2) 6= 0} = ރP ,

σ3 ∼ 2 M = {[ j · q0, q1, q2] | qi ∈ ރ,(q0, q1, q2) 6= 0} = ރP ,

T2 ∼ 2 M = {[1, 0, 0]} ∪ {[0, q1, q2] | qi ∈ ރ,(q1, q2) 6= 0} = pt ∪ S Hence, the type of X = pt is (4, 4, 0), and the type of X = S2 is (4, 4, 4). The proof of Theorem 1.1 is based on the next three lemmas. Lemma 3.4. If dim Mσ =6 for some involution σ ∈ T, then the Euler characteristic of M is 2 or 5. Proof. Let N ⊂ Mσ be the connected component of dimension 6. Note that all other connected components are isolated σ -fixed points, by the intersection theorem. In view of Corollary 2.1, the odd Betti numbers of M and N vanish, and the even Betti numbers satisfy b2(M) = b4(M) = b6(M) = b2(N) = b4(N). In particular, we have χ(M) − χ(N) = b2(M). By the Lefschetz fixed point formula for the Euler characteristic (see Theorem 2.4(1)), this difference is equal to the number of isolated σ -fixed points. Suppose isolated σ -fixed points do occur (otherwise, b2(M) = 0 and χ(M) = 2). Using Lemma 3.2(2), we see that an isolated σ -fixed point is an isolated T2- fixed point, of type (6, 2, 0) or (4, 4, 0). If some isolated σ -fixed point is of type (6, 2, 0), then M contains a 6-dimensional fixed point manifold different from N. In this case, M is homeomorphic to S8 or homotopy equivalent to ރP4; see Corollary 2.1(2). In particular, χ(M) = 2 or 5. We now consider the remaining case. Suppose all isolated σ -fixed points are of type (4, 4, 0). We fix a nontrivial involution σ1 ∈ T different from σ, and denote by σ F1 the 4-dimensional connected component of M 1 (note that F1 is unique and has nonempty intersection with N, by the intersection theorem). Each isolated σ -fixed point, being of type (4, 4, 0), is contained in F1, and hence the number d of isolated σ -fixed points is equal to χ(F1) − χ(F1 ∩ N). Since T acts nontrivially on F1, Theorem 2.3 tells us that χ(F1) ≤ 3. Using Lemma 3.2, we see that the connected T components of (F1∩N) 2 are necessarily 2-dimensional of type (6, 4, 2). It follows T from Lemma 3.1 that at least one of the connected components of (F1 ∩ N) 2 is 2 diffeomorphic to S . Hence, b2(M) = d = χ(F1) − χ(F1 ∩ N) ≤ 1, which in turn implies χ(M) = 2 or 5. Remark 3.5. Under the assumptions of Lemma 3.4, M is homeomorphic to S8 or ރP4. This follows from the proof above, together with the work of Smale [1961] on the high-dimensional Poincare´ conjecture, and Sullivan’s classification [1996] of homotopy complex projective spaces. TOPOLOGY OF POSITIVELY CURVED 8-MANIFOLDS WITH SYMMETRY 35

Lemma 3.6. If dim Mσ = 2 for some involution σ ∈ T, then χ(M) = 2. Proof. We first note that the assumption on dim Mσ implies that the signature of M vanishes, by Theorem 2.4(4). Since χ(M) ≡ sign(M) mod 2, the Euler characteristic χ(M) is even. By Lemma 3.4, we may assume that the dimension of the fixed point manifold Mτ is ≤ 4 for every nontrivial involution τ of T . Using Lemma 3.2, we see that every T2-fixed point component is an isolated fixed point of type (4, 4, 0) or (4, 2, 2). Let σ1 and σ2 denote the nontrivial involutions distinct from σ. Without loss σ of generality, we may assume that dim M 1 = 4. Let F1 denote the 4-dimensional σ connected component of M 1 . Since T acts nontrivially on F1, the universal cover 4 2 of F1 is homeomorphic to S or ރP by Theorem 2.3. Since all T2-fixed points are isolated, the involution σ acts on F1 with isolated fixed points. By Theorem 2.5, 2 F1 cannot be a cohomology ރP . Hence, χ(F1) ≤ 2. σ If dim M 2 < 4, then any T2-fixed point component is contained in F1 and hence σ χ(M) = χ(F1) = 2 by Theorem 1.3. So assume M 2 contains a 4-dimensional connected component F2. Arguing as above, we see that χ(F2) ≤ 2. Note that F1 and F2 intersect by the intersection theorem, and F1 ∩ F2 consists of isolated σ -fixed points. Hence, χ(M) = χ(F1) + χ(F2) − χ(F1 ∩ F2) ≤ 3. Since χ(M) is even (as pointed out above) and no less than 2 (by Theorem 1.3), the lemma follows. Lemma 3.7. If dim Mσ = 0 for some involution σ ∈ T, then χ(M) = 2. Proof. The proof is very similar to that of Lemma 3.6. By Theorem 2.4(4), the signature of M vanishes. In particular, χ(M) is even. Applying Lemma 3.2, we see that a connected component of M T2 is of type (6, 2, 0) or (4, 4, 0). If some component has type (6, 2, 0), then, since χ(M) is even, the Euler characteristic of M is equal to 2 by Lemma 3.4. T So, assume all components of M 2 are of type (4, 4, 0). Let σ1 ∈ T be a nontrivial involution distinct from σ , and let F1 denote the unique 4-dimensional connected σ T component of M 1 . Note that M 2 ⊂ F1. By Theorem 2.3, χ(F1) ≤ 3. Since T χ(M) is even, we get χ(M) = χ(M 2 ) = χ(F1) ≤ 2. Now, the lemma follows from Theorem 1.3. Proof of Theorem 1.1. Using the lemmas above, we may assume that dim Mσ = 4 for every nontrivial involution σ ∈ T. In view of Lemma 3.2, every connected T2- fixed point component X is of type (4, 4, 0), (4, 2, 2) or (4, 4, 4). In the first two cases, X is an isolated fixed point, whereas in the third case an inspection of the isotropy representation shows that X is of dimension two. Let σi ∈ T for i = 1, 2, 3 denote the nontrivial involutions, and let Fi denote the unique 4-dimensional connected component of Mσi . By the intersection theorem, 36 ANAND DESSAI any two of the Fi intersect. Since T acts nontrivially on Fi , we have χ(Fi ) ≤ 3 by 2 Theorem 2.3. If χ(Fi ) = 3 for some i, that is, if Fi is homeomorphic to ރP , then Fi contains a T2-fixed point component of positive dimension (see Theorem 2.5), which is necessarily of type (4, 4, 4). Hence, if none of the T2-fixed point components is of type (4, 4, 4), then χ(Fi ) ≤ 2 for all i, and X X χ(M) = χ(Fi ) − χ(Fi ∩ Fj ) ≤ 3 · 2 − 3 = 3. i i< j Since χ(M) ≥ 2 by Theorem 1.3, we are done in this case. In the other case, the intersection of the Fi contains a 2-dimensional T2-fixed point component X of type (4, 4, 4). It follows from Lemma 3.1 and Theorem 2.5 that X is diffeomorphic to ރP1. Hence χ(M) ≤ 3 · 3 − 2 · 2 = 5 with equality 2 holding if and only if each Fi is homeomorphic to ރP . Moreover, in the equality case, the T2-fixed point components distinct from X are all of type (4, 2, 2), and σ for each σi the fixed point manifold M i is the union of Fi and a 2-dimensional sphere. We claim that χ(M) 6= 4. Suppose to the contrary that χ(M) = 4. Then we may assume that at least one of the Fi (say, F1) has Euler characteristic equal to 3. σ Now, χ(M) = 4 implies that the fixed point manifold M 1 is the union of F1 and an isolated fixed point q (in fact, arguing as for X, we see that any σ1-fixed point 1 component of positive dimension distinct from F1 would be diffeomorphic to ރP , implying that χ(M) > 4). Note that q, being an isolated σ1-fixed point, must be of type (4, 4, 0). Hence, q belongs to F2 and F3. This implies that χ(F2)=χ(F3)=3. T2 0 On the other hand, F1 is the union of X and a point q , distinct from q, which is 0 of type (4, 4, 0) or (4, 2, 2). If q has type (4, 4, 0), then χ(F2) ≥ 4 or χ(F3) ≥ 4, 0 σ which contradicts χ(F2) = χ(F3) = 3. If q has type (4, 2, 2), then χ(M 2 ) ≥ 5 and χ(Mσ3 ) ≥ 5, which contradicts χ(M) = 4. Hence, χ(M) 6= 4. Since χ(M) ≤ 5 and χ(M) ≥ 2 (by Theorem 1.3), we get χ(M) = 2, 3, 5. Recall that, for an nonorientable even-dimensional manifold of positive curvature, a two-fold cover (the orientation cover) is simply connected [Synge 1936]. Hence, Theorem 1.1 implies this: Corollary 3.8. Let M be an nonorientable 8-dimensional manifold. If M admits a metric of positive curvature and symmetry rank ≥ 2, then χ(M) = 1.

4. Rationally elliptic manifolds

In this section, we apply Theorem 1.1 to rationally elliptic manifolds. Recall that a closed simply connected n-dimensional manifold M is rationally elliptic if its rational homotopy π∗(M)⊗ ޑ is of ﬁnite rank. Rational ellipticity imposes strong TOPOLOGY OF POSITIVELY CURVED 8-MANIFOLDS WITH SYMMETRY 37 topological constraints. For example, Halperin has shown that the Euler characteristic of a rationally elliptic manifold is nonnegative, and that all odd Betti numbers vanish if the Euler characteristic is positive; see [1977, Theorem 10, page 175]. By work of [Friedlander and Halperin 1979, Corollary 1.3], the sum of degrees of L generators of π2∗(M) ⊗ ޑ := i π2i (M) ⊗ ޑ is ≤ n, On the other hand, the class of rationally elliptic manifolds contains some interesting families, for example, Lie groups, homogeneous spaces, biquotients and manifolds of cohomogeneity one. Also, all simply connected manifolds presently known to admit a metric of nonnegative curvature are rationally elliptic. Using the information on the Euler characteristic given in Theorem 1.1, we obtain the following classiﬁcation result for rationally elliptic manifolds. I thank Mikiya Masuda for explaining to me the properties of ޚ/2ޚ-cohomology ރP4’s with T-action that are used in the proof. Theorem 1.2. Suppose M is a simply connected positively curved 8-dimensional manifold of symmetry rank ≥ 2. (1) If M is rationally elliptic, M has the rational cohomology ring and the rational homotopy type of a rank-one symmetric space, that is, of S8, ވP2 or ރP4. (2) If M is rationally elliptic and H ∗(M; ޚ) is torsion-free, then M is homeomorphic to S8, diffeomorphic to ވP2, or tangentially equivalent to ރP4. Proof. (1) We will show that M has the rational cohomology ring of S8, ވP2 or ރP4. From this, one easily deduces that M is formal. Hence, M has the same rational homotopy type as S8, ވP2 or ރP4. By Theorem 1.1, the Euler characteristic of M is 2, 3 or 5. Since M is rationally elliptic, the rational cohomology ring of M is concentrated in even degrees; see [Halperin 1977, Theorem 10, page 175]. If χ(M) = 2 or χ(M) = 3, then M has the rational cohomology ring of S8 or ވP2, respectively; this follows directly from Poincare´ duality. If χ(M) = 5, then the rational cohomology ring of M belongs to one of the following three cases:

(1) b2(M) = 0 and b4(M) = 3; 2 2 (2) b2(M) = b4(M) = 1, and x = 0 for a generator x of H (M; ޑ); 2 2 (3) b2(M) = b4(M) = 1, and x 6= 0 for a generator x of H (M; ޑ). According to [Friedlander and Halperin 1979, Corollary 1.3], the sum of degrees of generators of π2∗(M) ⊗ ޑ is ≤ 8. This excludes the ﬁrst two cases. In fact, in the ﬁrst case the minimal model of M must have three generators of degree 4, and in the second case the minimal model of M must have generators of degree 2, 4 2 and 6. So, only the third case can occur; that is, b2(M) = b4(M) = 1 and x 6= 0 for a generator x of H 2(M; ޑ). By Poincare´ duality, M has the rational cohomology ring of ރP4. 38 ANAND DESSAI

(2) Now assume that M is rationally elliptic and H ∗(M; ޚ) is torsion-free. If χ(M) = 2, then M is rationally a sphere by (1). Since H ∗(M; ޚ) is torsion-free, in this case M is an integral cohomology S8. Being simply connected, M is a homotopy sphere and hence homeomorphic to S8 by [Smale 1961]. If χ(M) = 3, then M is an integral-cohomology ވP2 by Poincare´ duality. We choose the orientation of M for which the signature of M is one. Note that M is 3-connected, and hence M is a Spin manifold. By the Atiyah–Hirzebruch vanishing theorem (see Theorem 2.4(5)), the Aˆ-genus of M vanishes (this follows also from the result of Lichnerowicz [1963] on the vanishing of the Aˆ-genus for Spin manifolds with positive scalar curvature). Since in dimension eight the space of Pontryagin numbers is spanned by the Aˆ-genus and the signature, the manifolds M and ވP2 have the same Pontryagin numbers. From [Smale 1962, Theorem 6.3], it follows that M admits a Morse function with three critical points. The classification results of Eells and Kuiper [1962, theorem on page 216] for these manifolds imply that the diffeomorphism type of M is determined by its Pontryagin numbers, up to connected sums with homotopy spheres. In particular, M and ވP2 are homeomorphic and diffeomorphic, up to connected sums, with a homotopy sphere. Recently, Kramer and Stolz used Kreck’s surgery theory to show that the action of the group of homotopy spheres on ވP2 via connected sum is trivial; see [2007, Theorem A]. Hence, M and ވP2 are diffeomorphic. Finally, we consider the case when χ(M) = 5. From (1), we know already that M is a rational cohomology ރP4. Below, we will show that M is in fact an integral cohomology ރP4. Assuming this for the moment, we now prove that M is tangentially equivalent to ރP4; that is, there exists a homotopy equivalence f : M → ރP4 such that f ∗(T ރP4) and TM are stably isomorphic. We first note that M and ރP4 are homotopy equivalent since M is an integral cohomology ރP4 and simply connected. In the early 1970s, Petrie conjectured that a smooth S1-manifold N that is homotopy equivalent to ރPn has the same Pontryagin classes as ރPn; that is, the total Pontryagin class p(ރPn) is mapped to p(N) under a homotopy equivalence N → ރPn. Petrie’s conjecture holds for n = 4 [James 1985]. Hence, the homotopy equivalence f : M → ރP4 maps the Pontryagin classes of ރP4 to the Pontryagin classes of M. In this situation M and ރP4 are tangentially equivalent [Petrie 1973, page 140]. We sketch the argument: Since H ∗(M; ޚ) is torsion-free, the condition on the Pontryagin classes implies that the complexified vector bundles TM ⊗ރ and f ∗(T ރP4)⊗ރ agree in complex K-theory. When M is a homotopy complex projective space of complex dimension 6≡ 1 mod 4, the complexification map KO(M) → K(M) is injective. Hence, the real vector bundles TM and f (T ރP4) are stably isomorphic (in fact, they are isomorphic, since they have up to sign the same Euler class). TOPOLOGY OF POSITIVELY CURVED 8-MANIFOLDS WITH SYMMETRY 39

To complete the proof, we need to show that M is an integral cohomology ރP4. We choose the orientation of M for which the signature of M is 1. Since H ∗(M; ޚ) is torsion-free, it follows from Poincare´ duality that M is a twisted ރP4; that is, 2i there are generators x2i ∈ H (M; ޚ) for i = 1, 2, 3, 4, and an integer m > 0, such that x8 is the preferred generator with respect to the chosen orientation, and

2 2 x2 · x6 = x4 = x8, x2 = m · x4, x2 · x4 = m · x6.

Let T denote the 2-dimensional torus that acts isometrically and effectively ∼ on M, let T2 = ޚ/2ޚ × ޚ/2ޚ denote the 2-torus in T, and let σ ∈ T2 be a nontrivial involution. By Theorem 2.4(4), the codimension of Mσ is 2 or 4. If the codimension of Mσ is 2, then M is an integral cohomology ރP4; that is, m = 1. This follows directly from the proof of Corollary 2.1. So, we are left with the case when dim Mσ = 4 for every nontrivial involution σ ∈ T. Let σi ∈ T for i = 1, 2, 3 denote the nontrivial involutions. From the discussion in the previous section (see the proof of Theorem 1.1), we recall the σ following facts: For each σi , the fixed point manifold M i is the union of a 4- 2 dimensional connected component Fi and a 2-dimensional sphere Si . Moreover, 2 Fi is homeomorphic to ރP , the three Fi intersect in a 2-dimensional T2-fixed point component X of type (4, 4, 4), and X is diffeomorphic to ރP1. We choose the orientation of Fi for which the signature of Fi is 1. Therefore the normal bundle of X in M is isomorphic as a real vector bundle to three copies of the Hopf bundle. In particular, the normal bundle is not spin and the restriction of the second Stiefel–Whitney class of M to X is nonzero. This shows ∗ 2 2 that the restriction homomorphism fi : H (M; ޚ) → H (Fi ; ޚ) induced by the 2 ∼ inclusion fi : Fi ,→ M maps x2 to an odd multiple of a generator x of H (Fi ; ޚ) = 2 2 ∗ H (ރP ; ޚ); that is, fi (x2) = a · x with a odd. It follows from the Lefschetz fixed point formula for the equivariant signature (see [Atiyah and Singer 1968]) and Theorem 2.4(4) that the Euler class e(νi ) of the normal bundle νi of Fi ,→ M 2 4 ∗ 2 is equal to the preferred generator x ∈ H (Fi ; ޚ). Hence, fi (x4) = e(νi ) = x . 2 = · = 2 By restricting the identity x2 m x4 to Fi , we see that m a is an odd square. In particular, M is a ޚ/2ޚ-cohomology ރP4. Next, we recall from the proof of Theorem 1.1 that M T2 is the union of X and 2 three points pi for i = 1, 2, 3, with pi ∈ Fi . We fix a lift ξ ∈ HT (M; ޚ) of x2 and denote by wi the restriction of ξ to pi . By the structure theorem for rational cohomology complex projective spaces [Hsiang 1975, Theorem VI.I, page 106], ∗ ∗ the kernel of the restriction homomorphism HT (M; ޑ) → HT (pi ; ޑ) is generated by (ξ − wi ). The following argument, due to Masuda, shows that m is equal to 1. Let fi ! : ∗ ∗+4 HT (Fi ; ޚ) → HT (M; ޚ) denote the equivariant Gysin map (or push-forward) 40 ANAND DESSAI induced by fi : Fi ,→ M. For properties of the Gysin map, see for example [Masuda 1981, pages 132–133]. 2 Claim 1: fi !(1) = (1/a ) (ξ − w j ) · (ξ − wk), where {i, j, k} = {1, 2, 3}.

Proof. Since p j and pk are not in Fi , the restriction of fi !(1) to each of these points must vanish. Hence, fi !(1) is divisible by (ξ − w j ) · (ξ − wk). Comparing degrees, we ﬁnd that fi !(1) = c·(ξ −w j )·(ξ −wk) for some rational constant c. By = · 2 restricting this identity to ordinary cohomology, we obtain x4 c x2 , and hence 2 c = 1/m = 1/a . X

2 2 Claim 2: wi − w j is divisible by a in H (BT ; ޚ). Proof. From the ﬁrst claim, we deduce that 1 f (1) − f (1) = (w − w ) · ξ − (w − w ) · w . i ! j ! a2 i j i j k ∗ ∗ Since (1, ξ) is part of a basis for the free H (BT ; ޚ)-module HT (M; ޚ), it follows 2 that (wi − w j ) is divisible by a . X

Recall that any two T-fixed points pi and p j with i 6= j are contained in a 2 T-invariant 2-dimensional sphere Sk that is fixed pointwise by the involution σk 2 2 (where {i, j, k} = {1, 2, 3}). Consider the T-action on Sk , and let mi j ∈ H (BT ; ޚ) denote the weight of the tangential T-representation at pi . Note that mi j is only defined up to sign and that mi j = ±m ji (here and in the following the notation α = ±β is a shortcut for α = β or α = −β).

Claim 3: ±a · mi j = wi − w j , and mi j is divisible by a.

Proof. First, note that the normal bundle of Fi in M restricted to pi has weights ±mi j and ±mik, where {i, j, k} = {1, 2, 3}. Hence, by restricting the identity given in the ﬁrst claim to the pi , we obtain the following identities in the polynomial ring ∗ ; H (BT ޚ): 2 ±a · m12 · m13 = (w1 − w2) · (w1 − w3), 2 ±a · m23 · m21 = (w2 − w3) · (w2 − w1), 2 ±a · m31 · m32 = (w3 − w1) · (w3 − w2).

Since mi j and m ji agree up to sign, ±a · mi j = wi − w j . Using Claim 2, mi j is divisible by a. X T 1 Suppose X ⊂ M . We can choose a subgroup S of T such that F1 is ﬁxed pointwise by S1. By Claim 3, the subgroup ޚ/a ޚ of S1 acts trivially on M. Since the T-action is effective, we get a = ±1. T T 0 If X 6⊂ M , then M consists of ﬁve isolated points {p, p, p1, p2, p3}, where 0 2 p, p ∈ X and pi ∈ Fi . Recall that Fi is homeomorphic to ރP . In particular, there is a unique T-invariant 2-dimensional sphere in Fi that contains p and pi . TOPOLOGY OF POSITIVELY CURVED 8-MANIFOLDS WITH SYMMETRY 41

2 Consider the T-action on this sphere and let mi ∈ H (BT ; ޚ) denote the weight 0 2 of the tangential T-representation at pi . Similarly, let mi ∈ H (BT ; ޚ) denote the 0 weight of the tangential T-representation that corresponds to p and pi . Note that 0 mi and mi are only defined up to sign. Let w and w0 denote the restriction of ξ to p and p0, respectively. Since any 2 torus action on a homotopy ރP is of linear type and x2 restricted to Fi is equal 2 to a times a generator of H (Fi ; ޚ), we get 0 0 (1) ±a · mi = wi − w and ± a · mi = wi − w. Now consider the circle subgroup S ,→ T defined by w = w0. Since T acts linearly S on Fi , the fixed point set Fi is the union of X and pi . For u ∈ H ∗(BT ; ޚ), let u¯ denote the restriction of u ∈ H ∗(BT ; ޚ) to H ∗(BS; ޚ). 0 0 Since w = w , it follows from equations (1) that mi and mi agree up to sign. Since T acts effectively on M and the weights mi j are divisible by a (see Claim 3), mi and 0 mi are both coprime to a. Suppose m = a2 is not equal to 1. Consider the action of ޚ/a ޚ ⊂ S. Since 0 mi j is divisible by a, and mi and mi are both coprime to a, the connected ޚ/a ޚ- fixed point component Z that contains p1 contains both p2 and p3 but does not ∗ ∗+4 contain X. Hence, the S-equivariant Gysin map f! : HS (Z; ޚ) → HS (M; ޚ) induced by the inclusion f : Z ,→ M vanishes after restricting to X. Applying the structure theorem for rational cohomology complex projective spaces [Hsiang 1975, Theorem VI.I, page 106] to M and Z, we find that f!(1) is divisible by (ξ¯ − w)2. Comparing degrees, we see that there is a rational constant C such that ¯ 2 f!(1) = C · (ξ − w) . By restricting this identity to the T-fixed point pi , we obtain 0 2 2 2 ¯ 2 ±mi ·mi = C ·(wi −w) . Using (1), we get C = ±1/a . Hence (1/a )·(ξ −w) ∈ 4 2 ¯ ¯ 4 HS (M; ޚ). Recall from Claim 1 that (1/a )(ξ −w1)·(ξ −w2) is also in HS (M; ޚ). Taking the difference of these two elements, we obtain 1 (w − w + 2 · (w − w)) · ξ¯ + (w2 − w · w ) ∈ H 4(M; ޚ). a2 2 1 1 1 2 S ¯ ∗ ∗ Since (1, ξ) is part of a basis of the free H (BS; ޚ)-module HS (M; ޚ), it follows 2 2 that (w2 − w1 + 2 · (w1 − w)) is divisible by a . Now (w2 − w1) is divisible by a 2 by Claim 3, and a is odd. Hence (w1 −w) is divisible by a . Using (1), we deduce 2 2 that a divides mi . This contradicts a 6= 1 since mi is coprime to a. Hence m = a is equal to 1. We close this section with an application to biquotients. Recall that any biquotient of a compact connected Lie group G is rationally elliptic and comes with a metric of nonnegative curvature induced from a biinvariant metric on G. Corollary 4.1. A simply connected 8-dimensional biquotient of positive curvature 8 4 2 and symmetry rank ≥ 2 is diffeomorphic to S , ރP , ވP or G2/ SO(4). 42 ANAND DESSAI

Proof. According to Theorem 1.2, a simply connected positively curved 8-dimensional biquotient with symmetry rank ≥ 2 is rationally singly generated. Rationally singly generated biquotients where classiﬁed by Kapovitch and Ziller in [2004, Theorem A]. In dimension 8, these are the homogeneous spaces given. Remark 4.2. From the classiﬁcation of homogeneous positively curved manifolds, it follows that G2/ SO(4) does not admit a homogeneous metric of positive curvature. We do not know whether G2/ SO(4) admits a positively curved metric with symmetry rank two.

5. Bordism type

In this section, we consider the bordism type of closed simply connected Riemann- ian 8-manifolds with positive curvature. We determine the Spin-bordism type and comment on the oriented bordism type for manifolds with symmetry rank ≥ 2. Spin ⊕ In dimension eight, the Spin-bordism group 8 is isomorphic to ޚ ޚ, and the Spin-bordism type is detected by Pontryagin numbers; see [Milnor 1963, page 201]. Since in this dimension the Pontryagin numbers are uniquely determined by the Aˆ-genus and the signature, it suffices to compute these numerical invariants. Proposition 5.1. Let M be an 8-dimensional Spin-manifold. If M admits a metric of positive curvature and symmetry rank ≥ 2, then χ(M) = 2 or 3, and M is Spin- bordant to S8 or ±ވP2. Proof. By the Atiyah–Hirzebruch vanishing theorem (see Theorem 2.4(5)) or al- ternatively by the theorem of Lichnerowicz [1963], the Aˆ-genus of M vanishes. Hence, it suffices to show that the signature of M is equal to the signature of S8 or ±ވP2; that is, we want to show that |sign(M)| ≤ 1. If some isometry in T acts with a fixed point component of codimension 2, this follows directly from Corollary 2.1 and Theorem 1.1 (in fact M is bordant to S8 since an integral-cohomology ރP4 is never spin). So assume that for any τ ∈ T the fixed point manifold

(∗) Mτ has no ﬁxed point component of codimension 2.

Recall from Theorem 1.1 that χ(M) = 2, 3 or 5. If χ(M) = 2, sign(M) = 0. To see this consider a subgroup S1 ⊂ T of positive fixed point dimension (that is, dim M S1> 0). By condition (∗), any connected component of M S1 is of dimension no more than 4. It follows from Theorem 2.4(1) that M S1 is S2 or an integral cohomology S4. Since the signature of M is the sum of the signatures of the connected components of M S1, the signature of M vanishes; see Theorem 2.4(3). If χ(M) = 3, then |sign(M)| = 1. The reasoning is similar to the one above. Choose an S1 subgroup of T such that the fixed point manifold M S1 has a connected TOPOLOGY OF POSITIVELY CURVED 8-MANIFOLDS WITH SYMMETRY 43 component F of dimension 2 or 4. Any such F is simply connected by [Synge 1936] and satisfies |sign(F)| ≤ χ(F) − 2. Since χ(M) = χ(M S1 ) = 3 and the signature of M is the sum of the signatures of the connected components of M S1 (taken with the appropriate orientation), we get |sign(M)| = 1. Finally, we claim that the case χ(M)=5 cannot occur. First note that in this case the signature of M is odd since sign(M) ≡ χ(M) mod 2. Let σi for i ∈ {1, 2, 3} denote the three nontrivial involutions in T. It follows from condition (∗) and σ Theorem 2.4(4) that M i contains a 4-dimensional connected component Fi (which 4 is unique by the intersection theorem). By Lemma 3.1, Fi is homeomorphic to S 2 or ރP . Since M is spin, the action of σi must be even; see Theorem 2.4(6). Hence σ M i is the union of Fi and isolated σi -fixed points. Using Lemma 3.2, we see that each connected component of M T2 has type (4, 4, 4) or (4, 4, 0). In particular, any T2-fixed point component is contained in some Fi . To derive a contradiction, we will compute the Euler characteristic. Consider the case when for one of the Fi , say F1, the Euler characteristic is equal to 3 2 and hence Fi is homeomorphic to ރP . Since σ2 acts nontrivially on F1, we get σ σ 2 = 3 = 2 ∪{ } 2 T2 F1 F1 S pt , where S and pt are connected components of M of type (4, 4, 4) and (4, 4, 0), respectively. Hence, one of the other Fi , say F2, contains S2 ∪ {pt}. This leads to the contradiction

T2 5 = χ(M) = χ(M ) = χ(F2) + χ(F3) − χ(F2 ∩ F3) ≤ 3 + 3 − 2 = 4. T So, assume χ(Fi ) = 2 for all i. Note that M 2 cannot contain a connected component of type (4, 4, 4) since otherwise χ(M) = 2 by a computation similar to the one above. Hence, each connected component of M T2 is of type (4, 4, 0). In particular, the Fi intersect pairwise in different points, which gives the contradiction X X 5 = χ(M) = χ(Fi ) − χ(Fi ∩ Fj ) + χ(F1 ∩ F2 ∩ F3) ≤ 6 − 3 = 3. i i< j In conclusion, we have shown that χ(M) 6= 5, which completes the proof of the theorem. A more natural and apparently more difficult problem is to understand the oriented bordism type of an 8-dimensional positively curved manifold M with symmetry rank ≥ 2. SO Wall [1960] has shown that the only torsion in the oriented bordism ring ∗ is 2-torsion, and that the oriented bordism type of a manifold is determined by Pon- SO tryagin and Stiefel–Whitney numbers. In dimension eight, 8 is isomorphic to ޚ⊕ޚ. Hence, in this dimension the oriented bordism type of an oriented manifold is determined by its Pontryagin numbers. In contrast to the case of Spin-manifolds, the Aˆ-genus does not have to vanish on positively curved 8-dimensional oriented manifolds with symmetry (consider for 44 ANAND DESSAI example ރP4). This makes the problem of determining the oriented bordism type more difficult. One way to attack this problem is to prove the stronger statement that for some orientation of M and some S1 ⊂ T , the S1-action has locally the same S1-geometry as a suitable chosen S1-action on one of the symmetric spaces S8, ވP2 or ރP4 (that two S1-manifolds have the same local S1-geometry just means that there exists an equivariant orientation-preserving diffeomorphism between the normal bundles of the S1-fixed point manifolds). Once this has been accomplished, one can glue the complements of the normal bundles together to get a new manifold W, with fixed point free S1-action, that is bordant to the difference of M and the symmetric space in question. As observed by Bott [1967], all Pontryagin numbers of a manifold with fixed point-free S1-action vanish and hence W is rationally zero- bordant. Since the oriented bordism ring has no torsion in degree 8, the manifold M is bordant to the symmetric space in question. This line of attack can be applied successfully at least if χ(M) 6= 5. Details will appear elsewhere. It is interesting to compare the results above with [Dessai and Tuschmann 2007], in which it is shown that there exists an infinite sequence of closed simply connected Riemannian 8-manifolds with nonnegative curvature and mutually distinct oriented bordism type.

Acknowledgments

Part of this work was carried out during a stay at the University of Pennsylvania, and I thank its mathematics department for its hospitality. I also thank Burkhard Wilking and Wolfgang Ziller for helpful discussions. A particular step in the proof of Theorem 1.2 relies on a result about ޚ/2ޚ- cohomology ރP4’s. I am grateful to Mikiya Masuda for sharing this insight with me.

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Received September 18, 2009. Revised February 10, 2010.

ANAND DESSAI DÉPARTEMENT DE MATHÉMATIQUES CHEMINDU MUSÉE 23 FACULTÉ DES SCIENCES UNIVERSITÉDE FRIBOURG,PÉROLLES 1700 FRIBOURG SWITZERLAND [email protected] http://homeweb1.unifr.ch/dessaia/pub/

PACIFIC JOURNAL OF MATHEMATICS Vol. 249, No. 1, 2011

STRONG KÄHLER WITH TORSION STRUCTURES FROM ALMOST CONTACT MANIFOLDS

MARISA FERNÁNDEZ,ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA

For an almost contact metric manifold N, we ﬁnd conditions under which either the total space of an S1-bundle over N or the Riemannian cone over N admit a strong Kähler with torsion (SKT) structure. In so doing, we construct new 6-dimensional SKT manifolds. Moreover, we study the geometric structure induced on a hypersurface of an SKT manifold and use it to construct new SKT manifolds via appropriate evolution equations. We also study hyper-Kähler with torsion (HKT) structures on the total space of an S1-bundle over manifolds with three almost contact structures.

1. Introduction

On any Hermitian manifold (M2n, J, h) there exists a unique Hermitian connection ∇ B with totally skew-symmetric torsion, which is called the Bismut connection after [Bismut 1989]. The torsion 3-form h(X, T B(Y, Z)) of ∇ B can be identiﬁed with the 3-form −J dF( · , · , · ) = −dF(J · , J · , J · ), where F( · , · ) = h( · , J · ) is the fundamental 2-form associated to the Hermitian structure (J, h). Hermitian structures with closed J dF are called strong Kähler with torsion (in ¯ 1 short, SKT) or pluriclosed [Egidi 2001]. Since ∂∂ acts as 2 d J d on forms of bidegree (1, 1), the latter condition is equivalent to ∂∂¯ F = 0. SKT structures have been recently studied by many authors, and they also have applications in type II string theory and in 2-dimensional supersymmetric σ -models [Gates et al. 1984; Strominger 1986; Ivanov and Papadopoulos 2001]. The class of SKT metrics includes of course the Kahler¨ metrics, but as in [Fino et al. 2004], we are interested on non-Kahler¨ geometry, so by an SKT metric we

MSC2000: primary 53C55; secondary 53C26, 22E25, 53C15. Keywords: Hermitian metric, hypercomplex structure, torsion, almost contact structure, Riemannian cone. Partially supported through Project MICINN (Spain) MTM2008-06540-C02-01/02, Project MIUR “Riemannian Metrics and Differentiable Manifolds” and by GNSAGA of INdAM.

49 50 MARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA will mean a Hermitian metric h whose fundamental 2-form F is ∂∂¯-closed but not d-closed. Gauduchon [1984] showed that on a compact complex surface, an SKT metric can be found in the conformal class of any given Hermitian metric, but in higher dimensions the situation is more complicated. SKT structures on 6-dimensional nilmanifolds, that is, on compact quotients of nilpotent Lie groups by discrete subgroups, were classified in [Fino et al. 2004; Ugarte 2007]. Simply connected examples of 6-dimensional SKT manifolds have been found in [Grantcharov et al. 2008] by using torus bundles, and recently Swann [2010] has reproduced them via the twist construction, by extending them to higher dimensions and finding new compact simply connected SKT manifolds. Moreover, Fino and Tomassini [2009] showed that the SKT condition is preserved by the blow-up construction. The odd-dimensional analogue of a Hermitian structure is given by a normal almost contact metric structure. Indeed, on the product N 2n+1× ޒ of a (2n+1)- dimensional almost contact metric manifold N 2n+1 by the real line ޒ, it is possible to define a natural almost complex structure, which is integrable if and only if the almost contact metric structure on N 2n+1 is normal [Sasaki and Hatakeyama 1961]. More generally, it is possible to construct Hermitian manifolds starting from an almost contact metric manifold N 2n+1 by considering a principal fiber bundle P with base space N 2n+1 and structural group S1, that is, an S1-bundle over N 2n+1; see [Ogawa 1963]. Indeed, by using the almost contact metric structure on N 2n+1 and the connection 1-form θ, Ogawa constructed an almost Hermitian structure (J, h) on P and found conditions under which J is integrable and (J, h) is Kahler.¨ In Section 2, we determine in Theorem 2.3 general conditions under which an S1-bundle over an almost contact metric (2n+1)-dimensional manifold N 2n+1 is SKT. We study the particular case when N 2n+1 is quasi-Sasakian, that is, when it has an almost contact metric structure for which the fundamental form is closed (Corollary 2.4). In this way, we are able to construct some new 6-dimensional SKT examples, starting from 5-dimensional quasi-Sasakian Lie algebras, and also from Sasakian ones. A Sasakian structure can be also seen as the analogue, in odd dimensions, of a Kahler¨ structure. Indeed, by [Boyer and Galicki 1999], a Riemannian manifold (N 2n+1, g) of odd dimension 2n+1 admits a compatible Sasakian structure if and only if the Riemannian cone N 2n+1 × ޒ+ is Kahler.¨ In Section 3, Theorem 3.1 gives the conditions that must be satisfied by the compatible almost contact metric structure on a Riemannian manifold (N 2n+1, g) in order that the Riemannian cone N 2n+1 × ޒ+ be SKT. We provide an example of an SKT manifold constructed as a Riemannian cone, and in Section 4 we consider the case when the Riemannian cone is 6-dimensional. This case is interesting since one can impose that the SKT STRONGKÄHLERWITHTORSIONSTRUCTURES 51 structure is in addition an SKT SU(3)-structure, and one can find relations with the SU(2)-structures studied by Conti and Salamon [2007]. In Section 5, we study the geometric structure induced naturally on any oriented hypersurface N 2n+1 of a (2n+2)-dimensional manifold M2n+2 carrying an SKT structure, and in Section 6, we use such structures in Theorem 6.4 to construct new SKT manifolds via appropriate evolution equations [Hitchin 2001; Conti and Salamon 2007] starting from a 5-dimensional manifold endowed with an SU(2)- structure. A good quaternionic analogue of Kahler¨ geometry is given by hyper-Kähler with torsion (in short, HKT) geometry. An HKT manifold is a hyper-Hermitian 4n manifold (M , J1, J2, J3, h) admitting a hyper-Hermitian connection with totally skew-symmetric torsion, that is, one in which the three Bismut connections associated with the three Hermitian structures (Jr , h) coincide for r = 1, 2, 3. This geometry was introduced by Howe and Papadopoulos [1996] and later studied in [Grantcharov and Poon 2000; Fino and Grantcharov 2004; Barberis et al. 2009; Barberis and Fino 2008; Swann 2010]. In the interesting special case in which the torsion 3-form of such a hyper- Hermitian connection is closed, the HKT manifold is called strong. In Section 7, Theorem 7.1 gives conditions under which an S1-bundle over a (4n+3)-dimensional manifold endowed with three almost contact metric structures is HKT and in particular when it is strong HKT.

2. SKT structures arising from S1-bundles

Consider a (2n+1)-manifold N 2n+1 endowed with an almost contact metric structure (I, ξ, η, g); that is, I is a tensor field of type (1, 1), ξ is a vector field, η is a 1-form, and g is a Riemannian metric on N 2n+1, satisfying together the conditions I 2 = − Id +η ⊗ ξ, η(ξ) = 1, g(IU, IV ) = g(U, V ) − η(U)η(V ) for any vector fields U and V on N 2n+1. Denote by ω the fundamental 2-form of (I, ξ, η, g); that is, ω is the 2-form on N 2n+1 given by ω( · , · ) = g( · , I · ). Given the tensor field I , consider its Nijenhuis torsion [I, I ], defined by (1) [I, I ](X, Y ) = I 2[X, Y ] + [IX, IY ] − I [IX, Y ] − I [X, IY ]. On the product N 2n+1 × ޒ, one can define a natural almost complex structure d d J X, f = IX + f ξ, −η(X) , dt dt where f is a Ꮿ∞-function on N 2n+1 × ޒ and t is the coordinate on ޒ. 52 MARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA

Deﬁnition 2.1 [Sasaki and Hatakeyama 1961]. We call an almost contact metric structure (I, ξ, η, g) on N 2n+1 normal if the almost complex structure J on N 2n+1 × ޒ is integrable, or equivalently if [I, I ](X, Y ) + 2dη(X, Y )ξ = 0 for any vector ﬁelds X, Y on N 2n+1.

By [Blair 1967, Lemma 2.1], one has iξ dη = 0 for a normal almost contact metric structure (I, ξ, η, g). Remark 2.2. The normality of the almost contact structure implies as well that I dη = dη. Indeed, d(η − i dt) = dη has no (0, 2)-part and therefore has no (2, 0)- part since dη is real. Thus, J dη = dη, but Jdη = I dη as well since iξ dη = 0. We recall that a Hermitian manifold (M, J, h) is SKT if and only if the 3-form J dF is closed, where F is the fundamental 2-form of (J, h). We will use the convention that J acts on r-forms β by

(Jβ)(X1,..., Xr ) = β(JX1,..., JXr ) for any vector fields X1,..., Xr . We now show general conditions under which an S1-bundle over an almost contact metric (2n+1)-dimensional manifold is SKT. Let (N 2n+1, I, ξ, η) be a (2n+1)-dimensional almost contact manifold, and let be a closed 2-form on N 2n+1 that represents an integral cohomology class on N 2n+1. From the well-known result of Kobayashi [1956], we can consider the circle bundle S1 ,→ P → N 2n+1 and the connection 1-form θ on P whose curvature form is dθ = π ∗(), where π : P → N 2n+1 is the projection. By using the almost contact structure (I, ξ, η) and the connection 1-form θ, one can define an almost complex structure J on P as follows [Ogawa 1963]. For any right-invariant vector field X on P, the vector field JX is given by ∗ (2) θ(JX) = −π (η(π∗ X)) and π∗(JX) = I (π∗ X) + θ(˜ X)ξ, where θ(˜ X) is the unique function on N 2n+1 such that (3) π ∗θ(˜ X) = θ(X). This definition can be extended to an arbitrary vector field X on P since X can P be written in the form X = j f j X j , with f j smooth functions on P, and X j P right-invariant vector fields. Then JX = j f j JX j . Ogawa [1963] showed that when (N 2n+1, I, ξ, η) is normal, the almost complex structure J on P defined by (2) is integrable if and only if dθ is J-invariant, that is, if J(dθ) = dθ or equivalently dθ(JX, Y )+dθ(X, JY ) = 0 for any vector fields X and Y on P. That is, dθ is a complex 2-form on P having bidegree (1, 1) with respect to J. STRONGKÄHLERWITHTORSIONSTRUCTURES 53

In terms of the 2-form , whose lift to P is the curvature of the circle bundle S1 ,→ P → N 2n+1, the previous condition means that is I -invariant, that is, I () = . Therefore iξ = 0. If {e1,..., e2n, η} is an adapted coframe on a neighborhood U on N 2n+1, that is, I e2 j−1 = −e2 j and I e2 j = e2 j−1 for 1 ≤ j ≤ n, then we can take {π ∗e1, . . . , π ∗e2n, π ∗η, θ} as a coframe in π −1(U). By using the coframe {π ∗e1, . . . , π ∗e2n}, we may write dθ = π ∗α + π ∗β ∧ π ∗η, where α is a 2-form in V2he1,..., e2ni, and β ∈ V1he1,..., e2ni. Suppose that N 2n+1 has a normal almost contact metric structure (I, ξ, η, g). We consider a principal S1-bundle P with base space N 2n+1 and connection 1-form θ, and endow P with the almost complex structure J (associated to θ) defined by (2). Since N 2n+1 has a Riemannian metric g, a Riemannian metric h on P compatible with J (see [Ogawa 1963]) is given by ∗ (4) h(X, Y ) = π g(π∗ X, π∗Y ) + θ(X) θ(Y ) for any right-invariant vector fields X and Y . This definition can be extended to any vector field on P. Theorem 2.3. Consider a (2n+1)-dimensional almost contact metric manifold (N 2n+1, I, ξ, η, g), and let be a closed 2-form on N 2n+1 that represents an integral cohomology class. Consider the circle bundle S1 ,→ P → N 2n+1 with connection 1-form θ, whose curvature form is dθ = π ∗() for the projection π : P → N 2n+1. The almost Hermitian structure (J, h) on P defined by (2) and (4) is SKT if and only if (I, ξ, η, g) is normal, dθ is J-invariant, and ∗ d(π (I (iξ dω))) = 0, (5) ∗ ∗ ∗ ∗ d(π (I (dω) − dη ∧ η)) = (−π (I (iξ dω)) + π ) ∧ π , where ω is the fundamental form of the almost contact metric structure (I, ξ, η, g). Proof. As we mentioned, a result of Ogawa [1963] asserts that the almost complex structure J is integrable if and only if (I, ξ, η, g) is normal and J(dθ) = dθ. Thus, (J, h) is SKT if and only if the 3-form J dF is closed. Using the first equality in (2), we find that the fundamental 2-form F on P is F(X, Y ) = h(X, JY ) ∗ = π g(π∗ X, π∗ JY ) + θ(X) θ(JY ) ∗ ∗ = π g(π∗ X, π∗ JY ) − θ(X) π η(π∗Y ). 54 MARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA

Therefore, taking into account that we are working with a circle bundle (whose ﬁber is thus 1-dimensional), we have F = π ∗ω + π ∗η ∧ θ, d F = π ∗(dω) + π ∗(dη) ∧ θ − π ∗η ∧ dθ, (6) J dF = J(π ∗(dω)) − J(π ∗(dη)) ∧ π ∗η − θ ∧ dθ since J(π ∗η) = θ and J is integrable, and so J(dθ) = dθ. Moreover, ∗ ∗ ∗ (7) J(π (dω)) = π (I (dω)) + π (I (iξ dω)) ∧ θ. Indeed, locally and in terms of the adapted basis {e1,..., e2n+1} with I e2 j−1 = −e2 j for 1 ≤ j ≤ n, I e2n+1 = 0, and η = e2n+1, we can write dω = α + β ∧ η, where the local forms α ∈ V3he1,..., e2ni and β ∈ V2he1,..., e2ni are generated only by e1,..., e2n. Furthermore, we have I α = I (dω) and β = iξ dω. Thus ∗ ∗ ∗ J(π (dω)) = J(π (α)) + J(π (iξ dω)) ∧ θ. ∗ ∗ ∗ Now, by using (2) and (3), we see that J(π (α)) = π (I α) and J(π (iξ dω)) = ∗ π (I (iξ dω)), which proves (7). As a consequence of Remark 2.2, ∗ ∗ ∗ ∗ (8) J(π (dη)) = π (I (dη)) − π (I (iξ dη)) ∧ θ = π (dη) since iξ dη = 0 and I dη = dη. Using (7) and (8), we get ∗ ∗ ∗ ∗ (9) J dF = π (I (dω)) + π (I (iξ dω)) ∧ θ − π (dη) ∧ π η − θ ∧ dθ. Therefore, ∗ ∗ ∗ d(J dF) = d(π (I (dω))) + d(π (I (iξ dω))) ∧ θ + π (I (iξ dω)) ∧ dθ − d(π ∗(dη)) ∧ π ∗η − π ∗(dη) ∧ dπ ∗η − dθ ∧ dθ. Consequently, d(J dF) = 0 if and only if ∗ d(π (I (iξ dω))) = 0, ∗ ∗ d(π (I (dω) − dη ∧ η)) = (π (−I (iξ dω)) + dθ) ∧ dθ. An almost contact metric manifold (N 2n+1, I, ξ, η, g) is quasi-Sasakian if it is normal and its fundamental form ω is closed. In particular, if dη = α ω, then the almost contact metric structure is called α-Sasakian. When α = −2, the structure is said to be Sasakian. By [Friedrich and Ivanov 2002, Theorem 8.2], an almost contact metric manifold (N 2n+1, I, ξ, η, g) admits a connection ∇c that preserves the almost contact metric STRONGKÄHLERWITHTORSIONSTRUCTURES 55 structure and has totally skew-symmetric torsion tensor if and only if the Nijenhuis tensor of I , given by (1), is skew-symmetric and ξ is a Killing vector ﬁeld. This connection is unique. In particular, on any quasi-Sasakian manifold (N 2n+1, I, ξ, η, g) there exists a unique connection ∇c with totally skew-symmetric torsion, such that

∇c I = 0, ∇cg = 0, ∇cη = 0.

Such a connection ∇c is uniquely determined by ∇c = ∇g + 1 ∧ (10) g( X Y, Z) g( X Y, Z) 2 (dη η)(X, Y, Z), ∇g 1 ∧ ∇c where is the Levi-Civita connection and 2 (dη η) is the torsion 3-form of . Corollary 2.4. Let (N 2n+1, I, ξ, η, g) be a quasi-Sasakian (2n+1)-manifold, and let be a closed 2-form on N 2n+1 that represents an integral cohomology class. Consider the circle bundle S1 ,→ P → N 2n+1 with connection 1-form θ whose curvature form is dθ = π ∗() for the projection π : P → N 2n+1. The almost Hermitian structure (J, h) on P deﬁned by (2) and (4) is SKT if and only if is I -invariant, iξ = 0, and (11) dη ∧ dη = − ∧ .

The Bismut connection ∇ B of (J, h) on P and the connection ∇c on N given by (10) are related by ∇ B = ∗ ∇c (12) h( X Y, Z) π g( π∗ X π∗Y, π∗ Z) for any vector fields X, Y, Z ∈ Ker θ. Proof. Since dω = 0, if we impose the SKT condition, then we get by using the previous theorem the equation (11). The Bismut connection ∇ B associated to the Hermitian structure (J, h) on P is ∇ B = ∇h − 1 (13) h( X Y, Z) h( X Y, Z) 2 dF(JX, JY, JZ) for any vector fields X, Y, Z on P, where ∇h is the Levi-Civita connection associated to h. Then, for any X, Y, Z in the kernel of θ, we have ∇ B = ∗ ∇h + 1 ∗ ∧ ∗ h( X Y, Z) π g( X Y, Z) 2 (π (dη) π η)(X, Y, Z). By [Ogawa 1963, Lemma 3] and the definition of ∇c, we get h(∇ BY, Z) = π ∗g(∇g π Y, π Z) + 1 (π ∗(dη) ∧ π ∗η)(X, Y, Z) X π∗ X ∗ ∗ 2 = ∗ ∇c π g( π∗ X π∗Y, π∗ Z) for any X, Y, Z in the kernel of θ. 56 MARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA

Remark 2.5. If the structure (I, ξ, η, g) is α-Sasakian, equation (11) reads ∧ = −α2 ω ∧ ω.

In the case of a trivial S1-bundle, that is, if we consider the natural almost Hermitian structure on the product N 2n+1 × ޒ, we get this: Corollary 2.6. Let (N 2n+1, I, ξ, η, g) be a (2n+1)-dimensional almost contact metric manifold. Impose on the product N 2n+1 ×ޒ the almost complex structure J given by d JX = I X for X ∈ Ker η and Jξ = − , dt and the metric h given by h = g + (dt)2. The Hermitian structure (J, h) is SKT if and only if (I, ξ, η, g) is normal, d(I (dω)) = d(dη ∧ η) and d(I (iξ dω)) = 0, where ω denotes the fundamental 2-form of the almost contact metric structure (g, I, ξ, η). Corollary 2.7. Let (N 2n+1, I, ξ, η, g) be a (2n+1)-dimensional quasi-Sasakian manifold with dη ∧ dη = 0. The Hermitian structure (J, h) on N 2n+1× ޒ is SKT. Moreover, its Bismut connection ∇ B coincides with the unique connection ∇c on N 2n+1 given by (10). Proof. In this case, since dω = 0 we get d(J dF) = −d(dη ∧ η). By using (12), B c 2n+1 we get h(∇X Y, Z) = g(∇X Y, Z) for any vector ﬁelds X, Y, Z on N . 2.1. Examples. We will present three examples of quasi-Sasakian Lie algebras satisfying the condition dη ∧ dη = 0. By applying Corollary 2.7, one gets an SKT structure on the product of the corresponding simply connected Lie group by ޒ. Example 2.8. Let s be the 5-dimensional Lie algebra with structure equations de1 = e13 + e23 + e25 − e34 + e35, de2 = 2e12 − 2e13 + e14 − e15 − e24 + e34 + e45, de3 = −e12 + e13 + e14 − e15 + 2e24 − 2e34 + e45, de4 = −e12 − e23 + e24 − e25 − e35, de5 = e12 − e13 − e24 + e34, where ei j = ei ∧ e j . On s, take the quasi-Sasakian structure (I, ξ, η, g) given by

5 1 2 3 4 12 34 P5 j 2 (14) η = e , I e = −e , I e = −e , ω = −e − e , g = j=1(e ) . This quasi-Sasakian structure satisﬁes the condition d(dη ∧ η) = 0. The Lie algebra s is 2-step solvable since the commutator

1 s = [s, s] = ޒhe1 − e4, e2 + e3, e1 − e2 + 2e3 − e5i STRONGKÄHLERWITHTORSIONSTRUCTURES 57

1 5 is abelian, where {e1,..., e5} denotes the dual basis of {e ,..., e }. Moreover, s − has trivial center, is irreducible and nonunimodular, since the trace of ade1 is 3.

Example 2.9. Consider the family of 2-step solvable Lie algebras sa for a ∈ޒ−{0}, given by de1 = a e23 + 3e25, de3 = a e34, de2 = −a e13 − 3e15, de4 = 0, 5 = − 1 2 34 de 3 a e . The almost contact metric structure (I, ξ, η, g) defined in (14) is quasi-Sasakian and satisfies the condition dη ∧ dη = 0. The second cohomology group of sa is generated by e12 and e45. Example 2.10. Another family of quasi-Sasakian Lie algebras that satisfies the condition dη ∧ dη = 0 is gb for b ∈ ޒ − {0}, with structure equations de1 = b(e13 + e14 − e23 + e24) + e25, de3 = 2e45, de2 = b(−e13 + e14 − e23 − e24) − e15, de4 = −2e35, de5 = −4b2e34, and endowed with the quasi-Sasakian structure given by (14). The second coho- 12 mology group of gb is generated by e . The Lie algebras gb are not solvable since the commutators are [gb, gb] = gb. The Lie groups underlying Examples 2.9 and 2.10 also satisfy the conditions of Corollary 2.4 with ∧ = 0, by just taking as connection 1-form the 1-form e6 such that de6 = λe12. Then, = λe12. With this expression of de6, we have d2e6 = 0, J(de6) = de6, and de6 ∧ de6 = 0. Therefore, equation (11) is satisfied. Observe that λ = 0 provides examples of trivial S1-bundles. The next example allows us to recover one of the 6-dimensional nilmanifolds found in [Fino et al. 2004]: Example 2.11. Consider the 5-dimensional nilpotent Lie algebra with structure equations de j = 0 for j = 1,..., 4, de5 = e12 + e34, and endowed with the quasi-Sasakian structure given by (14). If we consider the closed 2-form = e13 + e24 and apply Corollary 2.4, we see that there exists a nontrivial S1-bundle over the corresponding 5-dimensional nilmanifold. Since de5 ∧ de5 = − ∧ 6= 0, the total space of this S1-bundle is an SKT nilmanifold. More precisely, according to the classification given in [Fino et al. 2004] (see also 58 MARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA

[Ugarte 2007]), the nilmanifold is the one with underlying Lie algebra isomorphic to h3 ⊕ h3, where by h3 we denote the real 3-dimensional Heisenberg Lie algebra. Since the starting Lie algebra from Example 2.11 is Sasakian, it is natural to start with other 5-dimensional Sasakian Lie algebras to construct new SKT structures in dimension 6. A classiﬁcation of 5-dimensional Sasakian Lie algebras was obtained in [Andrada et al. 2009].

Example 2.12. Consider the 5-dimensional Lie algebra k3 with structure equations de1 = 0, de4 = 0, de2 = −e13, de5 = λe14 + µe23, de3 = e12, where λ, µ<0. By [Andrada et al. 2009], this algebra admits the Sasakian structure given by

I e1 = e4, I e2 = e3, η = e5, = − 1 2 − 1 2 − 1 2 − 1 2 + 2 g 2 λ(e1) 2 λ(e2) 2 µ(e3) 2 µ(e4) (e5) , and is isomorphic to ޒ n (h3 × ޒ); moreover, the corresponding solvable simply connected Lie group admits a compact quotient by a discrete subgroup. 14 23 Consider on k3 the closed 2-form =λe −µe . The form is I -invariant and satisﬁes ∧=−2λµe1234. Since e5 is the contact form and de5∧de5 =2λµe1234, we get again by Corollary 2.4 an SKT structure on a nontrivial S1-bundle over the 5-dimensional solvmanifold. We denote by e6 the connection 1-form. The orthonormal basis {α1 = e1, α2 = e4, α3 = e2, α4 = e3, α5 = e5, α6 = θ} for the SKT metric satisﬁes the equations

dα1 = dα2 = 0, dα3 = −α14, dα4 = α13, dα5 = λα12 + µα34, dα6 = λα12 − µα34, and the complex structure is given by J(X1) = X2, J(X3) = X4 and J(X5) = X6, { }6 { i }6 where Xi i=1 denotes the basis dual to α i=1. Since the fundamental 2-form is F = α12 + α34 + α56, the 3-form torsion T of the SKT structure is

T = λα12(α5 + α6) + µα34(α5 − α6).

Moreover, ∗T = λα12(α5 + α6) − µα34(α5 − α6), where ∗ denotes the metric’s Hodge operator; this implies that the torsion form is also coclosed. B i B The only nonzero curvature forms ( )j of the Bismut connection ∇ are

B 1 2 12 B 3 2 34 ( )2 = −2λ α and ( )4 = −2µ α . STRONGKÄHLERWITHTORSIONSTRUCTURES 59

A direct calculation shows that the 1-forms α5 and α6 and the 2-forms α12 and α34 are parallel with respect to the Bismut connection, which implies that ∇ B T = 0. Finally, Hol(∇ B) = U(1) × U(1) ⊂ U(3) since ∇ Bαi 6= 0 for i = 1, 2, 3, 4.

3. SKT structures arising from Riemannian cones

Let N 2n+1 be a (2n+1)-dimensional manifold endowed with an almost contact metric structure (I, ξ, η, g), and denote by ω its fundamental 2-form. The Riemannian cone of N 2n+1 is deﬁned as the manifold N 2n+1 ×ޒ+ equipped with the cone metric (15) h = t2g + (dt)2.

The cone N 2n+1× ޒ+ has a natural almost Hermitian structure deﬁned by (16) F = t2ω + tη ∧ dt.

The almost complex structure J on N 2n+1 × ޒ+ deﬁned by (F, h) is given by d JX = IX for X ∈ Ker η and Jξ = −t . dt In terms of a local orthonormal adapted coframe {e1,..., e2n} for g with n X (17) ω = − e2 j−1 ∧ e2 j , j=1 we have Je2 j−1 = −e2 j , Je2 j = e2 j−1 for j = 1,..., n, (18) J(te2n+1) = dt, J(dt) = −te2n+1.

The almost Hermitian structure (J, h) on N 2n+1× ޒ+ is Kahler¨ if and only if the almost contact metric structure (I, ξ, η, g) on N 2n+1 is Sasakian, that is, a normal contact metric structure. If we impose that the almost Hermitian structure (J, h) on N 2n+1 × ޒ+ is SKT, we can prove the following: Theorem 3.1. Consider a (2n+1)-dimensional almost contact metric manifold (N 2n+1, I, ξ, η, g). The almost Hermitian structure (J, h) given by (15) and (16) on the Riemannian cone (N 2n+1 ×ޒ+, h) is SKT if and only if (I, ξ, η, g) is normal and

(19) −4η ∧ ω + 2 I (dω) − 2dη ∧ η = d(I (iξ dω)), where ω denotes the fundamental 2-form of the almost contact metric structure (I, ξ, η, g). 60 MARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA

Proof. J is integrable if and only if the almost contact metric structure is normal. We compute J dF. We have

d F = 2tdt ∧ ω + t2dω + t dη ∧ dt, and so JdF = −2t2η ∧ ω + t2 J(dω) − t2dη ∧ η since Jω = ω, J(dt) = −tη and J dη = dη. Moreover, with respect to an adapted basis {e1,..., e2n+1} we can get, in a way similar to the proof of Theorem 2.3, that

(20) J dω = I (dω) + I (iξ dω) ∧ Jη.

2 2 2 As a consequence, we get J dF = −2t η∧ω+t I (dω)+t dt ∧ I (iξ dω)−t dη∧η. Therefore, by imposing that d(J dF) = 0, we obtain

−4η ∧ ω + 2I (dω) − 2dη ∧ η − d(I (iξ dω)) = 0, −2d(η ∧ ω) + d(I (dω)) − d(dη ∧ η) = 0.

Since the second equation is a consequence of the first, the Hermitian structure (F, h) on the Riemannian cone N 2n+1 × ޒ+ is SKT if and only if the almost 2n+1 contact metric structure (I, η, ξ, g, ω) on N satisfies equation (19). Remark 3.2. As a consequence of previous theorem, when n = 1, equation (19) is satisfied if and only if the 3-dimensional manifold N is Sasakian. On the other hand, if n > 1 and the almost contact metric structure on N 2n+1 is quasi-Sasakian (that is, dω = 0), then the structure has to be Sasakian, that is, dη = −2ω.

Example 3.3. Consider the 5-dimensional Lie algebras ga,b,c with structure equations b3 b de1 = a e23 + 2e25 + − 1 ab + + 2 e34 + be45, 2 2a a 2 = − 13 − 15 − 1 34 − 35 de a e 2e 2 bce be , 4 b2 de3 = − − e34, a a de4 = ce34, de5 = 2e12 + be14 − be23 + (2 + b2)e34, where a, b, c ∈ ޒ and a 6= 0. They are endowed with the normal almost contact metric structure (I, ξ, η, g, ω) with

I e1 = −e2, I e3 = −e4, η = e5, ω = −e12 − e34.

This structure satisﬁes (19), and therefore the Riemannian cones over the corresponding simply connected Lie groups are SKT. STRONGKÄHLERWITHTORSIONSTRUCTURES 61

4. SKT SU(3)-structures

Let (M6, J, h) be a 6-dimensional almost Hermitian manifold. An SU(3)-structure 6 on M is determined by the choice of a (3, 0)-form 9 = 9++ i9− of unit norm. If 9 is closed, then the underlying almost complex structure J is integrable and the manifold is Hermitian. We will denote the SU(3)-structure (J, h, 9) simply by (F, 9), where F is the fundamental 2-form, since from F and 9 we can reconstruct the almost Hermitian structure. Deﬁnition 4.1. We say that an SU(3)-structure (F, 9) on M6 is SKT if (21) d9 = 0 and d(J dF) = 0, where J is the associated complex structure. We will see the relation between SKT SU(3)-structures in dimension 6 and SU(2)-structures in dimension 5. First, we recall some facts about SU(2)-structures on a 5-dimensional manifold. An SU(2)-structure on a 5-dimensional manifold N 5 is an SU(2)-reduction of the principal bundle of linear frames on N 5. By [Conti and Salamon 2007, Proposition 1], these structures are in one-to-one correspondence with quadru- 5 plets (η, ω1, ω2, ω3), where η is a 1-form and ωi are 2-forms on N satisfying ωi ∧ω j = δi j v and v ∧η 6= 0 for some 4-form v, and ω2(X, Y ) ≥ 0 if iX ω3 = iY ω1, 5 where iX denotes the contraction by X. Equivalently, an SU(2)-structure on N can be viewed as the datum of (η, ω1, 8), where η is a 1-form, ω1 is a 2-form, and 8 = ω2 + i ω3 is a complex 2-form such that

η ∧ ω1 ∧ ω1 6= 0, 8 ∧ 8 = 0, ω1 ∧ 8 = 0, 8 ∧ 8 = 2ω1 ∧ ω1, and 8 is of type (2, 0) with respect to ω1. Conti and Salamon [2007] locally characterize an SU(2)-structure as follows. If 5 (η, ω1, ω2, ω3) is an SU(2)-structure on a 5-manifold N , then locally there exists an orthonormal basis of 1-forms {e1,..., e5} such that 12 34 13 24 14 23 5 ω1 = e + e , ω2 = e − e , ω3 = e + e , η = e . We can also consider the local tensor field I given by I e1 = −e2, I e2 = e1, I e3 = −e4, I e4 = e3, I e5 = 0. This tensor gives rise to a global tensor field of type (1, 1) on the manifold N 5, 5 defined by ω1(X, Y ) = g(X, IY ) for any vector fields X and Y on N , where g is the Riemannian metric on N 5 underlying the SU(2)-structure. The tensor field I satisfies I 2 = − Id +η ⊗ ξ, where ξ is the vector field on N 5 dual to the 1-form η. Therefore, given an SU(2)-structure (η, ω1, ω2, ω3) we also have an almost contact metric structure (I, ξ, η, g) on the manifold, where ω1 is its fundamental form. 62 MARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA

Remark 4.2. Notice that we have two more almost contact metric structures when we consider ω2 and ω3 as fundamental forms. 5 5 If N has an SU(2)-structure (η, ω1, ω2, ω3), the product N × ޒ has a natural SU(3)-structure given by

(22) F = ω1 + η ∧ dt and 9 = (ω2 + iω3) ∧ (η − i dt). By Corollary 2.6, the previous SU(3)-structure is SKT if and only if

d(I (dω1)) = d(dη ∧ η), dω2 = −3ω3 ∧ η, (23) d(I (iξ dω1)) = 0, dω3 = 3ω2 ∧ η, proving this: Theorem 4.3. Suppose N 5 is a 5-dimensional manifold endowed with an SU(2)- structure (η, ω1, ω2, ω3). The SU(3)-structure (F, 9) given by (22) on the product N 5× ޒ is SKT if and only if the equations (23) are satisﬁed. Example 4.4. On the 5-dimensional Lie algebras introduced in Examples 2.8, 2.9 and 2.10, consider the SU(2)-structure given by

12 34 13 24 14 23 ω = ω1 = e + e , ω2 = e − e , ω3 = e + e . For Example 2.8, we have

124 134 123 234 dω2 = −2ω3 ∧ η − 4(e − e ) and dω3 = 2ω2 ∧ η + 4(e + e ).

For Examples 2.9 and 2.10, we get dω2 = −3ω3 ∧ η and dω3 = 3 ω2 ∧ η. Therefore one gets an SKT SU(3)-structure on the product of the corresponding simply connected Lie groups by ޒ. We will study the existence of SKT SU(3)-structures on a Riemannian cone over 5 a 5-dimensional manifold N endowed with an SU(2)-structure (η, ω1, ω2, ω3). 5 Then N has an induced almost contact metric structure (I, ξ, η, g), and ω1 is its fundamental form. The Riemannian cone (N 5 × ޒ+, h) of (N 5, g) has a natural SU(3)-structure deﬁned by

2 2 F = t ω1 + tη ∧ dt and 9 = t (ω2 + iω3) ∧ (tη − idt). In terms of a local orthonormal coframe {e1,..., e5} for g such that

12 34 13 24 14 23 5 ω1 = −e − e , ω2 = −e + e , ω3 = −e − e , η = e , we have Je1 = −e2, Je2 = e1, Je3 = −e4, Je4 = e3, J(te5) = dt, J(dt) = −te5. STRONGKÄHLERWITHTORSIONSTRUCTURES 63

We recall that the SU(3)-structure (F, 9) on N 5× ޒ+ is integrable if and only 5 if the SU(2)-structure (η, ω1, ω2, ω3) on N is Sasaki–Einstein, or equivalently if and only if

dη = −2ω1, dω2 = −3ω3 ∧ η, dω3 = 3ω2 ∧ η.

For the Riemannian cones, we can prove the following

Corollary 4.5. Let N 5 be a 5-dimensional manifold endowed with an SU(2)- structure (η, ω1, ω2, ω3). The SU(3)-structure (F, 9) on the Riemannian cone (N 5× ޒ+, h) is SKT if and only if

−4η ∧ ω1 + 2I (dω1) − 2dη ∧ η = d(I (iξ dω1)),

(24) dω2 = 3ω3 ∧ η,

dω3 = −3ω2 ∧ η.

Proof. By imposing that d9 = 0, we get the conditions dω2 = −3 ω3 ∧ η and dω3 = 3 ω2 ∧η. By imposing d(J dF) = 0, we get, as in the proof of Theorem 3.1, equation (19) for ω = ω1.

5. Almost contact metric structure induced on a hypersurface

We study the almost contact metric structure induced naturally on any oriented hypersurface N 2n+1 of a (2n+2)-manifold M2n+2 equipped with an SKT structure. Let f : N 2n+1 → M2n+2 be an oriented hypersurface of a (2n+2)-dimensional manifold M2n+2 endowed with an SKT structure (J, h, F), and denote by ޕ the unitary normal vector ﬁeld. It is well known that N 2n+1 inherits an almost contact metric structure (I, ξ, η, g) such that η and the fundamental 2-form ω are given by

∗ ∗ (25) η = − f (iޕ F) and ω = f F, where F is the fundamental 2-form of the almost Hermitian structure; see, for instance, [Blair 2002].

Proposition 5.1. Suppose f : N 2n+1 → M2n+2 is an immersion of an oriented (2n+1)-dimensional manifold into a (2n+2)-dimensional Hermitian manifold. If the Hermitian structure (J, h) on M2n+2 is SKT, then the induced almost contact metric structure (I, ξ, η, g) on N 2n+1, with η and ω given by (25), satisﬁes

∗ (26) d(I dω − I ( f (iޕ d F)) ∧ η) = 0.

Proof. Locally we can choose an adapted coframe {e1,..., e2n+2} for the Hermit- ian structure so that the unitary normal vector ﬁeld ޕ is dual to e2n+2. Since the 64 MARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA almost complex structure J is given in this adapted basis by Je2 j−1 = −e2 j , Je2 j = e2 j−1 for j= 1,..., n, Je2n+1 = e2n+2, Je2n+2 = −e2n+1, it follows that the tensor ﬁeld I on N 2n+1 has I f ∗ei = f ∗ Jei for i = 1,..., 2n+1. That is,

I f ∗e2 j−1 = − f ∗e2 j , I f ∗e2 j = f ∗e2 j−1 for j = 1,..., n, I f ∗e2n+1 = 0.

However, I f ∗e2n+2 = 0 6= f ∗e2n+1 = − f ∗ Je2n+2. We compute f ∗ J dF. First, we decompose (locally and in terms of the adapted basis) the differential of F as

dF = α + β ∧ e2n+1 + γ ∧ e2n+2 + µ ∧ e2n+1 ∧ e2n+2, where the local forms

α ∈ V3he1,..., e2ni, β, γ ∈ V2he1,..., e2ni, µ ∈ V1he1,..., e2ni are generated only by e1,..., e2n. Then,

J dF = Jα + Jβ ∧ e2n+2 − Jγ ∧ e2n+1 + Jµ ∧ e2n+1 ∧ e2n+2.

Since f ∗e2n+2 = 0 and f ∗e2n+1 = η, we have f ∗ J dF = f ∗ Jα − ( f ∗ Jγ ) ∧ η. ∗ ∗ ∗ However, f (iޕ dF) = f γ + f µ ∧ η, which implies that ∗ ∗ ∗ I ( f (iޕ dF)) = I f γ = f Jγ. On the other hand, I dω = I d f ∗ F = I f ∗dF = I f ∗α = f ∗ Jα. We conclude that ∗ ∗ ∗ ∗ f J dF = f Jα − ( f Jγ ) ∧ η = I dω − I ( f (iޕ d F)) ∧ η. Now, if the Hermitian structure is SKT, then JdF is closed and the induced structure satisﬁes (26).

Remark 5.2. Notice that, using iޕ d F = ᏸޕ F − diޕ F, we can write (26) as ∗ d(I dω − I ( f (ᏸޕ F) + dη) ∧ η) = 0. ∗ Therefore, if f (ᏸޕ F) = 0, then the induced almost contact metric structure has to satisfy the equation d(I dω− I (dη)∧η) = 0. In the case of the product N 2n+1×ޒ, ∗ the condition f (ᏸޕ F) = 0 is satisﬁed. In the case of the Riemannian cone, we = + ∧ ∗ = have ᏸd/dt F 2tω dt η and therefore f (ᏸd/dt F) 2ω. In this way, we recover some of the equations obtained in Corollary 2.6 and Theorem 3.1. Now we study the structure that is induced naturally on any oriented hypersurface N 5 of a 6-manifold M6 equipped with an SKT SU(3)-structure. STRONGKÄHLERWITHTORSIONSTRUCTURES 65

Let f : N 5 → M6 be an oriented hypersurface of a 6-manifold M6 endowed with an SU(3)-structure (F, 9 = 9+ +i 9−), and denote by ޕ the unitary normal 5 vector ﬁeld. Then N inherits an SU(2)-structure (η, ω1, ω2, ω3) given by ∗ ∗ ∗ ∗ (27) η = − f (iޕ F), ω1 = f F, ω2 = − f (iޕ9−), ω3 = f (iޕ9+). Corollary 5.3. Let f : N 5 → M6 be an immersion of an oriented 5-dimensional manifold into a 6-dimensional manifold with an SU(3)-structure. If the SU(3)- structure is SKT, then the induced SU(2)-structure on N 5 given by (27) satisﬁes ∗ (28) d(I dω1 − I f (iޕ d F) ∧ η) = 0,

(29) d(ω2 ∧ η) = 0 and d(ω3 ∧ η) = 0.

Proof. Equation (28) follows from Proposition 5.1 by taking ω = ω1. Locally, we can choose an adapted coframe {e1,..., e5, e6} for the SU(3)-structure such that the unitary normal vector field ޕ is dual to e6. From (27), it follows that ∗ ∗ ω2 ∧η = f 9+ and ω3 ∧η = f 9−. If 9 = 9+ +i 9− is closed, then the induced structure satisfies (29). 5.1. A simple example. Consider the 6-dimensional nilmanifold M6 whose underlying nilpotent Lie algebra has structure equations de j = 0 for j = 1, 2, 3, 6, de4= e12, de5 = e14, and is endowed with the SU(3)-structure given by F = −e14 − e26 − e53 and 9 = (e1 − ie4) ∧ (e2 − ie6) ∧ (e5 − ie3). The oriented hypersurface with normal vector field dual to e2 is a 5-dimensional nilmanifold N 5 that by [Conti and Salamon 2007] has no invariant hypo structures. However, the SU(2)-structure on N 5, namely,

2 14 53 15 34 13 45 (30) η = e , ω1 = −e − e , ω2 = −e − e , ω3 = −e − e , satisﬁes (28) and (29). In Section 6, we will show that by using this SU(2)-structure and appropriate evolution equations, we can construct an SKT SU(3)-structure on the product of N 5 with an open interval.

6. SKT evolution equations

The goal here is to construct SKT SU(3)-structures by using appropriate evolution equations, starting from a suitable SU(2)-structure on a 5-dimensional manifold. We follow ideas of [Hitchin 2001] and [Conti and Salamon 2007].

Lemma 6.1. Let (η(t), ω1(t), ω2(t), ω3(t)) be a family of SU(2)-structures on a 5-dimensional manifold N 5 for t ∈ (a, b). The SU(3)-structure on M6 = N 5×(a, b) 66 MARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA

given by F = ω1(t)+η(t)∧dt and 9 = (ω2(t)+iω3(t))∧ η(t)−i dt satisﬁes the condition d9 = 0 if and only if (η(t), ω1(t), ω2(t), ω3(t)) is an SU(2)-structure such that, for any t in the open interval (a, b), ˆ ˆ d(ω2(t) ∧ η(t)) = 0, ∂t (ω2(t) ∧ η(t)) = −dω3(t), (31) ˆ ˆ d(ω3(t) ∧ η(t)) = 0, ∂t (ω3(t) ∧ η(t)) = dω2(t). Here, dˆ denotes the exterior differential on N 5 and d is the exterior differential on M6. We now present the additional evolution equations to be added to the last two of (31) in order to ensure that d Jd F = 0.

Proposition 6.2. Let (η(t), ω1(t), ω2(t), ω3(t)) be a family of SU(2)-structures on N 5 for t ∈ (a, b). The SU(3)-structure on M6 = N 5× (a, b) given by

(32) F = ω1(t) + η(t) ∧ dt and 9 = (ω2(t) + iω3(t)) ∧ (η(t) − i dt), has J dF closed if and only if (η(t), ω1(t), ω2(t), ω3(t)) satisfies the evolution equations ˆ ˆ ˆ d It dω1(t) − It (∂t ω1(t) + dη(t)) ∧ η(t) = 0, ˆ ˆ (33) ∂t It dω1(t) − It (∂t ω1(t) + dη(t)) ∧ η(t) ˆ ˆ ˆ = −d It (iξ dω1(t)) − It (iξ (∂t ω1(t) + dη(t))) ∧ η(t) , where ξ(t) denotes the vector field on N 5 dual to η(t) for each t ∈ (a, b). ˆ ˆ Proof. Since F = ω1(t) + η(t) ∧ dt, we have dF = dω1 + (∂t ω1 + dη) ∧ dt. Define {e1(t), . . . , e4(t), η(t)} to be a local adapted basis for the SU(2)-structure 1 4 (η(t), ω1(t), ω2(t), ω3(t)). Then, {e (t), . . . , e (t), η(t), dt} is an adapted basis for the SU(3)-structure (32), and J is given by Je1(t) = −e2(t), Je2(t) = e1(t), Jη(t) = dt, Je3(t) = −e4(t), Je4(t) = e3(t), J dt = −η(t).

5 For each t, the structures It induced on N are given by 1 2 2 1 It e (t) = −e (t), It e (t) = e (t), 3 4 4 3 It e (t) = −e (t), It e (t) = e (t), It η(t) = 0. Now, we can locally decompose a given τ(t) ∈ k(N 5) for t ∈ (a, b) as τ(t) = α(t) + β(t) ∧ η(t), where α(t) ∈ Vkhe1(t), . . . , e4(t)i and β(t) ∈ Vk−1he1(t), . . . , e4(t)i. Therefore,

Jτ(t) = Jα(t) + Jβ(t) ∧ Jη(t) = It α(t) + It β(t) ∧ dt k = It τ(t) − (−1) It (iξ(t)τ(t)) ∧ dt. STRONGKÄHLERWITHTORSIONSTRUCTURES 67

Applying this to J dF, we get ˆ ˆ J dF = Jdω1 − J(∂t ω1 + dη) ∧ η(t) ˆ ˆ ˆ = It dω1 − It (∂t ω1 + dη) ∧ η(t) + It (iξ(t)dω1) ∧ dt ˆ − It (iξ (∂t ω1 + dη)) ∧ η(t) ∧ dt. Finally, taking the differential of J dF, we get ˆ ˆ ˆ d J dF = d It dω1 − It (∂t ω1 + dη) ∧ η(t) ˆ ˆ + ∂t It dω1 − It (∂t ω1 + dη) ∧ η(t) ∧ dt ˆ ˆ ˆ + d It (iξ(t)dω1) − It (iξ (∂t ω1 + dη)) ∧ η(t) ∧ dt. Remark 6.3. Observe that the ﬁrst equation in (33) is exactly condition (28) for F = ω1(t) + η(t) ∧ dt. See Remark 5.2. As a consequence of Lemma 6.1 and Proposition 6.2, we get the following:

Theorem 6.4. For t ∈ (a, b), let (η(t), ω1(t), ω2(t), ω3(t)) be a family of SU(2)- structures on a 5-dimensional manifold N 5 such that ˆ ˆ (34) d(ω2(t) ∧ η(t)) = 0 and d(ω3(t) ∧ η(t)) = 0 for any t. If the evolution equations ˆ ˆ ˆ d(It dω1(t) − It (∂t ω1(t) + dη(t)) ∧ η(t)) = 0, ˆ ˆ ∂t (It dω1(t) − It (∂t ω1(t) + dη(t)) ∧ η(t)) ˆ ˆ ˆ (35) = −d(It (iξ dω1(t)) − It (iξ (∂t ω1(t) + dη(t))) ∧ η(t)), ˆ ∂t (ω2(t) ∧ η(t)) = −dω3(t), ˆ ∂t (ω3(t) ∧ η(t)) = dω2(t), are satisﬁed, then the SU(3)-structure on M = N × (a, b) given by

(36) F = ω1(t) + η(t) ∧ dt and 9 = (ω2(t) + iω3(t)) ∧ (η(t) − idt) is SKT. Example 6.5. Consider the Lie algebra with structure equations

de j = 0 for = 1, 2, 3, de4 = e12, de5 = e14, which underlies the 5-dimensional nilmanifold N 5 considered in Section 5.1. We endow it with the SU(2)-structure given by (30). It is easy to verify that

d(ω2 ∧ η) = d(ω3 ∧ η) = d(ω1 ∧ ω1) = 0. 68 MARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA

We evolve this SU(2)-structure by = − 14 − 53 = − + 3 1/3 15 − + 3 −1/3 34 ω1(t) e e , ω2(t) (1 2 t) e (1 2 t) e , = + 3 1/3 2 = − + 3 1/3 13 − + 3 −1/3 45 η(t) (1 2 t) e , ω3(t) (1 2 t) e (1 2 t) e , where t ∈ (−2/3, ∞). For any t ∈ (−2/3, ∞), the family (ω1(t), ω2(t), ω3(t), η(t)) satisfies equations (34) and the last two equations of (35). Moreover, it satisfies the conditions ˆ ˆ ˆ ∂t ω1(t) = 0, d(η(t)) = 0, iξ (d(ω1(t))) = 0, ∂t (It (dω1(t))) = 0, which implies that the evolution equations (33) are also satisfied. On the product N 5× ޒ, we consider the local basis of 1-forms 1 = + 3 1/3 1 2 = + 3 −1/3 4 3 = 5 β (1 2 t) e , β (1 2 t) e , β e , 4 = 3 5 = + 3 1/3 2 6 = β e , β (1 2 t) e , β dt. The structure equations are 1 = − 1 + 3 −1 16 4 = dβ 2 (1 2 t) β , dβ 0, 2 = + 3 −1 15 + 1 26 5 = − 1 + 3 −1 56 dβ (1 2 t) (β 2 β ), dβ 2 (1 2 t) β , dβ3 = β12, dβ6 = 0.

Locally, J is given by Jβ1 = −β2, Jβ3 = −β4 and Jβ5 = β6. The fundamental form F = −β12 − β34 + β56 satisﬁes d(J dF) = 0, and the (3, 0)-form 9 = (β1 + iβ2) ∧ (β3 + iβ4) ∧ (β5 − iβ6) is closed. Therefore, (F, 9) is a local SKT SU(3)-structure on N 5× ޒ. Remark 6.6. A Hermitian structure (J, h) on a 6-dimensional manifold M6 is called balanced if F∧F is closed, F being the associated fundamental 2-form. The paper [Fernandez´ et al. 2009] introduced the notion of balanced SU(2)-structures on 5-dimensional manifolds, together with appropriate evolution equations whose solution gives rise to a balanced SU(3)-structure in six dimensions. If M6 is compact, then a balanced structure cannot be SKT; see, for instance, [Fino et al. 2004]. The SU(2)-structure (30) from the previous example is also balanced, and it gives rise to a balanced metric on the product of N 5 with a open interval; see [Fernandez´ et al. 2009, (11)]. However, one can check directly that this solution is not SKT. If G is the nilpotent Lie group underlying N 5, the product G × ޒ has no left- invariant SKT structures and does not admit any left-invariant complex structures; however, we can ﬁnd a local SKT SU(3)-structure on it. STRONGKÄHLERWITHTORSIONSTRUCTURES 69

7. HKT structures

We will now find conditions under which an S1-bundle over a (4n+3)-dimensional manifold endowed with three almost contact metric structures is hyper-Kahler¨ with torsion (HKT, for short). Recall that a 4n-dimensional hyper-Hermitian manifold 4n 4n (M , J1, J2, J3, h) is a hypercomplex manifold (M , J1, J2, J3) endowed with a Riemannian metric h compatible with the complex structures Jr for r = 1, 2, 3; that is, h satisfies h(Jr X, Jr Y ) = h(X, Y ) for any r = 1, 2, 3 and any vector fields X and Y on M4n. 4n A hyper-Hermitian manifold (M , J1, J2, J3, h) is called HKT if and only if

(37) J1 dF1 = J2 dF2 = J3 dF3, where Fr denotes the fundamental 2-form associated to the Hermitian structure (Jr , h); see [Grantcharov and Poon 2000]. We consider a (4n+3)-dimensional manifold N 4n+3 endowed with three almost contact metric structures (Ir , ξr , ηr , g) for r = 1, 2, 3, and satisfying

Ik = Ii I j − η j ⊗ ξi = −I j Ii + ηi ⊗ ξ j , (38) ξk = Ii ξ j = −I j ξi , ηk = ηi I j = −η j Ii . By applying Theorem 2.3, we can construct hyper-Hermitian structures on S1- bundles over N 4n+3 and study when they are strong HKT. Theorem 7.1. Let N 4n+3 be a (4n+3)-dimensional manifold with three normal almost contact metric structures (Ir , ξr , ηr , g) for r = 1, 2, 3, and satisfying (38). Let be a closed 2-form on N 4n+3 that represents an integral cohomology class, 1 and that is Ir -invariant for every r = 1, 2, 3. Consider the circle bundle S ,→ P → N 4n+3 with a connection 1-form θ whose curvature form is dθ = π ∗(), where π : P → N is the projection. The hyper-Hermitian structure (J1, J2, J3, h) on P deﬁned by (2) and (4) is HKT if and only if ∗ ∗ ∗ ∗ ∗ ∗ π (I1(dω1)) − π (dη1) ∧ π η1 = π (I2(dω2)) − π (dη2) ∧ π η2 ∗ ∗ ∗ (39) = π (I3(dω3)) − π (dη3) ∧ π η3, ∗ = ∗ = ∗ π (I1(iξ1 dω1)) π (I2(iξ2 dω2)) π (I3(iξ3 dω3)), where ωr is the fundamental form of the almost contact structure (Ir , ξr , ηr , g). Moreover, the HKT structure is strong if and only if ∗ d(π (Ir (iξ dωr ))) = 0, (40) r ∗ − ∧ = ∗ − + ∗ ∧ ∗ d(π (Ir (dωr ) dηr ηr )) (π ( Ir (iξr dωr )) π ) π for every r = 1, 2, 3. 70 MARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA

Proof. The almost hyper-Hermitian structure (J1, J2, J3, h) on P deﬁned by (2) and (4) is hyper-Hermitian if and only (Ir , ξr , ηr , g) is normal and dθ is Jr -invariant for every r = 1, 2, 3. The HKT condition is equivalent to (37). By (9), we have

= ∗ + ∗ ∧ − ∗ ∧ ∗ − ∧ Jr d Fr π (Ir (dωr )) π (Ir (iξr dωr )) θ π (dηr ) π ηr θ dθ, where Fr is the fundamental 2-form of (Jr , h). Therefore, condition (37) is satisﬁed if and only if (39) holds. Finally, the Jr dFr are closed if and only if (40) holds. 4n+3 On N × ޒ, consider the almost Hermitian structures (Jr , Fr , h) deﬁned by

2 h = g + (dt) , Fr = ωr + ηr ∧ dt, (41) Jr (ηr ) = dt, Jr (X) = Ir (X) for X ∈ Ker ηr . By (38), we have

J1 J2 = J3 = −J2 J1,

J1η2 = I1η2 = −η3, J2η3 = I2η3 = −η1, J3η1 = I3η1 = −η2.

4n+3 Therefore, (Jr , Fr , h) for r = 1, 2, 3 is a hyper-Hermitian structure on N × ޒ if and only if the structures (Ir , ξr , ηr , g) are normal. Corollary 7.2. Let N 4n+3 be a (4n +3)-dimensional manifold endowed with three normal almost contact metric structures (Ir , ξr , ηr , g) for r = 1, 2, 3. On the 4n+3 product N × ޒ, consider the hyper-Hermitian structure (J1, J2, J3, h) deﬁned by (41). Then, (J1, J2, J3, h) is HKT if and only if

I1(dω1) − dη1 ∧ η1 = I2(dω2) − dη2 ∧ η2 = I3(dω3) − dη3 ∧ η3, = = I1(iξ1 dω1) I2(iξ2 dω2) I3(iξ3 dω3). The HKT structure is strong if and only if

= − ∧ = = d(Ir (iξr dωr )) 0 and d(Ir (dωr ) dηr ηr ) 0 for every r 1, 2, 3.

Moreover, if (J1, J2, J3, h) is such that dη1 ∧ η1 = dη2 ∧ η2 = dη3 ∧ η3 and one of the conditions

(a) dωr = 0 for any r = 1, 2, 3, that is, (Ir , ξr , ηr , g) is quasi-Sasakian for any r = 1, 2, 3; or

(b) dωi ∧ η j ∧ ηk 6= 0, where (i, j, k) is a permutation of (1, 2, 3), as well as = = = = I1(dω1) I2(dω2) I3(dω3) and I1(iξ1 dω1) I2(iξ2 dω2) I3(iξ3 dω3), is satisﬁed, then (J1, J2, J3, h) is HKT. In case (a), the HKT structure is strong. In = = case (b), the HKT structure is strong if and only if d(I1(dω1)) d(I1(iξ1 dω1)) 0. STRONGKÄHLERWITHTORSIONSTRUCTURES 71

Proof. By Theorem 7.1, the hyper-Hermitian structure (Jr , Fr , h) for r = 1, 2, 3 is HKT if and only if

I1(dω1) − dη1 ∧ η1 = I2(dω2) − dη2 ∧ η2 = I3(dω3) − dη3 ∧ η3, (42) = = I1(iξ1 dω1) I2(iξ2 dω2) I3(iξ3 dω3).

Locally, we write

3 3 X r X r (43) dωr = αr + βi ∧ ηi + γi j ∧ ηi ∧ η j + ρr η1 ∧ η2 ∧ η3, i=1 i< j=1

r r where ρr are smooth functions, while αr , βi , and γi j are respectively 3-forms, T3 2-forms, and 1-forms in i=1 Ker ηi . By ﬁrst using the normality of the three almost contact metric structures, and = = then that iξr dηr 0 and Ir (dηr ) dηr , locally we can write

dη1 = A1 + B1 ∧ η2 − I1 B1 ∧ η3 + C1η2 ∧ η3,

(44) dη2 = A2 + B2 ∧ η1 + I2 B2 ∧ η3 + C2η1 ∧ η3,

dη3 = A3 + B3 ∧ η1 − I3 B3 ∧ η2 + C3η1 ∧ η2, where Ir Ar = Ar . Here, the Ar and Br are respectively 2-forms and 1-forms in T3 i=1 Ker ηi , while the Cr are smooth functions. We have

Jr (dFr ) = Jr (dωr ) + Jr (dηr ∧ dt) = Jr (dωr ) − dηr ∧ ηr .

Therefore, by using (43) and (44), we obtain

1 1 1 J1(dF1) = I1α1 + I1β1 ∧ dt − A1 ∧ η1 − I1β3 ∧ η2 − I1β2 ∧ η3 1 1 − I1γ13 ∧ η2 ∧ dt + I1γ12 ∧ η3 ∧ dt + B1 ∧ η1 ∧ η2 − I1 B1 ∧ η1 ∧ η3 1 + I1γ23 ∧ η2 ∧ η3 + ρ1η2 ∧ η3 ∧ dt − C1η1 ∧ η2 ∧ η3, 2 2 2 J2(dF2) = I2α2 + I2β2 ∧ dt − I2β3 ∧ η1 − A2 ∧ η2 + I2β1 ∧ η3 2 2 2 + I2γ23 ∧ η1 ∧ dt + I2γ12 ∧ η3 ∧ dt − B2 ∧ η1 ∧ η2 + I2γ13 ∧ η1 ∧ η3

+ I2 B2 ∧ η2 ∧ η3 − ρ2 η1 ∧ η3 ∧ dt + C2η1 ∧ η2 ∧ η3, 3 3 3 J3(dF3) = I3α3 + I3β3 ∧ dt + I3β2 ∧ η1 − I3β1 ∧ η2 − A3 ∧ η3 3 3 3 + I3γ23 ∧ η1 ∧ dt − I3γ13 ∧ η2 ∧ dt + I3γ12 ∧ η1 ∧ η2 − B3 ∧ η1 ∧ η3

+ I3 B3 ∧ η2 ∧ η3 + ρ3η1 ∧ η2 ∧ dt − C3η1 ∧ η2 ∧ η3. 72 MARISA FERNÁNDEZ, ANNA FINO, LUIS UGARTE AND RAQUEL VILLACAMPA

The conditions (42) are satisﬁed if and only if

1 1 2 2 3 3 γ12 = γ13 = γ12 = γ23 = γ13 = γ23 = 0,

ρr = 0, C1 = −C2 = C3, 1 2 3 (45) I1α1 = I2α2 = I3α3, I1β1 = I2β2 = I3β3 , 2 3 1 3 1 2 A1 = I2β3 = −I3β2 , A2 = −I1β3 = I3β1 , A3 = I1β2 = −I2β1 , 3 2 1 B1 = −B2 = I3γ12, −I1 B1 = −B3 = I2γ13, I2 B2 = I3 B3 = I1γ23. r Since Ir Ar = Ar the coefficients βi for r 6= i = 1, 2, 3 must satisfy the conditions i i Ii (β j − Ikβ j ) = 0 for all i, j, k = 1, 2, 3 with i 6= j, j 6= k and k 6= i. 1 = 2 = 3 = The last three equations in (45) are satisfied if and only if γ23 γ13 γ12 0. Thus, finally, we obtain

P3 r dωr = αr + i=1 βi ∧ ηi , dηi = Ai + λ η j ∧ ηk, i i 0 = Ii (β − Ikβ ) for all i, j, k = 1, 2, 3 with i 6= j, j 6= k and k 6= i, (46) j j I1α1 = I2α2 = I3α3, 2 3 1 3 1 2 A1 = I2β3 = −I3β2 , A2 = −I1β3 = I3β1 , A3 = I1β2 = −I2β1 for any even permutation of (1, 2, 3). The expression for d(J1dF1) is = + ∧ − ∧ d(J1d F1) d(I1(dω1) I1(iξ 1dω1) dt) d((dη1) η1) = + ∧ − ∧ d(I1(dω1)) d(I1(iξ 1dω1)) dt dη1 dη1 = − ∧ + ∧ d(I1(dω1) dη1 η1) d(I1(iξ 1dω1)) dt, and thus the HKT structure is strong if and only if − ∧ = = d(I1(dω1) dη1 η1) 0 and d(I1(iξ 1dω1)) 0. i To prove the last part of the corollary it sufﬁces to consider coefﬁcients βr = 0 if r 6= i in expression (43). Example 7.3. Consider the 7-dimensional Lie group G = SU(2)n ޒ4, with structure equations 1 = − 1 25 − 1 36 − 1 47 5 = 67 de 2 e 2 e 2 e , de e , 2 = 1 15 + 1 37 − 1 46 6 = − 57 de 2 e 2 e 2 e , de e , 3 = 1 16 − 1 27 + 1 45 7 = 56 de 2 e 2 e 2 e , de e . 4 = 1 17 + 1 26 − 1 35 de 2 e 2 e 2 e , STRONGKÄHLERWITHTORSIONSTRUCTURES 73

By [Fino and Tomassini 2008], G admits a compact quotient M7 = 0\G by a uniform discrete subgroup 0, and is endowed with a weakly generalized G2-structure. By [Barberis and Fino 2008], M7 × S1 admits a strong HKT structure. We can 7 show that M has three normal almost contact metric structures (Ir , ξr , ηr , g) for r = 1, 2, 3 that are given by

1 2 3 4 5 6 7 I1e = e , I1e = e , I1e = e , η1 = e , 1 3 2 4 5 7 6 I2e = e , I2e = −e , I2e = −e , η2 = e , 1 4 2 3 6 7 5 I3e = e , I3e = e , I3e = e , η3 = e and that satisfy the conditions of Corollary 7.2(a).

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Received September 23, 2009.

MARISA FERNÁNDEZ UNIVERSIDADDEL PAÍS VASCO DEPARTAMENTO DE MATEMÁTICAS FACULTAD DE CIENCIAY TECNOLOGÍA APARTADO 644 48080 BILBAO SPAIN [email protected] STRONGKÄHLERWITHTORSIONSTRUCTURES 75

ANNA FINO UNIVERSITÀ DI TORINO DIPARTIMENTO DI MATEMATICA VIA CARLO ALBERTO 10 I-10123 TORINO ITALY annamaria.ﬁ[email protected]

LUIS UGARTE UNIVERSIDADDE ZARAGOZA DEPARTAMENTO DE MATEMÁTICAS-IUMA CAMPUS PLAZA SAN FRANCISCO 50009 ZARAGOZA SPAIN [email protected]

RAQUEL VILLACAMPA CENTRO UNIVERSITARIO DE LA DEFENSA ACADEMIA GENERAL MILITAR CRTA. DE HUESCAS/N 50090 ZARAGOZA SPAIN [email protected]

PACIFIC JOURNAL OF MATHEMATICS Vol. 249, No. 1, 2011

CONNECTIONS BETWEEN FLOER-TYPE INVARIANTS AND MORSE-TYPE INVARIANTS OF LEGENDRIAN KNOTS

MICHAEL B.HENRY

We define an algebraic/combinatorial object on the front projection 6 of a Legendrian knot, called a Morse complex sequence, abbreviated MCS. This object is motivated by the theory of generating families and provides new connections between generating families, normal rulings, and augmentations of the Chekanov–Eliashberg DGA. In particular, we place an equivalence relation on the set of MCSs on 6 and construct a surjective map from the equivalence classes to the set of chain homotopy classes of augmentations of L6, where L6 is the Ng resolution of 6. In the case of Legendrian knot classes admitting representatives with two-bridge front projections, this map is bijective. We also exhibit two standard forms for MCSs and give explicit algorithms for finding these forms. The definition of an MCS, the equivalence relation, and the statements of some of the results originate from unpublished work of Petya Pushkar.

1. Introduction

Legendrian knot theory is a rich refinement of smooth knot theory with deep connections to low-dimensional topology, symplectic and contact geometry, and singularity theory. In this article, we investigate connections between Legendrian knot invariants derived from symplectic field theory and from the theory of generating families. Specifically, we relate augmentations, derived from symplectic field theory, and Morse complex sequences, derived from generating families. A Legendrian knot K in ޒ3 is a smooth knot whose tangent space sits in the standard contact structure ξ on ޒ3, where ξ is the kernel of the 1-form dz−ydx. Legendrian knot theory is the study of Legendrian knots up to isotopy through Legendrian knots. We begin by recalling existing connections between Legendrian knot invariants. The Chekanov–Eliashberg differential graded algebra (abbreviated CE-DGA) of a Legendrian knot K is a differential graded algebra (Ꮽ(L), ∂) associated to the

MSC2000: primary 57M25, 57R17; secondary 53D40. Keywords: Legendrian knots, augmentations, normal rulings, generating families, contact homology.

77 78 MICHAEL B. HENRY xy-projection L of K . The CE-DGA is derived from the symplectic field theory of [Eliashberg 1998; Eliashberg et al. 2000] and is developed in the latter and [Chekanov 2002a]. The homology of (Ꮽ(L), ∂) is a Legendrian invariant and, if we consider (Ꮽ(L), ∂) up to a certain algebraic equivalence, the resulting DGA class is also a Legendrian invariant. Geometrically, the CE-DGA is Floer theoretic in nature. See [Chekanov 2002b; Etnyre et al. 2002] for a more detailed introduction. An augmentation is a type of algebra homomorphism from the CE-DGA of a Legendrian knot to the base field ޚ2. We denote the set of augmentations of (Ꮽ(L), ∂) by Aug(L). There is a natural algebraic equivalence relation on Aug(L) and we denote the set of equivalence classes of augmentations of L by Augch(L). The cardinality of Augch(L) is a Legendrian isotopy invariant. A second source of Legendrian invariants is the theory of generating families; see any of [Pushkar and Chekanov 2005; Jordan and Traynor 2006; Ng and Traynor 2004; Traynor 1997; Traynor 2001]. A generating family for a Legendrian knot K encodes the xz-projection of K as the Cerf diagram of a one-parameter family of functions Fx . In particular, let W be a smooth manifold and F : ޒ × W → ޒ be a smooth function. Let Ꮿx ⊂ {x}×W denote the set of critical points of Fx = F(x, · ) S and let ᏯF = x∈ޒ Ꮿx . If the rank of the matrix of second derivatives of F is maximal at all points in ᏯF , then 6F = {(x, F(x, w)) | (x, w) ∈ ᏯF } is the xz- projection of an immersed Legendrian submanifold in (ޒ3, ξ) and we say F is a generating family for this submanifold; see Figure 1. If we restrict our attention to generating families that are sufficiently nice outside a compact set of the domain, then it can be shown that the existence of a generating family for a Legendrian knot is a Legendrian isotopy invariant; see [Jordan and Traynor 2006]. Throughout this article, we will let 6 denote the xz-projection of a Legendrian knot and call it the front projection of K . Every Legendrian knot can be Legen- drian isotoped in an arbitrarily small neighborhood of itself so that the singularities of 6 are left cusps, right cusps, and transverse double points and so that the x- coordinates of these singularities are all distinct. We say a front is σ-generic if its singularities are arranged in this manner. From a generating family, we may derive a combinatorial object called a graded normal ruling. Suppose a Legendrian knot K with σ -generic front 6 admits a generating family F : ޒ × W → ޒ. The generating family may be chosen so that Fx is a Morse function whose critical points have distinct critical values for all but finitely many values of x. By placing an appropriate metric g on ޒ × W, we may construct the Morse–Smale chain complex (Cx , ∂x , gx ) of the pair (Fx , gx ).A graded normal ruling is a combinatorial object on 6 that encodes a certain pairing of the generators of (Cx , ∂x , gx ) as x varies. Chekanov and Pushkar [2005] work with a more general object called a pseudoinvolution; their Section 12 explains in detail the connection between generating families and pseudoinvolutions, including CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 79 z 6

x Fx

Figure 1. A generating family for a Legendrian unknot. the restrictions placed on the types of generating families considered and on the metric g. Many connections exist between augmentations, generating families, and graded normal rulings; see any of [Pushkar and Chekanov 2005; Fuchs 2003; Fuchs and Ishkhanov 2004; Fuchs and Rutherford 2008; Kalm´ an´ 2006; Ng and Sabloff 2006; Sabloff 2005]. For a fixed Legendrian knot K with Lagrangian projection L and front projection 6, the following results are known. Theorem 1.1. The CE-DGA (Ꮽ(L), ∂) admits a graded augmentation if and only if 6 admits a graded normal ruling. In particular, there exists a many-to-one map from the set of graded augmentations of (Ꮽ(L), ∂) to the set of graded normal rulings of 6. The reverse implication of the first statement was proved by Fuchs [2003]. Fuchs and Ishkhanov [2004], and independently Sabloff [2005], proved the forward implication. Ng and Sabloff [2006] proved the second statement. Theorem 1.2. A Legendrian knot K admits a generating family if and only if 6 admits a graded normal ruling. Chekanov and Pushkar [2005] proved the forward direction and state the converse without proof. Fuchs and Rutherford [2008] then proved the converse. By encoding the Morse theory data inherent in a generating family, we hope to refine the many-to-one map in Theorem 1.1. We encode this data in a finite sequence of chain complexes. The resulting algebraic object is called a Morse complex sequence. Geometrically, it should be thought of as a sequence from the 1-parameter family of chain complexes (Cx , ∂x , gx ) from a generating family. In this article we will not work explicitly with generating families, though they provide important geometric intuition. We let MCS(6) denote the set of MCSs of 6 and let MCS[(6) denote the set of MCSs of 6 up to a natural equivalence. 80 MICHAEL B. HENRY

(a) (b) (c) Figure 2. Cusps and crossings in the Ng resolution procedure.

In 1999, Petya Pushkar began a program to combinatorialize the Morse theory data coming from a generating family. The work with pseudoinvolutions in [Pushkar and Chekanov 2005] may be considered the ﬁrst step in this program. In emails to D. Fuchs in 2001 and 2008, Pushkar outlines his “Spring Morse theory,” which encodes the sequence (Cx , ∂x , gx ) coming from a generating family and provides an equivalence relation on the resulting objects. The equivalence relation is the result of understanding the evolution of one-parameter families of functions and metrics. The ideas behind Morse complex sequences and the equivalence relation we deﬁne in this article originate with Petya Pushkar.

1a. Results. Given a Legendrian knot K with σ-generic front projection 6, we form the Ng L6 resolution of 6 by resolving the cusps and crossings as indicated in Figure 2. The CE-DGA of L6 is equal to the CE-DGA of an xy-projection for the Legendrian knot class of K . Therefore we may use L6 to compare objects defined on 6 with objects derived from the CE-DGA of K . Given 6 and L6, we have the following results. Theorem 1.3. For a fixed Legendrian knot K with σ-generic front projection 6 and Ng resolution L6, there exists a surjective map ch 9b : MCS[(6) → Aug (L6). Theorem 1.4. If 6 has exactly two left cusps, then the map 9b above is a bijection. In his 2008 email, Pushkar announced, without proof, results very similar to Theorem 1.3. In the case of a front projection with exactly two left cusps, we can explicitly ch calculate |Aug (L6)| = |MCS[(6)|. The language in this corollary is defined in Section 7. Corollary 1.5. Suppose 6 has exactly two left cusps, and let N(6) denote the set of graded normal rulings on 6. For each N ∈ N(6), define ν(N) to be the number of graded departure-return pairs in N. Then

ch X ν(N) |Aug (L6)| = |MCS[(6)| = 2 . N∈N(6) In Section 6e, we describe two standard forms for MCSs on 6. The algorithms used to ﬁnd these forms allow us to understand 9b, while avoiding the involved CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 81 algebraic arguments required to prove Theorem 1.3. We will use the SR-form to calculate bounds on the number of MCS classes associated to a ﬁxed graded normal ruling. The A-form of an MCS Ꮿ allows us to easily compute the augmentation class 9(b [Ꮿ]).

Theorem 1.6. Every MCS is equivalent to an MCS in SR-form and an MCS in A-form.

1b. Outline of the rest of the article. In Section 2, we provide the necessary background material in Legendrian knot theory. The front and Lagrangian projections of a Legendrian knot are used to develop combinatorial descriptions of the Chekanov– Eliashberg DGA, augmentations, and graded normal rulings. The deﬁnition of a Morse complex sequence (MCS) is given in Section 3, along with an equivalence relation on MCSs. Section 4 reviews properties of differential graded algebras, DGA morphisms and DGA chain homotopies and applies them to the case of the CE-DGA and augmentations. We also sketch the proof that Augch(L) is a Legen- drian knot invariant. In Section 5 we use a variation of the splash construction ﬁrst developed in [Fuchs 2003] to write down the boundary map of the CE-DGA of a “dipped” version of L6 as a system of local matrix equations. This gives us local control over augmentations and chain homotopies of augmentations. A number of lemmas are proved involving extending augmentations to dipped diagrams. In Section 6 we develop the connections between MCSs and augmentations. We use the lemmas on dipped diagrams from Section 5 and the lemmas concerning chain homotopies from Section 4 to explicitly construct 9b and prove Theorem 1.3. In Section 6e, we describe two standard forms for MCSs on 6 and prove Theorem 1.6 using explicit algorithms. In Section 7 we prove Theorem 1.4 and Corollary 1.5.

2. Background

We assume the reader is familiar with the basic concepts in Legendrian knot theory, including front and Lagrangian projections, and the classical invariants. Through- out this article, 6 and L denote the front and Lagrangian projections of a knot K , respectively. Legendrian knots with nonzero rotation number do not admit augmentations, generating families, and graded normal rulings. Thus we will always assume the rotation number of K is 0. An in-depth survey of Legendrian knot theory can be found in [Etnyre 2005].

2a1. The Ng resolution. In [2003], Ng algorithmically constructs a Legendrian isotopy of K such that the Lagrangian projection L0 of the resulting Legendrian knot K 0 is topologically similar to 6. Deﬁnition 2.1 gives a combinatorial description of the Lagrangian projection produced by Ng’s resolution algorithm. 82 MICHAEL B. HENRY

Definition 2.1. Given a Legendrian knot K with front projection 6, we form the Ng resolution L6 by smoothing the left cusps as in Figure 2(a), smoothing and twisting the right cusps as in Figure 2(b), and resolving the double points as in Figure 2(c). 0 The projection L6 is regularly homotopic to L , and the CE-DGA of L6 is equal 0 to the CE-DGA of L . Given that 6 and L6 are combinatorially very similar, the Ng resolution algorithm provides a natural first step towards finding connections between generating families and the CE-DGA. 2b. The Chekanov–Eliashberg DGA. Chekanov [2002a] and Eliashberg [2000] develop a differential graded algebra, henceforth referred to as the CE-DGA, that has led to the discovery of several new Legendrian isotopy invariants. The definitions and statements in this section originated in [Chekanov 2002a]; see also [Etnyre et al. 2002].

2b1. The algebra. Label the crossings of a Lagrangian projection L by q1,..., qn. Let A(L) denote the ޚ2 vector space freely generated by the elements of Q = {q1,..., qn}. The algebra Ꮽ(L) is the unital tensor algebra TA(L). We consider Ꮽ(L) to be a based algebra since the algebra basis Q is part of the data of Ꮽ(L). An element of Ꮽ(L) looks like the sum of noncommutative words in the letters qi .

2b2. The grading. We define a ޚ-grading |qi | on the generators qi and extend it Ql = Pl | | to all monomials in Ꮽ(L) by requiring j=1 qi j j=1 qi j . We isotope K slightly so that the two strands meeting at each crossing of L are orthogonal. Let γi be a path in K that begins on the overstrand of L at qi and follows L until it first returns to qi . We let r(γi ) denote the fractional winding number of the tangent 2 space of γi with respect to the trivialization {∂x , ∂y} of the tangent space of ޒ . The grading of a crossing qi is defined to be |qi | = 2r(γi ) − 1/2. The grading is well-defined since we have assumed K has rotation number 0. If L6 is the Ng resolution of a front projection 6, then we can calculate the grading of the crossings of L6 from 6 using a Maslov potential. Definition 2.2. A strand in 6 is a smooth path in 6 from a left cusp to a right cusp. A Maslov potential on 6 is a map µ from the strands of 6 to ޚ satisfying the relation shown in Figure 3(a). Given a crossing q in 6, the grading of the corresponding resolved crossing q in L6 is computed by |q| = µ(T )−µ(B), where T and B are the strands crossing at q and T has smaller slope. The crossings created by resolving right cusps have grading 1. 2b3. The differential. The differential of the CE-DGA counts certain disks in the xy-plane with polygonal corners at the crossings of L. We begin by decorating each corner of qi with a + or − sign as in Figure 3(b). CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 83

i+1 i+1 − + + i i − (a) (b) Figure 3. Left: a Maslov potential near left and right cusps. Right: the Reeb sign of a crossing.

Figure 4. A convex immersed polygon contributing q3q2q1 to ∂q5. Crossings are labeled from left to right.

2 Deﬁnition 2.3. Let D be the unit disk in ޒ and let ᐄ = {x0,..., xn} be a set of distinct points along ∂ D in counterclockwise order. A convex immersed polygon is a continuous map f : D → ޒ2 such that (1) f is an orientation-preserving immersion on the interior of D; (2) the restriction of f to ∂ D \ ᐄ is an immersion whose image lies in L; and

(3) the image of each xi ∈ ᐄ under f is a crossing of L, and the image of a neighborhood of xi covers a convex corner at the crossing. Call a corner of an immersed polygon positive if it covers a + sign and negative otherwise. 3 Remark 2.4. Given a crossing qi , let γi be the path in ޒ that begins and ends on K , has constant x and y coordinates, and projects to the crossing qi . Such paths are called Reeb chords. We deﬁne a height function h : Q → ޒ+ on the crossings of L by deﬁning h(qi ) to be length of the path γi . In the Ng resolution algorithm, we may arrange the Legendrian isotopy so that the height function on L0 is strictly increasing as we move from left to right along the x-axis. Thus the height function provides an ordering on the crossings of L6 corresponding to the ordering coming from the x-axis.

Lemma 2.5. Let f be a convex immersed polygon and let γi be the Reeb chord that lies over the corner f (xi ). Then X X h(γi ) − h(γ j ) = Area( f (D)).

xi positive x j negative As a consequence, every convex immersed polygon has at least one positive corner and in the case of an immersed polygon with a single positive corner, the 84 MICHAEL B. HENRY height of the positive corner is greater than the height of each of the negative corners. The differential ∂qi of the generator qi is a mod 2 count of convex immersed polygons with a single positive corner at qi . ˜ ; Definition 2.6. Let 1(qi q j1 ,..., q jk ) be the set of convex immersed polygons with a positive corner at qi and negative corners at q j1 ,..., q jk . The negative corners are ordered by the counterclockwise order of the marked points along ∂ D. ; ˜ ; Let 1(qi q j1 ,..., q jk ) be 1(qi q j1 ,..., q jk ) modulo smooth reparameterization. Definition 2.7. The differential ∂ on the algebra Ꮽ(L) is defined on a generator qi ∈ Q by the formula = X ; ··· ∂qi #(1(qi q j1 ,..., q jk ))q j1 q jk , ; 1(qi q j1 ,...,q jk ) where #(1( ··· )) is the mod 2 count of the elements in 1( ··· ). We extend ∂ to all of Ꮽ(L) by linearity and the Leibniz rule.

The convex immersed polygon contributing the monomial q3q2q1 to ∂q5 is given in Figure 4. Given the differential and grading deﬁned above, (Ꮽ(L), ∂) is a differential graded algebra. Theorem 2.8 [Chekanov 2002a]. The differential ∂ satisﬁes (1) |∂q| = |q| − 1 modulo 2r(K ), and (2) ∂ ◦ ∂ = 0.

Remark 2.9. In the case of an Ng resolution L6, we noted in Remark 2.4 that the heights of the crossings increase as we move from left to right along the x- axis. Thus the negative corners of a convex immersed polygon contributing to ∂ in (Ꮽ(L6), ∂) always appear to the left of the positive corner. In Section 5, this fact will allow us to find all convex immersed polygons contributing to ∂. Chekanov [2002a] defines an algebraic equivalence on DGAs, called stable tame isomorphism, and proves that, up to this equivalence, the CE-DGA is a Legendrian isotopy invariant. In general, it is difficult to determine if two CE-DGAs are stable tame isomorphic. Definition 2.10. Given two algebras Ꮽ and Ꮽ0, a grading-preserving identifica- 0 tion of their generating sets Q and Q , and a generator q j for Ꮽ, an elementary isomorphism φ is an graded algebra map defined by 0 qi if i 6= j, φ(qi ) = 0 0 0 q j + u if i = j and u is a term in Ꮽ not containing q j . A composition of elementary isomorphisms is called a tame isomorphism.A tame isomorphism of DGAs between (Ꮽ, ∂) and (Ꮽ0, ∂0) is a tame isomorphisms 8 that is also a chain map, that is, 8 ◦ ∂ = ∂0 ◦ 8. CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 85

Definition 2.11. Given a DGA (Ꮽ, ∂) with generating set Q, we define the degree i stabilization Si (Ꮽ, ∂) to be the differential graded algebra generated by the set Q ∪ {e1, e2}, where |e1| = i and |e2| = i − 1 and with the differential extended by ∂e1 = e2 and ∂e2 = 0. Definition 2.12. Two DGAs (Ꮽ, ∂) and (Ꮽ0, ∂0) are stable tame isomorphic if there exist stabilizations S ,..., S and S0 ,..., S0 and a tame isomorphism i1 im j1 jn ψ : S (... S (Ꮽ) . . .) → S0 (... S0 (Ꮽ0)...) i1 im j1 jn of DGAs such that the composition of maps is a chain map. A stable tame isomorphism preserves the homology of the DGA and so the homology of (Ꮽ(L), ∂) is also a Legendrian isotopy invariant, which is called the Legendrian contact homology of K .

2c. Augmentations. Chekanov [2002a] implicitly deﬁnes a class of DGA chain maps called augmentations.

Deﬁnition 2.13. An augmentation is an algebra map : (Ꮽ(L), ∂) → ޚ2 with the properties that (1) = 1, that ◦∂ = 0, and that |qi | = 0 if (qi ) = 1. We let Aug(L) denote the set of augmentations of (Ꮽ(L), ∂). A tame isomorphism between DGAs induces a bijection on the corresponding sets of augmentations. Stabilizing a DGA may double the number of augmentations, depending on the grading of the new generators. It is possible to normalize the number of augmentations by an appropriate power of two and obtain an integer Legendrian isotopy invariant; see [Mishachev 2003; Ng and Sabloff 2006].

2d. Graded normal rulings. The geometric motivation for a graded normal ruling on 6 comes from examining the one-parameter family of Morse–Smale chain complexes that comes from a suitably generic generating family F for 6. Pushkar and Chekanov [2005, Section 12] provide a detailed explanation of the connection between generating families and graded normal rulings, including the restrictions placed on the types of generating families considered. In their language, a graded normal ruling is a positive Maslov pseudoinvolution. As combinatorial objects, rulings originated in the work of Chekanov [2002b] and Fuchs [2003]. Deﬁnition 2.14. A ruling on the front diagram 6 is a one-to-one correspondence between the set of left cusps and the set of right cusp and, for each corresponding pair of cusps, two paths in 6 that join them. The paths are such that (1) any two paths in the ruling meet only at crossings or at cusps; and (2) the two paths joining corresponding cusps meet only at the cusps, and hence their interiors are disjoint. 86 MICHAEL B. HENRY

(a) (b) Figure 5. Left: the three possible conﬁgurations of a normal switch. Right: a graded normal ruling on the standard Legendrian trefoil.

The two paths joining corresponding cusps are called companions of each other. Together the two paths bound a disk in the plane called the ruling disk. At a crossing, two paths either pass through each other or one path lies entirely above the other. In the latter case, we call the crossing a switch. Deﬁnition 2.15. We say a ruling is graded if each switched crossing has grading 0, where the grading comes from a Maslov potential as deﬁned in Section 2b2. A ruling is normal if at each switch the two paths at the crossing and their companion strands are arranged as in Figure 5(a). Let N(6) denote the set of graded normal rulings of a Legendrian knot with front projection 6. Figure 5(b) gives a graded normal ruling of the standard Legendrian trefoil. Theorem 2.16 [Pushkar and Chekanov 2005]. If K and K 0 are Legendrian isotopic and 6 and 60 are σ -generic, then 6 and 60 admit the same number of graded normal rulings. In a normal ruling, there are two types of unswitched crossings. A departure is an unswitched crossing in which, to the left of the crossing, the two ruling disks are either disjoint or one is nested inside the other. A return is an unswitched crossing in which the two ruling disks partially overlap to the left of the crossing.

3. Deﬁning Morse complex sequences

As indicated in the introduction, the ideas behind Morse complex sequences and the equivalence relation we define in this section originate with Petya Pushkar. Equivalence classes of Morse complex sequences correspond in the language of his 2008 email to “combinatorial generating families”. A Morse complex sequence is a finite sequence of chain complexes and a set of chain maps relating consecutive chain complexes. Its definition is geometrically motivated by the sequence of Morse–Smale chain complexes (Cx , ∂x , gx ) coming from a suitably generic generating family F and metric g. The local moves used to define an equivalence relation are found by considering two-parameter families of t t function/metric pairs (Fx , gx ). Hatcher and Wagoner [1973] describe possible re- 0 1 lationships between the sequences (Cx , ∂x , gx ) and (Cx , ∂x , gx ). In his 2001 email, CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 87

y6 y5 y4 y3

y1 y2 3 2 1 0 Figure 6. The graphical presentation of an ordered chain complex. The sloped lines from y4 to y3 and y2 indicate that ∂y4 = y2 + y3.

Pushkar identiﬁes the relationships that are necessary for working with Legendrian front projections.

3a. Ordered chain complexes. We begin by deﬁning the chain complexes that comprise a Morse complex sequence.

Deﬁnition 3.1. An ordered chain complex is a ޚ2 vector space C with ordered basis y1 < y2 < ··· < ym, a ޚ grading |y j | on y1,..., ym, and a linear map ∂ : C → C, such that (1) ∂ ◦ ∂ = 0,

(2) |∂y j | = |y j | − 1, and P (3) ∂y j = i< j a j,i yi , where a j,i ∈ ޚ2. We denote an ordered chain complex by (C, ∂) when the ordered basis and grading are understood. We let h∂y j | yi i denote the contribution of the generator yi to ∂y j , that is, h∂y j | yi i = a j,i .

Remark 3.2. The m × m lower triangular matrix Ᏸ deﬁned by (Ᏸ) j,i = a j,i for j > i is a matrix representative of the map ∂. Indeed, ∂2 = 0 implies Ᏸ2 = 0 and with respect to the basis {ym} the ޚ2 coefﬁcients of ∂y j are given by e j Ᏸ, where e j is the j-th standard basis row vector. In Section 6, we use the matrix representatives of a sequence of ordered chain complexes to associate an augmentation to an MCS. In Figure 6 we give an example of the graphical encoding of an ordered chain complex (C, ∂) used in [Barannikov 1994]. The vertical lines indicate the gradings of the generators y1,..., y6, the height of the vertices on the vertical lines indicates the ordering of the generators, and the sloped lines connecting vertices represent the boundary map ∂.

3a1. Matrix notation. All of the matrices in this article have entries in ޚ2 and matrix operations are done mod 2. For k > l, we let Ᏼk,l denote a square matrix with 1 in the (k, l) position and zeros everywhere else. We let Ek,l = I + Ᏼk,l , where I denotes the identity matrix. We let Pi+1,i denote the square permutation 88 MICHAEL B. HENRY matrix obtained by interchanging rows i and i + 1 of the identity matrix. Finally, we let Ji−1 denote the matrix obtained by inserting two columns of zeros after − T column i 1 in the identity matrix. The matrix Ji−1 is the transpose of Ji−1. 3b. Morse complex sequences. Consecutive ordered chain complexes in a Morse complex sequence are related by one of the following four chain maps. Definition 3.3. Suppose (C, ∂) and (C0, ∂0) are ordered chain complexes with or- ··· 0 ··· 0 dered generating sets y1 < < yn and y1 < < ym, respectively. We define four types of chain isomorphisms. 0 0 (1) Suppose n = m, 1 ≤ l < k ≤ n, and |yk| = |yl |. We say (C, ∂) and (C , ∂ ) are related by a handleslide between k and l if and only if the linear extension of the map on generators defined by 0 yi if i 6= k, φ1(yi ) = 0 0 yk + yl if i = k is a chain isomorphism from (C, ∂) to (C0, ∂0). As matrices, this means Ᏸ = 0 −1 Ek,l Ᏸ Ek,l . (2) Suppose n = m and 1 ≤ k < n. We say (C, ∂) and (C0, ∂0) are related by interchanging critical values at k if and only if the linear extension of the map on generators defined by y0 if i ∈/ {k, k + 1},  i = 0 = φ2(yi ) yk+1 if i k,  0 yk if i = k + 1 is a chain isomorphism from (C, ∂) to (C0, ∂0). As matrices, this means Ᏸ = 0 −1 Pk+1Ᏸ Pk+1. = − ≤ h 0 0 | 0 i = 0 0 (3) Suppose n m 2, 1 k < m and ∂ yk+1 yk 1. We say (C, ∂) and (C , ∂ ) are related by the birth of two generators at k if and only if (C, ∂) is chain isomorphic to the quotient of (C0, ∂0) by the acyclic subcomplex generated by { 0 0 0 } yk+1, ∂ yk+1 by the map [y0] if i < k, φ (y ) = i 3 i [ 0 ] + yi+2 if i > k 1. The matrix Ᏸ is computed explicitly as follows. Denote by y0 < y0 < y0 < ··· < y0 the generators of C0 satisfying k+1 u1 u2 us h∂0 y0 | y0 i = 1. Let y0 < ··· < y0 < y0 denote the generators of C0 such k vr v1 k h 0 0 | 0i = = that ∂ yk+1 y 1. Let E Ek,vr ... Ek,v1 Eu1,k+1 ... Eus ,k+1 I . Then, as = 0 −1 T matrices, Ᏸ Ji−1 EᏰ E Ji−1. CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 89

We say (C, ∂) and (C0, ∂0) are related by the death of two generators at k if the roles of (C, ∂) and (C0, ∂0) are exchanged in the map above. In particular, 0 0 n = m+2, h∂yk+1 | yki = 1 for some 1 ≤ k < n, and (C , ∂ ) is chain isomorphic to the quotient of (C, ∂) by the acyclic subcomplex generated by {yk+1, ∂yk+1} 0 0 by the map φ4 given by yi 7→ [yi ] if i < k and yi 7→ [yi+2] otherwise. In the matrix equation in (3), EᏰ0 E−1 represents a series of handleslide moves 0 0 on the chain complex (C , ∂ ). The generators yk+1 and yk form a trivial acyclic 0 −1 = 0 −1 T subcomplex in EᏰ E , and Ᏸ Ji−1 EᏰ E Ji−1 is the result of quotienting out this subcomplex. A Morse complex sequence encodes possible algebraic changes resulting from a generic deformation between two Morse functions without critical points. A generating family for a Legendrian knot is such a deformation. Deﬁnition 3.4. A Morse complex sequence Ꮿ is a ﬁnite sequence of ordered chain j ··· j complexes (C1, ∂1)...(Cm, ∂m) with ordered generating sets y1 < < yn j for each (C j , ∂ j ) for 1 ≤ j ≤ m, and τ j ∈ ޚ⊕ޚ for 1 ≤ j < m satisfying the following:

(1) For (C1, ∂1) and (Cm, ∂m), 1 1 m m n1 = nm = 2, h∂1 y2 | y1 i = 1, h∂m y2 | y1 i = 1.

In particular, both (C1, ∂1) and (Cm, ∂m) have trivial homology.

(2) |n j+1 − n j | ∈ {0, 2} for each 1 ≤ j < m.

(3) If n j = n j+1 − 2, then τ j = (k, 0) for some k, and (C j , ∂ j ) and (C j+1, ∂ j+1) are related by the birth of two generators at k.

(4) If n j = n j+1 + 2, then τ j = (k, 0) for some k, and (C j , ∂ j ) and (C j+1, ∂ j+1) are related by the death of two generators at k.

(5) If n j = n j+1, then either

• τ j = (k, 0) for some k and (C j , ∂ j ), and (C j+1, ∂ j+1) are related by interchanging critical values at k, or • τ j = (k, l) for some 1 ≤ l < k ≤ n j , and (C j , ∂ j ) and (C j+1, ∂ j+1) are related by a handleslide between k and l.

3c. Associating a marked front projection to an MCS. We may encode an MCS graphically using a front projection and certain vertical line segments. The vertical marks encode the handleslides occurring in the chain isomorphisms of type (1) and (3) in Deﬁnition 3.3. Deﬁnition 3.5. A handleslide mark on a σ-generic front projection 6 with Maslov potential µ is a vertical line segment in the xz-plane with endpoints on 6. We require that the line segment not intersect the crossings or cusps of 6 and the endpoints sit on strands of 6 with the same Maslov potential. A marked front 90 MICHAEL B. HENRY

x1 x2 x3 x4 x5 x6 x7

Figure 7. An example of an MCS and its associated marked front projection.

(a) (b) (c) (d)

Figure 8. The four possible tangles used in Section 3c. projection is a σ -generic front projection with a collection of handleslide marks; see Figure 7.

Let (C1, ∂1) ··· (Cm, ∂m) and τ1, . . . , τm−1 ∈ ޚ × ޚ be an MCS Ꮿ. We build the marked front projection for Ꮿ inductively beginning with a single left cusp and building to the right by adjoining tangles of the types in Figure 8. For each 1 ≤ j < m, we adjoin a tangle from Figure 8. At each step, we assume the strands of our tangle are numbered from top to bottom. If (C j , ∂ j ) and (C j+1, ∂ j+1) are related by a handleslide between k and l, then we adjoin a tangle of type (a) with a handleslide mark between strands k and l. If (C j , ∂ j ) and (C j+1, ∂ j+1) are related by interchanging critical values at k, then we adjoin a tangle of type (b) with a crossing between strands k and k + 1. If (C j , ∂ j ) and (C j+1, ∂ j+1) are related by the death of two generators at k, then we adjoin a tangle of type (c) where the right cusp connects strands k and k + 1. If (C j , ∂ j ) and (C j+1, ∂ j+1) are related by the birth of two generators at k, then we adjoin a tangle of type (d) where the left cusp sits above strand k − 1. We will refer to the marks arising from handleslides between consecutive chain complexes as explicit handleslide marks. At the cusps corresponding to births or deaths, we also place marks in a small neighborhood of the cusp. These marks CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 91 correspond to the composition of handleslides represented by the matrix product E in Definition 3.3(3). We call these implicit handleslide marks and use dotted vertical lines to distinguish them from the explicit handleslide marks. The front projection we have constructed is σ -generic and has a natural Maslov potential coming from the index of the generators in (C1, ∂1) ··· (Cm, ∂m). In addition, the handleslide marks connect strands of the same Maslov potential. Hence, we have constructed a marked front projection. Figure 7 gives an MCS and its associated marked front projection. Remark 3.6. (1) If a marked front projection is associated to Ꮿ, then by revers- ing the construction defined above, (C1, ∂1) ··· (Cm, ∂m) and τ1, . . . , τm−1 are uniquely reconstructed from the cusps, crossings and handleslide marks of the marked front projection using Definition 3.4(1) and the chain isomorphisms from Definition 3.3. Thus, a marked front projection may be associated with at most one MCS. h j | j i (2) By Definition 3.3, ∂ j yi+1 yi is equal to 1 if (C j , ∂ j ) and (C j+1, ∂ j+1) are related by the death of two generators at i, and equal to 0 if they are related by interchanging critical values at i.

(3) In an MCS Ꮿ, the homologies of (C j , ∂ j ) and (C j+1, ∂ j+1) are isomorphic. Thus, by Definition 3.4(1), all of the homologies in Ꮿ are trivial. Definition 3.7. Given a σ -generic front projection 6, an MCS Ꮿ is in the set MCS(6) if and only if the marked front projection associated to Ꮿ has a front projection that is planar isotopic to 6 by a planar isotopy through σ-generic front projections. Such an isotopy pulls back to a Legendrian isotopy in ޒ3. Pushkar (in the emails to Fuchs) defines a “spring sequence” to be a sequence of ordered chain complexes over a commutative ring ޅ with consecutive complexes connected by maps similar to those defined in Definition 3.3. Originally defined in [Pushkar and Chekanov 2005, Section 12], these chain complexes are called M- complexes. Pushkar encodes a spring sequence by adding vertical marks, called “springs”, to the front projection of an associated Legendrian knot.

3d. An equivalence relation on MCS(6). In this section, we describe a set of local moves used to deﬁne an equivalence relation on MCS(6). Since an MCS is completely determined by its associated marked front projection, these moves are deﬁned as graphical changes in the handleslide marks. The equivalence relation encodes local algebraic changes resulting from a generic deformation of a 1-parameter family of Morse functions. Such deformations are studied extensively in [Hatcher and Wagoner 1973]. Figures 9, 10, and 11 describe local changes in the handleslide marks of a marked front projection. We call these MCS moves. Other strands may appear 92 MICHAEL B. HENRY

1 2

3 4

5 6

7 8

9 10

Figure 9. MCS moves 1–10. We also allow reﬂections about the horizontal axis. in a local neighborhood of an MCS move, although we assume no other crossings or cusps appear. The ordering of implicit handleslide marks at a birth or death is irrelevant; thus we will not consider analogues of moves 1–6 for implicit handleslide marks. In moves 11–14, there may be other implicit marks that the indicated explicit handleslide mark commutes past without incident. Additional implicit marks may also appear at the birth or death in moves 15 and 16. MCS move 17 requires explanation. Let Ꮿ ∈ MCS(6) and suppose (C, ∂) is an ordered chain complex in Ꮿ with generators y1 < ··· < ym and a pair of generators | | = | | + ··· yl < yk such that yl yk 1. Let yu1 < yu2 < < yus denote the generators h | i = ··· of C satisfying ∂y yk 1; see the left three arrows in Figure 11. Let yvr < < yv1 < yi denote the generators of C appearing in ∂yl ; see the right two arrows in = ··· ··· Figure 11. Let E Ek,vr Ek,v1 Eu1,l Eus ,l . Then over ޚ2, we have (1) Ᏸ = EᏰE−1. The MCS move in Figure 11 says that we may either introduce or remove the handleslides represented by E and this move is local in the sense that it does not change any of the other chain complexes in Ꮿ. The next proposition shows that all of the MCS moves are local in this sense.

Proposition 3.8. Suppose Ꮿ = (C1, ∂1)...(Cm, ∂m) is an MCS on 6 and that in the interval [a, b] of the x-axis we modify the marked front projection of Ꮿ by one of CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 93

11 12

13 14

15 16

Figure 10. MCS moves 11–16. We also allow reﬂections about the horizontal and vertical axes.

yu3

yu2 yu 1 17 yk yl

yv1

yv2

Figure 11. MCS move 17. the MCS moves in Figures 9, 10, or 11. Then the resulting marked front projection Ꮿ0 determines an MCS. In addition, the ordered chain complexes of Ꮿ and Ꮿ0 agree outside of the interval [a, b]. Proof. As marked front projections, Ꮿ and Ꮿ0 agree outside of [a, b] and so by Remark 3.6(1) the chain complexes they determine agree to the left of [a, b]. The chain complexes to the right of [a, b] depend on the handleslide marks of each MCS and the chain complex to the immediate right of [a, b], so it sufﬁces to show that Ꮿ and Ꮿ0 determine the same chain complex to the immediate right of [a, b]. This follows from the matrix equations in Deﬁnition 3.3 and the following matrix equations:

Move 1: Ek,l Ek,l = I for k > l. = 6= Moves 2–5: Ek1,l1 Ek2,l2 Ek2,l2 Ek1,l1 for k1 > l1 and k2 > l2 with k1 l2 and k2 6= l1.

Move 6: Ea,b Eb,c = Eb,c Ea,c Ea,b for a > b > c.

Moves 7 and 10: Ek,l Pi+1,i = Pi+1,i Ek,l for k > l with k, l ∈/ {i + 1, i}.

Moves 8 and 9: Ek,i Pi+1,i = Pi+1,i Ek,i+1 for k > i + 1. 94 MICHAEL B. HENRY

Moves 15 and 16: Suppose the explicit handleslide mark occurs between strands k and i + 1. If E (respectively F) represents the sequence of implicit handleslide moves in the formula for the differential of the chain complex of Ꮿ 0 (respectively Ꮿ ) occurring after the right cusp, then F = EEk,i+1. In moves 11–13, the explicit handleslide mark does not involve the generators of the acyclic subcomplex that is quotiented out, so the chain complexes of Ꮿ and Ꮿ0 to the right of [a, b] are equal. Move 14 is a composition of moves 6, 13, and 15. Indeed, moves 15 and 6 allow us to commute the explicit mark past the implicit mark and then move 13 pushes the explicit mark past the cusp. Finally, the localness of move 17 was detailed in the discussion surrounding equation (1). Definition 3.9. We say Ꮿ and Ꮿ0 in MCS(6) are equivalent, denoted Ꮿ ∼ Ꮿ0, if 0 there exists a finite sequence Ꮿ = Ꮿ1, Ꮿ2,..., Ꮿn = Ꮿ ∈ MCS(6) where for each 1 ≤ i < n the marked front projections of Ꮿi and Ꮿi+1 are related by an MCS move. We let MCS[(6) denote the equivalence classes of MCS(6)∼, and we let [Ꮿ] denote an equivalence class in MCS[(6). 3e. Associating a normal ruling to an MCS. Definition 3.10 [Barannikov 1994]. An ordered chain complex (C, ∂) with generators y1,..., ym is in simple form if the following hold:

• For all i, either ∂yi = 0 or ∂yi = y j for some j.

• If ∂yi = ∂yk = y j , then k = i. Lemma 3.11 [Barannikov 1994, Lemma 2]. Let (C, ∂) be an ordered chain complex with generators y1,..., ym. Then after a series of handleslide moves as in Definition 3.3(1), we can reduce (C, ∂) to simple form. In addition, this simple form is unique. Barannikov proves this lemma by using an inductive construction to find the simple forms of each of the subcomplexes (Ck, ∂) generated by y1,..., yk for k ≤ m. Remark 3.12. The following two observations follow directly from Lemma 3.11. (1) If (C, ∂) and (C0, ∂0) are chain isomorphic by a handleslide move, that is, = 0 −1 Ᏸ Ek,l Ᏸ Ek,l , then they have the same simple form. (2) For consecutive generators in (C, ∂), a handleslide move does not change the value of h∂y j+1 | y j i. Thus, if h∂y j+1 | y j i = 1, then ∂y j+1 = y j in the simple form for (C, ∂). Definition 3.13. Suppose that (C, ∂) is an ordered chain complex with generators y1,..., ym and trivial homology. By Lemma 3.11, there exists a fixed-point free involution τ :{1,..., m} → {1,..., m} such that in the simple form of (C, ∂), either ∂yi = yτ(i) or ∂yτ(i) = yi for all i. We call τ the pairing of (C, ∂). CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 95

The next lemma assigns a normal ruling to an MCS. Pushkar and Chekanov [2005, Section 12.4] prove this lemma for a generating family using the language of pseudoinvolutions. Lemma 3.14. The pairings determined by the simple forms of the ordered chain complexes in an MCS Ꮿ = (C1, ∂1) ··· (Cm, ∂m) determine a graded normal ruling NᏯ on 6.

Proof. Each (C j , ∂ j ) has trivial homology, and thus, a pairing τi . Given a pair of consecutive singularities xi and xi+1 of 6, we deﬁne a pairing on the strands of the tangle 6 ∩ (xi , xi+1) using the pairing of a chain complex of Ꮿ between xi and xi+1. There may be several chain complexes of Ꮿ between xi and xi+1. How- ever, these chain complexes differ by handleslide moves and so by Remark 3.12(1) their pairings are identical. Remark 3.12(2) justiﬁes that two strands entering a cusp are paired. It remains to check that the switched crossings are graded and normal. The normality follows from [Barannikov 1994, Lemma 4]. The two strands meeting at a switch exchange companion strands and the Maslov potentials of two companioned strands differ by 1. Hence, two strands meeting at a switch must have the same Maslov potential and so the grading of a switched crossing is 0.

If Ꮿ1 and Ꮿ2 are equivalent by a single MCS move, then by Proposition 3.8 the chain complexes in Ꮿ1 and Ꮿ2 are equal outside of a neighborhood of the MCS move. Thus, we may use chain isomorphic chain complexes in Ꮿ1 and Ꮿ2 to determine the graded normal rulings NᏯ1 and NᏯ1 . As a consequence we have the following. ∼ = Proposition 3.15. If Ꮿ1 Ꮿ2 then NᏯ1 NᏯ2 .

We let N[Ꮿ] ∈ N(6) denote the graded normal ruling associated to the MCS class [Ꮿ].

3f. MCSs with simple births. A simple birth in an MCS is a birth with no implicit handleslides. In the language of Deﬁnition 3.3(3)), this says E = I . From now on we will restrict our attention to MCSs with only simple births and MCSs equivalence moves that do not involve implicit handleslide marks at births. The connection between MCSs and augmentations is clearer under this assumption. For the sake of simplicity, the results stated in the introduction were given in terms of MCS[(6). The corresponding results proved in the remainder of the article are given in terms of MCSs with simple births. In light of Proposition 3.17 below, there is no harm in this ambiguity.

Deﬁnition 3.16. Let MCSb(6) be the set of MCSs in MCS(6) that have only 0 simple births. Two MCSs Ꮿ, Ꮿ ∈ MCSb(6) are equivalent if they are equivalent 96 MICHAEL B. HENRY

0 in MCS(6) by a sequence of MCS moves Ꮿ = Ꮿ1, Ꮿ2,..., Ꮿn = Ꮿ such that Ꮿi ∈ MCSb(6) for all i. We let MCS[ b(6) denote the equivalence classes of MCSb(6)∼, and we let [Ꮿ] denote an equivalence class in MCS[ b(6).

Proposition 3.17. The inclusion map i : MCSb(6) → MCS(6) determines a bi- ˆ jection i : MCS[ b(6) → MCS[(6) by [Ꮿ] 7→ [i(Ꮿ)]. We sketch the argument for injectivity. Suppose [i(Ꮿ)]=[i(Ꮿ0)] in MCS(6) and 0 ᐆ = {i(Ꮿ) = Ꮿ1, Ꮿ2,..., Ꮿn−1, i(Ꮿ ) = Ꮿn} is a sequence of MCS moves. Since i(Ꮿ) and i(Ꮿ0) have simple births, each implicit handleslide appearing at a birth in an MCS in ᐆ is introduced and eventually eliminated using one of moves 14, 15, or 16. We may modify the sequence of MCS moves in ᐆ inductively beginning with Ꮿ1 and Ꮿ2 so that Ꮿi ∈ MCSb(6) for all i. We do this by replacing occurrences of move 16 at lefts cusps with move 1 and removing the occurrences of move 15 at left cusps. As a result, we must also change the occurrences of move 14 to a composition of moves 6 and 13 and possibly include additional moves 2–5. The map iˆ is surjective since MCS move 15 may be used to ﬁnd a representative of [Ꮿ] ∈ MCS[(6) with simple births. Remark 3.18. If we assume all births are simple, then we do not need to indicate the implicit handleslide marks at deaths in the associated marked front projection, since they can be determined by reconstructing the ordered chain complexes as in Remark 3.6(1). Thus, from now on, marked front projections will include only explicit handleslide marks and we will no longer indicate implicit handleslide marks in the MCS moves.

4. Chain homotopy classes of augmentations

We begin by defining DGA morphisms and chain homotopy on arbitrary DGAs and then restrict to the case of CE-DGAs. Definition 4.1 and Lemma 4.2 follow directly from [Kalm´ an´ 2005, Section 2.3]. Definition 4.1. Let (Ꮽ, ∂) and (Ꮾ, ∂0) be differential graded algebras over a commutative ring R and graded by a cyclic group 0.A DGA morphism ϕ : (Ꮽ, ∂) → (Ꮾ, ∂0) is a grading-preserving algebra homomorphism satisfying ϕ ◦ ∂ = ∂0 ◦ ϕ. Given two DGA morphisms ϕ, ψ : (Ꮽ, ∂) → (Ꮾ, ∂0), a chain homotopy between ϕ and ψ is a linear map H : (Ꮽ, ∂) → (Ꮾ, ∂0) such that (1) |H(a)| = |a| + 1 for all a ∈ Ꮽ, (2) H(ab) = H(a)ψ(b) + (−1)|a|ϕ(a)H(b) for all a, b ∈ Ꮽ, and (3) ϕ − ψ = H ◦ ∂ + ∂0 ◦ H. We refer to condition (2) as the derivation product property. CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 97

Augmentations are DGA morphisms between the CE-DGA and the DGA whose only nonzero chain group is a copy of ޚ2 in grading 0. From [Kalm´ an´ 2005, Lemma 2.18], a chain homotopy between augmentations is determined by its action on the generators of the CE-DGA. Lemma 4.2. Let (Ꮽ(L), ∂) be the CE-DGA of the Lagrangian projection L with generating set Q, and let 1, 2 ∈ Aug(L). If a map H : Q → ޚ2 has support on generators of grading −1, then H can be uniquely extended by linearity and the derivation product property to a map, also denoted H, on all of (Ꮽ(L), ∂) that has support on elements of grading −1. Moreover, if the extension satisﬁes 1(q) − 2(q) = H ◦ ∂(q) for all q ∈ Q, then 1 − 2 = H ◦ ∂ on all of Ꮽ(L), and thus H is a chain homotopy between 1 and 2.

We say 1 and 2 are chain homotopic and write 1 ' 2 if a chain homotopy H exists between 1 and 2. Lemma 4.3 [Félix et al. 1995]. The relation ' is an equivalence relation. ch We let Aug (L) denote Aug(L)' and [] denote an augmentation class in Augch(L). The following lemma will be necessary for our later work connecting augmentation classes and MCS classes. Lemma 4.4. Let L and L0 be Lagrangian projections with CE-DGAs (Ꮽ(L), ∂) and (Ꮽ(L0), ∂0), respectively. Let f : (Ꮽ(L), ∂) → (Ꮽ(L0), ∂0) be a DGA morphism. Then f induces a map F : Augch(L0) → Augch(L) by [] 7→ [ ◦ f ]. If g : (Ꮽ(L), ∂) → (Ꮽ(L0), ∂0) is also a DGA morphism and H is a chain homotopy between f and g, then F = G. Proof. We begin by checking F is well-deﬁned, that is, ◦ f is an augmentation and ' 0 implies ◦ f ' 0 ◦ f . Since f is a degree-preserving chain map and ◦ ∂0 = 0, the equality ◦ f (q) = 1 implies |q| = 1 and ◦ f ◦ ∂ = ◦ ∂0 ◦ f = 0. Therefore ◦ f is an augmentation. If H is a chain homotopy between and 0, then H ◦ f is a chain homotopy between ◦ f and 0 ◦ f . Suppose g : (Ꮽ(L), ∂) → (Ꮽ(L0), ∂0) is also a DGA morphism and H is a chain 0 homotopy between f and g. The map H = ◦ H : (Ꮽ(L), ∂) → ޚ2 is a chain homotopy between ◦ f and ◦ g. Thus, F = G since

F([]) = [ ◦ f ] = [ ◦ g] = G([]). In the case of the CE-DGA, the number of chain homotopy classes is a Legen- drian invariant. Proposition 4.5. If L and L0 are Lagrangian projections of Legendrian isotopic knots K and K 0, then there is a bijection between Augch(L) and Augch(L0). Thus, |Augch(L)| is a Legendrian invariant of the Legendrian isotopy class of K . 98 MICHAEL B. HENRY

We sketch the proof of Proposition 4.5. Since K and K 0 are Legendrian isotopic, (Ꮽ(L), ∂) and (Ꮽ(L0), ∂0) are stable tame isomorphic. Thus, we need only consider the case of a single elementary isomorphism and a single stabilization. Suppose (Si (Ꮽ), ∂) is an index i stabilization of a DGA (Ꮽ, ∂). If i 6= 0, then ∈ Aug(Ꮽ) extends uniquely to an augmentation ˜ ∈ Aug(Si (Ꮽ, ∂)) by sending both e1 and e2 to 0. If i = 0, then |e1| = 0 and ∈ Aug(Ꮽ) extends to two different augmentations in S0(Ꮽ, ∂); the first, denoted ˜, sends both e1 and e2 to 0 and the second sends e1 to 1 and e2 to 0. However, these two augmentations are chain homotopic by the chain homotopy H that sends e2 to 1 and all of the other generators to 0. Regardless of i, the map defined by [] 7→ [˜] is the desired bijection. Finally, if φ : (Ꮽ, ∂) → (Ꮾ, ∂0) is an elementary isomorphism between two DGAs, then the map 8 : Augch(Ꮾ) → Augch(Ꮽ), [0] 7→ [0 ◦ φ] is the desired bijection. In the next section, we will concentrate on understanding the chain homotopy classes of a fixed Lagrangian projection. By using a procedure called “adding dips”, we may reformulate the augmentation condition ◦∂ and the chain homotopy condition 1 − 2 = H ◦ ∂ as a system of matrix equations. Understanding the chain homotopy classes of augmentations then reduces to understanding solutions to these matrix equations.

5. Dipped resolution diagrams

Fuchs [2003] modiﬁes the Lagrangian projection of a Legendrian knot so that the differential in the CE-DGA is easy to compute, at the cost of increasing the number of generators. This technique has proved to be very useful; see [Fuchs and Ishkhanov 2004; Fuchs and Rutherford 2008; Ng and Sabloff 2006; Sabloff 2005]. We will use the version of this philosophy implemented in [Sabloff 2005].

5a. Adding dips to a Ng resolution diagram. By acting on L6 by a series of Lagrangian Reidemeister type II moves, we can limit the types of convex immersed polygon appearing in the differential of the CE-DGA. A dip, denoted D, is the collection of crossings created by performing type II moves on the strands of L6 as in Figure 12. At the location of the dip, label the strands of L6 from bottom to top with the integers 1,..., n. For all k > l, there is a type II move that pushes strand k over strand l. If k < i, then k crosses over l before i crosses over any strand. If l < j < k, then k crosses over l before k crosses over j. The notation (k, l) ≺ (i, j) denotes that k crosses over l before i crosses over j. The a-lattice of D is composed of all crossings to the right of the vertical line of symmetry, and the b-lattice is composed of all crossings to the left. The label ak,l denotes the crossing in the a-lattice of strand k over strand l where k > l; see CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 99

4 3 2 1

b3,1 a4,2

Figure 12. Adding a dip to L6 and labeling the resulting crossings. Crossings b3,1 and a4,2 are indicated.

Figure 13. The four possible inserts in a sufﬁciently dipped dia- d gram L6: (1) parallel lines, (2) a single crossing, (3) a resolved right cusp, and (4) a resolved left cusp. In each case, any number of horizontal strands may exist.

Figure 12. The label bk,l is similarly defined. The gradings of the crossings in a dip are computed by |bk,l | = µ(k) − µ(l) and |ak,l | = |bk,l | − 1, where µ is a Maslov potential on 6. The type II moves may be arranged so when strand k is pushed over strand l, a crossing q has height less than bk,l if and only if q appears to the left of the dip, or q = br,s or q = ar,s, where r − s ≤ k −l. Similarly, a crossing q has height less than ak,l if and only if q appears to the left of the dip, or q = br,s or q = ar,s, where r − s ≤ k − l. Definition 5.1. Given a Legendrian knot K with front projection 6 and Ng reso- d lution L6, a dipped diagram, denoted L6, is the result of adding some number of dips to L6. We require that during the process of adding dips, we not allow a dip to be added between the crossings of an existing dip and we not add dips in the loop of a resolved right cusp. d We let D1,..., Dm denote the m dips of L6, ordered from left to right with respect to the x-axis. For consecutive dips D j−1 and D j , we let I j denote the tangle between D j−1 and D j . We define I1 to be the tangle to the left of D1 and d Im+1 to be the tangle to the right of Dm. We call I1,..., Im+1 the inserts of L6. d Definition 5.2. We say a dipped diagram L6 is sufficiently dipped if each insert I1,..., In+1 is isotopic to one of those in Figure 13. d Remark 5.3. If we slide a dip of L6 left or right without passing it by a crossing, resolved cusp or another dip, then the resulting dipped diagram is topologically 100 MICHAEL B. HENRY identical to the original. In particular, they determine the same CE-DGA. Thus, we will not distinguish between two dipped diagrams that differ by such a change.

5b. The CE-DGA on a sufficiently dipped diagram. A convex immersed polygon cannot pass completely through a dip. Therefore, we may calculate the boundary d map of (Ꮽ(L6), ∂) by classifying convex immersed polygons that lie between consecutive dips or that sit entirely within a single dip. This classification is aided d by our understanding of the heights of the crossings in L6. d Let L6 be a sufficiently dipped diagram of L6 with dips D1,..., Dm and inserts I1,..., Im+1. Let qi ,..., qM and z j ,..., zN denote the crossings found in the inserts of type (2) and (3), respectively. The subscripts on q and z correspond to the subscripts of the inserts in which the crossings appear. Remark 5.4. In Section 5a, we labeled the crossings in the a-lattice and b-lattice of a dip using a labeling of the strands at the location of the dip. When calculating ∂ on the crossings in D j , it will make our computations clearer and cleaner if we change slightly the labeling we use for the strands in D j−1 and D j . If I j is of type (3) and the right cusp occurs between strands i + 1 and i, we will label the strands of D j−1 by 1,..., n and the strands of D j by 1,..., i − 1, i + 2,..., n. If I j is of type (4) and the left cusp occurs between strands i + 1 and i, we will label the strands of D j−1 by 1,..., i − 1, i + 2,..., n and the strands of D j by 1,..., n. If I j is of type (1) or (2), we do not change the labeling of the strands. These changes are local, so the labeling of the strands in D j may vary depending on whether we are calculating ∂ on the crossings in D j or D j+1.

Fix an insert I j with j 6= m+1, and label the strands of D j and D j−1 as indicated in Remark 5.4. For i ∈{ j, j−1}, Ai (respectively Bi ) is the strictly lower triangular square matrices with rows and columns labeled the same as the strands of Di and k,l k,l entries given by (Ai )k,l = ai (respectively (Bi )k,l = bi ) for k > l. The following ˜ formulas deﬁne a matrix A j−1 using the a j−1-lattice and possible crossings in I j . The subscript j − 1 has been left off the a to make the text more readable. ˜ (a) A j−1 = Ᏼ2,1 for j = 1, where dim(Ᏼ2,1) = 2. ˜ (b) Suppose I j is of type (1). Then A j−1 = A j−1.

(c) Suppose I j is of type (2), A j has dimension n, and q j involves strands i + 1 ˜ × ˜ u,v and i. Then A j−1 is the n n matrix with (u, v) entry a j−1 deﬁned by au,v if u, v∈ / {i, i + 1}, a˜ u,v = 0 if u = i + 1, v = i, i,v i+1,v i,v i+1,v i,v a˜ = a + q j a , a˜ = a , u,i u,i+1 u,i+1 u,i u,i+1 a˜ = a , a˜ = a + a q j . CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 101

Figure 14. Disks contributing to ∂zr and ∂qs.

(d) Suppose I j is of type (3), A j has dimension n, and z j involves strands i + 1 ˜ and i. Then A j−1 is an n × n matrix with rows and columns numbered − + + ˜ u,v 1,..., i 1, i 2,..., n 2 and (u, v) entry a j−1 deﬁned by au,v if u < i or v > i + 1,  u,v u,v u,i i+1,v u,i+1 i+1,v a˜ = a + a a + a z j a  u,i i,v u,i+1 i,v + a z j a + a z j z j a if u > i + 1 > i > v.

(e) Suppose I j is of type (4), A j has dimension n and the resolved left cusp + ˜ × ˜ u,v strands are i 1 and i. Then A j−1 is the n n matrix with (u, v) entry a j−1 deﬁned by au,v if u, v∈ / {i, i + 1},  a˜ u,v = 1 if u = i + 1, v = i, 0 if otherwise. d We can now give matrix equations for the boundary map of (Ꮽ(L6), ∂). Lemma 5.5. The CE-DGA boundary map ∂ of a sufﬁciently dipped diagram of a Ng resolution L6 is computed as follows: = i+1,i + (1) ∂qs as−1 for a crossing qs between strands i 1 and i. = + i+1,i + (2) ∂zr 1 ar−1 for a crossing zr between strands i 1 and i, where zr is a crossing between strands i + 1 and i. 2 (3) ∂ A j = A j for all j. ˜ (4) ∂ B j = (I + B j )A j + A j−1(I + B j ) for all j.

Here I is the identity matrix of appropriate dimension, and the matrices ∂ A j and ∂ B j are the result of applying ∂ to A j and B j entry by entry. This result appears in [Fuchs and Rutherford 2008] in the language of “splashed diagrams”. In the next three sections we justify these formulas. d 5b1. Computing ∂ on qs and zr . The height ordering on the crossings of L6 ensures that qs is the rightmost convex corner of any nontrivial disk in ∂qs. Thus, the disk in Figure 14 is the only disk contributing to ∂qs. The case of ∂zr is similar. 102 MICHAEL B. HENRY

(a) (b) 5,3 3,2 5,2 Figure 15. Left: the disk a j a j appearing in ∂a j . Right: the 4,3 3,2 4,2 4,2 disks b j a j and a j appearing in R(∂b j ).

5b2. Computing ∂ on the A j lattice. The combinatorics of the dip D j along with d k,l the height ordering on the crossings of L6 require that disks in ∂a j sit within the A j lattice. In addition, a convex immersed polygon sitting within the A j lattice must have exactly two negative convex corners; see Figure 15(a). From this we k,l P k,i i,l 2 compute ∂a j = l

k,l 5b3. Computing ∂ on the B j lattice. The disks in ∂b j are of two types. The ﬁrst k,l type sit within D j and include the lower right corner of b j as their positive convex k,l corner. We denote these disks by R(∂b j ). Figure 15(b) shows the two types of k,l convex immersed polygons found in R(∂b j ). From this, we compute

k,l k,l X k,i i,l R(∂b j ) = a j + b j a j . l

k,l k,l Regardless of the type of I j , R(∂b j ) + L(∂b j ) is the (k, l)-entry in the matrix ˜ (I + B j )A j + A j−1(I + B j ). In Section 5c, we study the extension of augmentations across individual type II moves and then across dips, and thus, understand the connections between aug- d mentations on L6 and augmentations on a sufﬁciently dipped diagram L6. In Section 6, we use an argument from [Fuchs and Rutherford 2008] to associate an d augmentation of L6 to an MCS of 6. CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 103

4,2 4,3 3,2 a j−1 a j−1b j (a)

1 3,1 b j (b)

k,l Figure 16. Disks contributing to L(∂b j ) when I j is of type (1) or (4). The disks in (a) may occur in an insert of any type.

Figure 17. Disks contributing to L(∂ B j ) when I j is of type (2).

5c. Extending an augmentation across a dip. If the Lagrangian projection L2 is obtained from L1 by a single type II move introducing new crossings a and b with |b| = i and |a| = i −1, then there exists a stable tame isomorphism from (Ꮽ(L2), ∂) 0 to (Ꮽ(L1), ∂ ). As we demonstrated in Proposition 4.5, such a stable tame isomorphism induces a map from Aug(L1) to Aug(L2). By understanding this map we may keep track of augmentations as we add dips to L6. This careful analysis will 104 MICHAEL B. HENRY

Figure 18. Disks contributing to L(∂ B j ) when I j is of type (3). The boundary of each disk is highlighted. allow us in Section 6 to assign an MCS class of 6 to an augmentation class of L6 ch and, hence, prove the surjectivity of the map 9b : MCS[ b(6) → Aug (L6). 0 The stable tame isomorphism from (Ꮽ(L2), ∂) to (Ꮽ(L1), ∂ ) was first explicitly written down in [Chekanov 2002a]. We will use the formulation found in [Sabloff 2005]. The stable tame isomorphism is a single stabilization Si introducing generators α and β followed by a tame isomorphism ψ. The tame isomorphism 0 ψ : (Ꮽ(L2), ∂) → Si (Ꮽ(L1), ∂ ) is defined as follows. Let {x1,..., xN } denote the generators with height less than h(a) and labeled so that h(x1) < ··· < h(xN ). Let {y1,..., yM } denote the generators with height greater than h(b) and labeled so that h(y1) < ··· < h(yM ). Recall that ∂ lowers height, so we may write ∂b as ∂b = a + v, where v consists of words in the letters x1,..., xN . We begin by 0 defining a vector space map H on Si (Ꮽ(L1), ∂ ) by 0 if w ∈ Ꮽ(L ),  1 H(w) = 0 if w = Qβ R with Q ∈ Ꮽ(L1) and R ∈ Si (Ꮽ(L1)),  Qβ R if w = QαR with Q ∈ Ꮽ(L1) and R ∈ Si (Ꮽ(L1)). 0 We build ψ up from a sequence of maps ψi : (Ꮽ(L2), ∂) → Si (Ꮽ(L1), ∂ ) for 0 ≤ i ≤ M:  β if w = b,  yi + H ◦ ψi−1(∂yi ) if w = yi , ψ0(w) = α + v if w = a, ψi (w) = ψ − (w) otherwise. w otherwise, i 1 CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 105

If we extend the map ψM by linearity from a map on generators to a map on all of 0 (Ꮽ(L2), ∂), then the resulting map ψ : (Ꮽ(L2), ∂) → Si (Ꮽ(L1), ∂ ) is the desired DGA isomorphism; see [Chekanov 2002a; Sabloff 2005]. If i 6= 0, then an augmentation ∈ Aug(L1) is extended to an augmentation ∈ Aug(Si (L1)) by (β) = (α) = 0. If i = 0, then we may choose to extend ∈ Aug(L1) either by (β) = (α) = 0 or by (β) = 1 and (α) = 0. Given the tame isomorphism ψ deﬁned above, we would like to understand how our choice of an extension from Aug(L1) to Aug(Si (L1)) affects the induced map from 0 Aug(L1) to Aug(L2). Recall ψ : (Ꮽ(L2), ∂) → Si (Ꮽ(L1), ∂ ) induces a bijection 0 9 : Aug(Si (L1)) → Aug(L2) by 7→ ◦ ψ. Let = ◦ ψ. From the formulas 0 0 0 for ψ we note that (xi ) = (xi ) for all xi , (b) = (β), and (a) = (v), where ∂b = a + v. The next two lemmas follow from [Sabloff 2005, Lemma 3.2] and 0 detail the behavior of on y1,..., yM .

Lemma 5.6. If we extend an augmentation ∈ Aug(L1) to an ∈ Aug(Si (L1)) by 0 0 (β) = (α) = 0, the augmentation ∈ Aug(L2) given by = ◦ ψ satisﬁes (1) 0(b) = 0, (2) 0(a) = (v), where ∂b = a + v, and (3) 0 = on all other crossings. 0 Proof. We are left to show = on y1,..., yM . This follows from observing that (β) = 0 implies ◦ H ◦ ψ(∂y j ) = 0 for all j.

Lemma 5.7. Suppose we extend an augmentation ∈ Aug(L1) to an augmentation ∈ Aug(Si (L1)) by (β) = 1 and (α) = 0. Suppose in (Ꮽ(L2), ∂) the generator a { } appears in the boundary of each of the generators y j1 ,..., y jl . Suppose that for ∈ { } ∈ y y j1 ,..., y jl , each disk contributing a to ∂y has the form QaR, where Q, R 0 Ꮽ(L2) and Q and R do not contain a or b. Then the augmentation ∈ Aug(L2) satisﬁes the following: (1) 0(b) = 1; (2) 0(a) = (v), where ∂b = a + v; ∈ { } 0 = (3) for each y y j1 ,..., y jl , we have (y) (y) if and only if the generator a appears in an even number of terms in ∂y that are of the form QaR with (Q) = (R) = 1; (4) 0 = on all other crossings.

Proof. Cases (1), (2), and (4) follow as in Lemma 5.6. Let y ∈ {y1,..., yM }. Note that 0(y) = (y) + ◦ H ◦ ψ(∂y). Suppose a does not appear in ∂y. Then H ◦ψ(∂y) = 0, so 0(y) = (y). Suppose a does appear in ∂y. By assumption, each disk contributing a to ∂y has the form QaR, where Q, R ∈ Ꮽ(L2) and Q and R do not contain a or b. Thus H ◦ψ(QaR) = Qβ R for each disk of the form QaR in ∂y, 106 MICHAEL B. HENRY and H ◦ ψ is 0 on all of the other disks in ∂y. Now ◦ H ◦ ψ(QaR) = (Q)(R), so ◦ H ◦ ψ(∂y) = 0 if and only if a appears in an even number of terms in ∂y that are of the form QaR with (Q) = (R) = 1. Now we follow an augmentation through the complete process of adding a dip d to L6. Suppose L6 is a dipped diagram of L6 with dips D1,..., Dm. We would d like to add a new dip D to L6 away from D1,..., Dm. We will denote the resulting d0 d dipped diagram by L6 . We want to understand how n augmentation in Aug(L6) d0 extends to one in Aug(L6 ) using the stable tame isomorphisms deﬁned above. This work is motivated by the construction deﬁned in [Sabloff 2005, Section 3.3]. d 0 d0 We set the following abbreviated notation. Let L = L6 and L = L6 . We will assume that, with respect to the ordering on dips coming from the x-axis, D is not the leftmost dip in L0. Let I denote the insert in L0 bounded on the right by D. Let D j denote the dip bounding I on the left. At the location of the dip D, label d the strands of L6 from bottom to top with the integers 1,..., n. The creation of 0 the dip D gives a sequence of Lagrangian projections L, L2,1,..., Ln,n−1 = L , where Lk,l denotes the result of pushing strand k over strand l. Let ∂k,l denote the boundary map of the CE-DGA of Lk,l . Let Dk,l denote the partial dip in Lk,l . In each Lagrangian projection Lk,l , the insert I and dip D j sit to the left of the partial dip Dk,l . Suppose (r, s) denotes the type II move immediately preceding the type II move (k, l). Then Lk,l is the result of pushing strand k over strand l in Lr,s. d 5c1. Extending ∈ Aug(L6) by 0. Suppose we have extended ∈ Aug(Lr,s) so 0 k,l 0 k,l k,l k,l that (b ) = 0. Lemma 5.6 implies (a ) = (v) where ∂k,l b = a + v, and 0 = on all other crossings. If the insert I is of one of the four types in k,l k,l Figure 13, we can describe the disks in v. Note that a is the only disk in ∂k,l b with the lower right corner of bk,l as its positive convex corner; thus we need only understand the disks with the upper left corner of bk,l as their positive convex corner. In the proof of Lemma 5.5, we let L(∂bk,l ) denote the disks in ∂bk,l that include the upper left corner of bk,l as their positive convex corner. Here ∂bk,l d0 refers to the boundary map in the CE-DGA of L6 . The order in which we perform the type II moves that create D ensures that when the crossing bk,l is created, all k,l k,l of the disks in L(∂b ) appear in ∂k,l b . In fact, the restrictions placed on convex k,l immersed polygons by the height function imply that any disk appearing in ∂k,l b must also appear in ∂bk,l . Thus, we have v = L(∂bk,l ) and so 0(ak,l )=(L(∂bk,l )), ˜ which is equal to the (k, l)-entry in the matrix (A j (I + B)). Since is an algebra 0 ˜ ˜ map and (B) = 0, we see that (A j (I +B)) = (A j ). Thus we have the following: d Lemma 5.8. Suppose L6 is a dipped diagram of the Ng resolution L6. Suppose we use the dipping procedure to add a dip D between the existing dips D j and d0 D j+1 and thus create a new dipped diagram L6 . Suppose the insert I between D j CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 107

d and D is of one of the four types in Figure 13. Let ∈ Aug(L6). Then if at every type II move in the creation of the dip D we choose to extend so that 0 is 0 on the new crossing in the b-lattice, then the stable tame isomorphism from Section 5c 0 d0 maps to ∈ Aug(L6 ) satisfying • 0(B) = 0, 0 ˜ • (A) = (A j ), and • 0 = on all other crossings.

d Deﬁnition 5.9. If ∈ Aug(L6), then we say that is extended by 0 if after each type II move in the creation of D, we extend so that sends the new crossing of the b-lattice of D to 0. ˜ ˜ In the next corollary, we investigate (A j ) by revisiting the deﬁnition of A j from Section 5b. The assumptions on (q) and (z) in cases (2) and (4) simplify the ˜ matrices (A j ) considerably, although verifying case (4) is still a slightly tedious matrix calculation.

d Corollary 5.10. Suppose we are in the setup of Lemma 5.8. Let ∈ Aug(L6), and 0 d0 let ∈ Aug(L6 ) be the extension of described in Lemma 5.8. 0 (1) If I is of type (1), then (A) = (A j ). (2) Suppose I is of type (2) with crossing q between strands i+1 and i. If (q)=0, 0 then (A) = Pi+1,i (A j )Pi+1,i . (3) Suppose I is of type (3) with crossing z between strands i + 1 and i.

• Let i + 1 < u1 < u2 < ··· < us denote the strands at dip D j that satisfy u1,i us ,i (a j ) = · · · = (a j ) = 1. + • i 1,v1 let vr < ··· < v1 denote the strands at dip D j that satisfy (a j ) = i+1,v2 i+1,vr (a j ) = · · · = (a j ) = 1; and • = let E Ei,vr ... Ei,v1 Eu1,i+1 ... Eus ,i+1. = i+1,i = 0 = −1 T If (z) 0 and (a j ) 1, then (A) Ji−1 E(A j )E Ji−1 as matrices. (4) Suppose I is of type (4) and the resolved birth is between strands i + 1 0 and i. Then, as matrices, (A) is obtained from (A j ) by inserting two rows (columns) of zeros after row (column) i − 1 in (A j ) and then changing the (i + 1, i) entry to 1.

Except for case (1), these matrix equations correspond to the chain maps that connect consecutive ordered chain complexes in an MCS with simple births; see Deﬁnition 3.3. Thus we see the ﬁrst hint of an explicit connection between MCSs and augmentations. 108 MICHAEL B. HENRY

i+1,i Figure 19. Disks appearing in ∂i+1,i with a as a negative convex corner.

d 5c2. Extending ∈ Aug(L6) by Ᏼi+1,i . In this section, we consider extending d 0 d0 0 ∈ Aug(L6) to an augmentation ∈ Aug(L6 ) such that (B) = Ᏼi+1,i . Recall Ᏼi+1,i denotes a square matrix with 1 in the (i+1, i) position and zeros everywhere else. During the type II move that pushes strand i +1 over strand i, we will choose to extend so that 0(bi+1,i ) = 1. By carefully using Lemma 5.7, we are able to determine the extended augmentation 0. d Deﬁnition 5.11. Suppose ∈ Aug(L6). We say that is extended by Ᏼi+1,i if µ(i + 1) = µ(i) and after each type II move in the creation of a new dip D, we 0 d0 0 extend so that the extended augmentation ∈ Aug(L6 ) satisﬁes (B) = Ᏼi+1,i . Understanding 0 when 0(bi+1,i ) = 1 requires that we understand all of the i+1,i crossings y such that a appears in ∂i+1,i y. If we add the following restrictions on I , D, and the pair (i + 1, i), then we can identify all disks containing ai+1,i as a negative convex corner. Suppose • |bi+1,i | = 0, • I is of type (1),

• D occurs to the immediate left of a crossing q in L6 between strands i + 1 and i. i+1,i i+1,l Given these conditions, a only appears in ∂i+1,i q and ∂i+1,i a for l < i; see Figure 19. Applying Lemma 5.7, we conclude that after the type II move that pushes strand i + 1 over strand i, the augmentation 0 satisﬁes • 0(bi+1,i ) = 1; • 0(ai+1,i ) = (v), where ∂b = a + v; • 0(q) 6= (q); + • 0 i+1,l i 1,l 0 i,l for all l < i, (a ) = (a j ) if and only if (a ) = 0; and • 0 = on all other crossings. In Section 5c1 we showed that v = L(∂bi+1,i ) if I is an insert of type (1)–(4), d0 where ∂ is the boundary in L6 . Since I is an insert of type (1), we conclude that 0 i+1,i i+1,i (a ) = (a j ). Suppose we continue creating the dip D, and with each new type II move we extend the augmentation so that it sends the new crossing in the b-lattice to 0. By CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 109

Lemma 5.8, we need only compute the value of the extended augmentation on the new crossing in the a-lattice. Consider the type II move pushing strand k over l where (i + 1, i) ≺ (k, l). By Lemma 5.8, we have

0 k,l k,l k,l X k,p 0 p,l (a ) = (v) = (L(∂b )) = (a j ) + (a j ) (b ). l

0 p,l 0 k,l k,l Since (b ) = 1 if and only if (p, l) = (i +1, i), we know that (a ) = (a j ) if k,i+1 d 0 d0 and only if l 6= i or (a j ) = 0. Thus, if we extend ∈ Aug(L6) to ∈ Aug(L6 ) 0 0 so that (B) = Ᏼi+1,i , then satisﬁes

(ai+1,l ) + 0(ai,l ) if k = i + 1 and l < i,  j 0 k,l = k,i k,i+1 (a ) (a j ) + (a j ) if k > i + 1 and l = i,  k,l (a j ) otherwise. 0 = −1 This equation is equivalent to (A) Ei+1,i (A j )Ei+1,i . Pulling this all together, we have the following lemma. d Lemma 5.12. Suppose L6 is a dipped diagram of the Ng resolution L6. Let q d be a crossing in L6 corresponding to a resolved crossing of 6 and with |q| = 0. Suppose we add a dip D to the right of the existing dip D j and just to the left of q, d0 thus creating a new dipped diagram L6 . Suppose the insert I between D and D j d is of type (1). Let ∈ Aug(L6). If is extended by Ᏼi+1,i , then the stable tame 0 d0 0 isomorphism from Section 5c maps to ∈ Aug(L6 ), where satisﬁes 0 • (B) = Ᏼi+1,i , • 0 = −1 (A) Ei+1,i (A j )Ei+1,i , • 0(q) 6= (q), and • 0 = on all other crossings. We now have sufﬁcient tools to begin connecting MCSs and augmentations.

6. Relating MCSs and augmentations

ch In this section, we construct a surjection 9b : MCS[ b(6) → Aug (L6) and deﬁne a simple construction that associates an MCS Ꮿ to an augmentation ∈ Aug(L6) such that 9(b [Ꮿ]) = []. We also detail two algorithms that use MCS moves to place an arbitrary MCS in one of two standard forms.

d 6a. Augmentations on sufficiently dipped diagrams. Suppose L6 is a sufficiently dipped diagram of L6 with dips D1,..., Dm and inserts I1,..., Im+1, and let qi ,..., qM and z j ,..., zN denote the crossings found in the inserts of type (2) and (3), respectively. The subscripts on q and z correspond to the subscripts of 110 MICHAEL B. HENRY the inserts in which the crossings appear. The formulas in Lemma 5.5 allow us to write the augmentation condition ◦∂ as a system of local equations involving the d dips and inserts of L6. d Lemma 6.1. An algebra homomorphism : Ꮽ(L6) → ޚ2 on a sufficiently dipped d diagram L6 of a Ng resolution L6 with (1) = 1 satisfies ◦ ∂ = 0 if and only if i+1,i = + (1) (as−1 ) 0, where qs is a crossing between strands i 1 and i, k+1,k = + (2) (ar−1 ) 1, where zr is a crossing between strands i 1 and i, 2 (3) (A j ) = 0, and ˜ −1 (4) (A j ) = (I + (B j ))(A j−1)(I + (B j )) . We will primarily concern ourselves with the following types of augmentations. d Definition 6.2. Given a sufficiently dipped diagram L6, we say an augmentation d ∈ Aug(L6) is occ-simple if

• sends all of the crossings of type qs and zr to 0,

• either (B j ) = 0 or (B j ) = Ᏼk,l for some k > l for I j of type (1), and

• (B j ) = 0 for I j of type (2), (3), or (4). d d Augocc(L6) denotes the set of all such augmentations in Aug(L6). We say d ∈ Augocc(L6) is minimal occ-simple if (B j ) = Ᏼk,l for some k > l for all I j of m type (1). We let Augocc(L6) denote the set of all minimal occ-simple augmentations over all possible sufﬁciently dipped diagrams of L6.

The matrices (A1), . . . , (Am) of an occ-simple augmentation determine a sequence of ordered chain complexes. This will be made explicit in Lemma 6.4. The following result is a consequence of Lemma 6.1, Definition 6.2, and the ˜ definition of A j−1 given in Section 5b. Recall that the matrices Ek,l , Pi+1,i , Ji , and Ᏼk,l were defined in Section 3a1. The proof of Lemma 6.3 is essentially identical to the proof of Corollary 5.10. d d Lemma 6.3. If L6 is a sufficiently dipped diagram and ∈ Augocc(L6), then the following hold for each insert I j : = = −1 (1) If I j is of type (1), then either (A j ) (A j−1) or (A j ) Ek,l (A j−1)Ek,l for some k > l.

(2) If I j is of type (2), with crossing q between strands i + 1 and i, then (A j ) = −1 Pi+1,i (A j−1)Pi+1,i . (3) Suppose I j is of type (3) with crossing z between strands i + 1 and i.

• Let i + 1 < u1 < u2 < ··· < us denote the strands at dip D j−1 that satisfy u1,i = · · · = us ,i = (a j−1) (a j−1) 1; CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 111

+ • ··· i 1,v1 = let vr < < v1 denote the strands at dip D j−1 that satisfy (a j−1 ) i+1,v2 = · · · = i+1,vr = (a j−1 ) (a j−1 ) 1; and • = ··· ··· let E Ei,vr Ei,v1 Eu1,i+1 Eus ,i+1. = −1 T Then (A j ) Ji−1 E(A j−1)E Ji−1. (4) Suppose I j is of type (4) and the resolved birth is between strands i +1 and i. Then, as matrices, (A j ) is obtained from (A j−1) by inserting two rows (columns) of zeros after row (column) i − 1 in (A j−1) and then changing the (i + 1, i) entry to 1. The equations in Lemma 6.3 are identical to those found in Deﬁnition 3.3, with the exception of the ﬁrst case of a type (1) insert. Indeed, given a front diagram 6 with resolution L6, MCSs on 6 correspond to minimal occ-simple augmentations of L6. We assign a minimal occ-simple augmentation Ꮿ to an MCS Ꮿ using an argument of Fuchs and Rutherford [2008].

Lemma 6.4. The set MCSb(6) of MCSs of 6 with simple births is in bijection m with the set Augocc(L6) of minimal occ-simple augmentations. d Proof. We assign an MCS to a minimal occ-simple augmentation ∈ Aug(L6), d m where L6 is a sufficiently dipped diagram of L6. Let ∈ Augocc(L6). For each dip D j , we form an ordered chain complex (C j , ∂ j ) as follows. Let t j denote the x- d coordinate of the vertical lines of symmetry of D j in L6. We label the m j points d ∩ { } × j j of intersection in L6 ( t j ޒ) by y1 ,..., ym j and we let C j be a ޚ2 vector j j j j space generated by y1 ,..., ym j . We label the generators y1 ,..., ym j based on j ··· j their y-coordinate so that y1 > > ym j . Each generator is graded by the Maslov potential of its corresponding strand in 6. The grading condition on and the fact 2 that (A j ) = 0 implies that (A j ) is a matrix representative of a differential on C j . Thus (C j , ∂ j ) with Ᏸ j = (A j ) is an ordered chain complex. Recall that the notation Ᏸ j was established in Remark 3.2. The relationship between consecutive differentials ∂ j−1 and ∂ j is defined in Lemma 6.3. Since is minimal occ-simple, = −1 = each insert I j of type (1) satisfies Ᏸ j Ek,l Ᏸ j−1 Ek,l , where (B j ) Ᏼk,l and k > l. Thus, (C j−1, ∂ j−1) and (C j , ∂ j ) are related by a handleslide between k and l. Since the matrix equations in Lemma 6.3 that satifies are equivalent to the matrix equations in Definition 3.3, the sequence (C1, ∂1), . . . , (Cm, ∂m) forms an MCS Ꮿ on 6. This process is invertible. Let Ꮿ ∈ MCSb(6) be a Morse complex sequence of the front projection 6 with chain complexes (C1, ∂1) ··· (Cm, ∂m). We will use the marked front projection associated to Ꮿ to define the placement of dips creating d d L6. Afterwards, we define an algebra homomorphism Ꮿ : Ꮽ(L6) → ޚ2 and show that it is an augmentation. Figure 20 gives an example of this process. Add a dip to L6 just to the right of the corresponding location of each resolved d cusp, resolved crossing or handleslide mark. The resulting dipped diagram L6 is 112 MICHAEL B. HENRY

x y

x D1 D2 D3 D4 D5 D6 D7 D8

Figure 20. Assigning an augmentation to an MCS.

sufﬁciently dipped with m dips. We deﬁne a ޚ2-valued map Ꮿ on the crossings of d L6 as follows.

• Ꮿ(qs) = 0 for all crossings qs coming from a resolved crossing of 6.

• Ꮿ(zr ) = 0 for all crossings zr coming from a resolved right cusp.

• Ꮿ(A j ) = Ᏸ j for all j.

• If I j is of type (1), then there exists k > l such that Ꮿ has a handleslide mark in I j between strands k and l. Let Ꮿ(B j ) = Ᏼk,l .

• If I j is of type (2), (3), or (4), then let Ꮿ(B j ) = 0.

We define Ꮿ(1) = 1 and extend Ꮿ by linearity to an algebra homomorphism on d Ꮽ(L6). If Ꮿ is an augmentation, then it is minimal occ-simple by construction. We must show Ꮿ ◦ ∂ = 0 and that Ꮿ(q) = 1 implies |q| = 0 for all crossings q d of L6. k,l j j If Ꮿ(a j ) = 1, then h∂ j yk | yl i = 1, where the notation for the generators of (C j , ∂ j ) corresponds to the notation in Definition 3.4. Thus µ(k) = µ(l) + 1, CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 113 where µ(i) denotes the Maslov potential of the strand in 6 corresponding to the j k,l generator yi . Recall |a j | = µ(k) − µ(l) − 1 and so µ(k) = µ(l) + 1 implies k,l k,l |a j | = 0. If Ꮿ(b j ) = 1, then in the marked front projection of Ꮿ, a handleslide k,l mark occurs in I j between strands k and l. Thus µ(k) = µ(l), and |b j | = 0 since k,l |b j | = µ(k) − µ(l). It remains to show that Ꮿ ◦ ∂ = 0. Let qr denote a crossing corresponding to a resolved crossing of 6 between strands i +1 and i. By Remark 3.6(2), the (i +1, i) i+1,i = = entry of the matrix ∂r−1 is 0, and hence Ꮿ(ar−1 ) 0. Therefore Ꮿ(∂qr ) i+1,i = Ꮿ(ar−1 ) 0. Let zs denote a crossing corresponding to a resolved right cusp of 6 between strands i+1 and i. By Remark 3.6(2), the (i+1, i) entry of the matrix ∂s−1 i+1,i = = + i+1,i = is 1, and hence Ꮿ(as−1 ) 1. Therefore Ꮿ(∂zs) 1 Ꮿ(as−1 ) 0. Since each 2 Ᏸ j is the matrix representative of a differential, we see that Ꮿ(∂ A j )=Ꮿ(A j ) =0. Verifying Ꮿ ◦ ∂ = 0 on each b-lattice B j is equivalent to verifying Lemma 6.1(4) and this can be done with Lemma 6.3. Indeed, the matrix equations in Lemma 6.3 correspond to the matrix equations in Definition 3.3 relating consecutive chain complexes in Ꮿ. Thus, Ꮿ(∂ B j ) = 0 for all j and so Ꮿ is a minimal occ-simple d augmentation on L6. ch 6b. Defining 9 : MCSb(6) → Aug (L6). Given Ꮿ ∈ MCSb(6), Lemma 6.4 gives an explicit construction of an augmentation Ꮿ on a sufficiently dipped di- d agram L6. In this section, we will use this construction to build a map 9 : ch MCSb(6) → Aug (L6). In order to do so, we must understand the possible d stable tame isomorphisms between Ꮽ(L6) and Ꮽ(L6) coming from sequences of Lagrangian Reidemeister moves. In [2005], Kalm´ an´ proves that two sequences of Lagrangian Reidemeister moves may induce inequivalent maps between the respective contact homologies. However, if we restrict our sequences of moves to adding and removing dips, then the we can avoid this problem and give a well- ch defined map 9 : MCSb(6) → Aug (L6). The following results allow us to modify a sequence of Lagrangian Reidemeister moves by removing canceling pairs of moves and commuting pairs of moves that are far away from each other. These modifications do not change the resulting bijection on augmentation chain homotopy classes. Proposition 6.5 [Kálmán 2005]. If we perform a type II move followed by a type II−1 move that creates and then removes two crossings b and a, then the induced DGA morphism from (Ꮽ(L), ∂) to (Ꮽ(L), ∂) is equal to the identity. If we perform a type II−1 move followed by a type II move that removes and then recreates two crossings b and a, then the induced DGA morphism from (Ꮽ(L), ∂) to (Ꮽ(L), ∂) is chain homotopic to the identity.

Proposition 6.6 [Kálmán 2005]. Suppose L1 and L2 are related by two consecutive moves of type II or II−1. We will call these moves A and B. Suppose the 114 MICHAEL B. HENRY crossings involved in A and B form disjoint sets. Then the composition of DGA morphisms constructed by performing move A and then move B is chain homotopic to the composition of DGA morphisms constructed by performing move B and then move A. Proposition 6.6 follows from [Kalm´ an´ 2005, case 1 of Theorem 3.7]. d d 6b1. Dipping/undipping paths for L6. Suppose L6 has dips D1,..., Dm. Let t1,..., tm denote the x-coordinates of the vertical lines of symmetry of the dips d D1,..., Dm in L6. d Definition 6.7. A dipping/undipping path for L6 is a finite-length monomial w in { ± ± ± ±} the elements of the set s1 ,..., sn , t1 ,..., tm . We require that w satisfies the following:

(1) Each si in w denotes a point on the x-axis away from the dips D1,..., Dm; and + (2) As we read w from left to right, the appearances of si alternate between si and − + − si , beginning with si and ending with si . The appearances of ti alternate − + − − between ti and ti , beginning with ti and ending with ti and each ti is required to appear at least once. Each dipping/undipping path w is a prescription for adding and removing dips d + from L6. In particular, si tells us to introduce a dip in a small neighborhood of si . − The letter si tells us to remove the dip that sits in a small neighborhood of si . The order in which these type II−1 moves occur is the opposite of the order used + + − d in si . The elements ti and ti work similarly. We perform these moves on L6 by reading w from left to right. The conditions we have placed on w ensure that we are left with L6 after performing all of the prescribed dips and undips. = − − ··· − Let w0 tm tm−1 t1 . Then w0 tells us to undip D1,..., Dm beginning with Dm and working to D1. Each w determines a stable tame isomorphism ψw from d ch d ch Ꮽ(L6) to Ꮽ(L6) that determines a bijection 9w : Aug (L6) → Aug (L6). We ch can now define a map 9 : MCSb(6) → Aug (L6). : → ch = [ ] Definition 6.8. Define 9 MCSb(6) Aug (L6) by 9(Ꮿ) 9w0 ( Ꮿ ). The definition of 9 is independent of dipping/undipping paths: d = Lemma 6.9. If w is a dipping/undipping paths for L6, then 9w 9w0 , and thus [ ] = [ ] = 9w( Ꮿ ) 9w0 ( Ꮿ ) 9(Ꮿ). Proof. By Proposition 6.6, we may reorder type II and II−1 moves that are “far d apart” without changing the chain homotopy type of the resulting map from Ꮽ(L6) + − and Ꮽ(L6). If si appears in w, then the next appearance of si is the letter si to + − + −1 the right of si . The letters between si and si represent type II and II moves − + − that are far away from si and si . Thus, we may commute si past the other letters CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 115

− + + so that si immediately follows si . By our ordering of the type II moves in si and −1 − −1 the type II moves in si , we may remove the type II and II moves in pairs. By Proposition 6.5, this does not change the chain homotopy type of the resulting map d from Ꮽ(L6) and Ꮽ(L6). In this manner, we remove all pairs of letters of the form + − si and si from w. By the same argument we remove pairs of letters of the form + − 0 ti , ti . The resulting dipping/undipping path, denoted w , only contains the letters − − 0 = t1 ,..., tm . We may reorder these letters so that w w0. By Proposition 6.6, this rearrangement does not change the chain homotopy type of the resulting map from d = Ꮽ(L6) and Ꮽ(L6); thus 9w 9w0 by Lemma 4.4.

6c. Associating an MCS to ∈ Aug(L6). We now prove that 9 : MCSb(6) → ch Aug (L6) is surjective. We do so by constructing a map that assigns to each augmentation in Aug(L6) an MCS in MCSb(6). The construction follows from our work in Lemma 5.12 and Corollary 5.10.

ch Lemma 6.10. The map 9 : MCSb(6) → Aug (L6) is surjective. ch Proof. Let [] ∈ Aug (L6) and let denote a representative of []. We will ﬁnd an MCS mapping to [] by constructing a minimal occ-simple augmentation and applying Lemma 6.4. We add dips to L6 beginning at the leftmost resolved left cusp and working to the right. As we add dips, we extend using Corollary 5.10 and Lemma 5.12. In a slight abuse of notation, we will let also denote the extended augmentation at every step. Begin by adding the dip D1 just to the right of the leftmost resolved left cusp. 2,1 = This requires a single type II move. We extend by requiring (b1 ) 0. In the 2,1 = + 2,1 2,1 = dipped projection, ∂(b1 ) 1 a1 . Hence, (a1 ) 1. Suppose we have added dips D1,..., D j−1 to L6 and let x j denote the next resolved cusp or crossing appearing to the right of D j−1. We introduce the dip D j and extend as follows.

• If x j is a resolved left cusp, right cusp, or crossing with (x j ) = 0, then we introduce D j just to the right of x j and extend by 0 as in Deﬁnition 5.9.

• If x j is a resolved crossing between strands i +1 and i and (x j ) = 1, then we introduce D j just to the left of x j and extend by Ᏼi+1,i as in Deﬁnition 5.11. Note that (x j ) = 0 after the extension.

In the ﬁrst case, Corollary 5.10 details the relationship between (A j ) and (A j−1). In the second, Lemma 5.12 details the relationship between (A j ) and (A j−1). The resulting augmentation is minimal occ-simple by construction. Let ˜ denote d the resulting augmentation on L6. By Lemma 6.4, ˜ has an associated MCS Ꮿ. = [˜] By deﬁnition, 9(Ꮿ) 9w0 ( ). The dipping/undipping path w0 tells us to undip t1,..., tn in reverse order. In particular, this is the inverse of the process we used to 116 MICHAEL B. HENRY

Figure 21. The dots in the right ﬁgure indicate the augmented crossings of ∈ Aug(L6). The left ﬁgure shows the resulting MCS Ꮿ from the construction in Lemma 6.10.

˜ d = [˜] = [ ] : → ch create on L6. Thus 9(Ꮿ) 9w0 ( ) and so 9 MCSb(6) Aug (L6) is surjective.

Remark 6.11. The marked front projection for Ꮿ is constructed as follows. Let = q j1 ,..., q jl denote the resolved crossings of L6 satisfying (q) 1. Then Ꮿ has a handleslide mark just to the left of each of the crossings q j1 ,..., q jl in 6; see Figure 21. Each handleslide mark begins and ends on the two strands involved in the crossing.

ch 6d. Deﬁning 9b : MCS[ b(6) → Aug (L6). We now prove Theorem 1.3.

ch Theorem 6.12. The map 9b : MCS[ b(6) → Aug (L6) deﬁned by 9(b [Ꮿ]) = 9(Ꮿ) is a well-deﬁned, surjective map.

Proof. Given Ꮿ1 ∼ Ꮿ2 we construct explicit chain homotopies between 9(Ꮿ1) and 9(Ꮿ2). We begin by restating the chain homotopy property 1 − 2 = H ◦ ∂ as a system of local equations. d d Let 1, 2 ∈ Aug(L6), where L6 is a sufﬁciently dipped diagram of the Ng d resolution L6. Let Q denote the set of crossings of L6. By Lemma 4.2, a map H : Q → ޚ2 that has support on the crossings of grading −1 may be extended by d linearity and the derivation product property to a map H : (Ꮽ(L6), ∂) → ޚ2 with d support on Ꮽ−1(L6). Then H is a chain homotopy between 1 and 2 if and only if 1 − 2 = H ◦ ∂ on Q.

d d Lemma 6.13. Let 1, 2 ∈ Aug(L6), where L6 is a sufﬁciently dipped diagram of d the Ng resolution L6. Suppose the linear map H : (Ꮽ(L6), ∂) → ޚ2 satisﬁes the derivation product property and has support on crossings of grading −1. Then H is a chain homotopy between 1 and 2 if and only if

(1) 1 − 2 = H ◦ ∂ for all qr and zs, −1 (2) 1(A j ) = (I + H(A j ))2(A j )(I + H(A j )) for all A j , ˜ (3) 1(B j ) = (I + 1(B j ))H(A j ) + H(A j−1)(I + 2(B j )) ˜ + 2(B j ) + H(B j )2(A j ) + 1(A j−1)H(B j ) for all B j . CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 117

2 Proof. Since ∂ A j = A j for each A j , the derivation property of H implies that 1 −2(A j ) = H ◦∂(A j ) if and only if 1 −2(A j ) = H(A j )2(A j )+1(A j )H(A j ). By rearranging terms, we obtain part (2). ˜ Since ∂ B j = (I + B j )A j + A j−1(I + B j ) for each B j , the derivation property of H implies 1 − 2(B j ) = H ◦ ∂(B j ) is equivalent to ˜ 1 − 2(B j ) = (I + 1(B j ))H(A j ) + H(A j−1)(I + 2(B j ))+ ˜ + H(B j )2(A j ) + 1(A j−1)H(B j ).

By rearranging terms, we obtain part (3). We will use the matrix equations of Lemma 6.13 to show that 9 maps equivalent MCSs to the same augmentation class. We will restrict our attention to occ-simple augmentations and in nearly every situation build chain homotopies that satisfy H(B j ) = 0 for all j. These added assumptions simplify the task of constructing chain homotopies. The next corollary follows immediately from Lemma 6.13 and Deﬁnition 6.2. d Corollary 6.14. Suppose L6, 1, 2, and H are as in Lemma 6.13. If 1 and 2 are occ-simple and H(B j ) = 0 for all j, then H is a chain homotopy between 1 and 2 if and only if

(1) 1 − 2 = H ◦ ∂ for all qr and zs, −1 (2) 1(A j ) = (I + H(A j ))2(A j )(I + H(A j )) for all A j ,

(3) 1(B j ) + (I + 1(B j ))H(A j ) = H(A j−1)(I + 2(B j )) + 2(B j ) for B j with I j of type (1), ˜ (4) H(A j ) = H(A j−1) for B j with I j of type (2), (3), or (4). ch We can now prove that 9b : MCS[ b(6) → Aug (L6) deﬁned by 9(b [Ꮿ]) = 9(Ꮿ) is well-deﬁned.

Lemma 6.15. If Ꮿ1 and Ꮿ2 are equivalent, then 9(Ꮿ1) = 9(Ꮿ2).

Proof. It is sufﬁcient to suppose Ꮿ1 and Ꮿ2 differ by a single MCS move. Let ∈ d1 ∈ d2 Ꮿ1 Aug(L6 ) and Ꮿ2 Aug(L6 ) be as deﬁned in Lemma 6.4. Since each MCS d1 d2 move is local, we may assume L6 and L6 have identical dips outside of the region of the MCS move.

We compare the chain homotopy classes of Ꮿ1 and Ꮿ2 by extending Ꮿ1 and d Ꮿ2 to augmentations on a third dipped diagram L6. In the case of MCS moves d d1 d2 1–16, we form L6 by adding 0, 1, or 2 additional dips to L6 and L6 . MCS move 17 may require the addition of many more dips. The dotted lines in Figures 22, 23 d1 d2 and 24 indicate the locations of dips in L6 and L6 and the additional dips needed d to form L6. The index j denotes the dip D j . Table 1 indicates which dotted lines d in Figures 22 and 23 represent new dips added to form L6. 118 MICHAEL B. HENRY

1 2

j j j j 3 4

j j j j 5 6

j j j j 7 8

j j j j 9 10

j j j j

Figure 22. MCS equivalence moves 1–10 with dip locations indicated.

11 12

j j j j 13 14

j j j j 15 16

j j j j Figure 23. MCS equivalence moves 11–16 with dip locations indicated.

j j

Figure 24. MCS move 17 with dip locations indicated.

d1 d2 As we add dips to L6 and L6 , we extend Ꮿ1 and Ꮿ2 by 0. By Lemma 5.8, we can keep track of the extensions of Ꮿ1 and Ꮿ2 after each dip. In fact, the ˜ ˜ ∈ d extensions will be occ-simple. We let Ꮿ1 , Ꮿ2 Aug(L6) denote the extensions CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 119

d1 d2 Adding dips to L6 and L6 d1 d2 Move L6 L6 1 j, j + 1 – 6 j + 1 – 7–14 j + 2 j 15 – j 16 – j Table 1. Entries indicate the dotted lines in Figures 22 and 23 that d correspond to dips added to form L6.

of Ꮿ1 and Ꮿ2 . We use the matrix equations in Corollary 6.14 and Lemma 6.13 ˜ ˜ to construct chain homotopies between Ꮿ1 and Ꮿ2 . In particular, for all MCS moves except move 6 and 17, the chain homotopy H is given in Table 2. The chain homotopy H for MCS move 6 sends all of the crossings to 0 except in the case of A j−1 and A j , where = + + H(A j−1) 1(B j−1) 2(B j−1) 1(B j−1)e2(B j−1), H(A j ) = H(A j−1) + 2(B j−1) + H(A j−1)2(B j−1). The case of MCS move 17 is slightly more complicated. In particular, just to the left of the region of the MCS move, the chain complex (C j , ∂ j ) in Ꮿ1 and Ꮿ2 | | = | | + ··· has a pair of generators yl < yk such that yl yk 1. Let yu1 < yu2 < < yus h | i = ··· denote the generators of C j satisfying ∂y yk 1. Let yvr < < yv1 < yi denote the generators of C j appearing in ∂ j yl . MCS move 17 introduces the handleslide marks Ek,vr ,... Ek,v1 , Eu1,l ,..., Eus ,l . We assume that Ꮿ2 includes these marks d1 and Ꮿ1 does not. Then we introduce r + s dips to the right of D j in L6 . As we d1 add dips to L6 , we extend Ꮿ1 by 0. By Lemma 5.8 we can keep track of the ˜ ˜ extensions of Ꮿ1 after each dip. The chain homotopy H between Ꮿ1 and Ꮿ2 is ≤ ≤ + − = Pi defined as follows. For 1 i r s 1, we have H(A j+i ) k=1 Ꮿ2 (B j+k) and H(B j+r+s) = Ᏼk,l , and all of the other crossings are sent to 0. Since each H defined above has few nonzero entries, it is easy to check that H has support on generators with grading −1. Thus we need only check that the ˜ − ˜ = extension of H by linearity and the derivation product property satisfies Ꮿ1 Ꮿ2 H ◦ ∂. In the case of MCS moves 11–16, this is equivalent to checking that H solves the matrix equations in Corollary 6.14. In the case of MCS move 17, we must check that H solves the matrix equations in Lemma 6.13. We leave this task to the reader. The maps H given above and in Table 2 were constructed using the following ˜ ˜ process. The augmentations Ꮿ1 and Ꮿ2 are equal on crossings outside of the region 120 MICHAEL B. HENRY

Chain Homotopy H between 1 and 2

Move H(A j ) H(A j+1)

1 2(B j ) 0 2 1(B j ) + 2(B j ) 0 3 1(B j ) + 2(B j ) 0 4 1(B j ) + 2(B j ) 0 5 1(B j ) + 2(B j ) 0 7 1(B j ) 1(B j ) 8 1(B j ) 2(B j+2) 9 1(B j ) 2(B j+2) 10 1(B j ) 1(B j ) 11 1(B j ) 2(B j+2) 12 1(B j ) 2(B j+2) 13 1(B j ) 2(B j+2) 14 1(B j ) 2(B j+2) 15 1(B j ) 0 16 1(B j ) 0 Table 2. In all cases, H = 0 on all other crossings.

˜ − ˜ = = of the MCS move; thus Ꮿ1 Ꮿ2 0 on these crossings. Hence, H 0 satisfies ˜ − ˜ = ◦ Ꮿ1 Ꮿ2 H ∂ on these crossings. Within the region of the MCS moves, we can ˜ ˜ ˜ = ˜ write down explicit matrix equations relating Ꮿ1 and Ꮿ2 . In particular, Ꮿ1 Ꮿ2 ˜ ˜ to the left of the MCS move, and within the region of the move, Ꮿ1 and Ꮿ2 are related by the matrix equations in Lemma 6.3. From these equations, we are able to define H. [˜ ] = [˜ ] = We now show that Ꮿ1 Ꮿ2 implies 9(Ꮿ1) 9(Ꮿ2). Let v denote any d dipping/undipping path from L6 to L6. By Lemma 6.9, the definition of 9 : ch MCSb(6) → Aug (L6) is independent of dipping/undipping paths. We calculate d1 d2 9(Ꮿ1) (respectively 9(Ꮿ2)) using the path that travels from L6 (respectively L6 ) d to L6 by adding dips as specified above to create L6 and then traveling along v. ˜ ˜ ˜ ' ˜ The first segment of this path maps Ꮿ1 to Ꮿ1 , and Ꮿ2 to Ꮿ2 . Since Ꮿ1 Ꮿ2 , the ˜ ˜ stable-tame isomorphism associated to the path v maps Ꮿ1 and Ꮿ2 to the same chain homotopy class. Thus, 9(Ꮿ1) = 9(Ꮿ1), as desired.

Finally, note that Lemma 6.15 and Lemma 6.10 imply that 9b : MCS[ b(6) → ch Aug (L6), [Ꮿ] 7→ 9(Ꮿ) is a well-deﬁned, surjective map. Thus we have proved Theorem 6.12. 6e. Two standard forms for MCSs. An MCS Ꮿ encodes both a graded normal d ruling NᏯ on 6 and an augmentation Ꮿ on L6. In this section, we formulate two CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 121

Figure 25. Sweeping a handleslide mark into V from the left. The grey marks are those contained in the original V . algorithms using MCS moves that highlight these connections. When combined the algorithms provide a map from MCSb(6) to Aug(L6) that, when passed to equivalence classes, corresponds to 9b. The upside is that this map does not require dipped diagrams or keeping track of chain homotopy classes of augmentations. We are also able to reprove the many-to-one correspondence between augmentations and graded normal rulings found in [Ng and Sabloff 2006]. 6e1. Sweeping collections of handleslide marks. The algorithms we deﬁne sweep handleslide marks from left to right in 6. During this process, we let V denote the collection of handleslide marks being swept. Suppose V sits away from the crossings and cusps of 6. Near V , label the strands of 6 from bottom to top with the integers 1,..., n. We deﬁne vk,l ∈ ޚ2 to be 1 if and only if the collection V includes a handleslide mark between strands k and l, k > l. We abuse notation slightly by allowing vk,l to denote the handleslide mark as well. We order the 0 marks in V as follows. If k < k and vk,l = vk0,l0 = 1, then vk0,l0 appears to the 0 left of vk,l in V . If l < l and vk,l = vk,l0 = 1, then vk,l0 appears to the left of vk,l in V . The following moves are used in the SR and A-algorithms and describe interactions between V and handleslide marks, cusps, and crossings. Move I (sweeping a handleslide from the left of V into/out of V ). Suppose a handleslide mark h sits to the left of V between strands k and l, with k > l. We use MCS moves 2–6 to commute h past the handleslides in V . In order to commute h past a handleslide of the form vl,i , we must use MCS move 6, and thus create a new handleslide mark h0 between strands k and i. The ordering on V allows us to commute h0 to the right so that it becomes properly ordered with the other marks 0 in V . If vk,i = 1, then h cancels the handleslide vk,i by MCS move 1. We continue this process of commuting h past each of the strands that begin on strand l. Once 122 MICHAEL B. HENRY

Figure 26. Simple switches. we have done so, we can commute h past the other marks in V , without introducing new marks, until it becomes properly ordered in V ; see Figure 25. Since each of MCS moves 1–6 is reversible, we are able to sweep an existing handleslide mark vk,l out of V to the left using the same process. Move II (sweeping a handleslide from the right of V into/out of V ). The process to sweep a handleslide from the right of V into/out of V is analogous to the process in move I. Move III (sweeping V past a crossing q between strands i + 1 and i assuming vi+1,i = 0). Sweep all of the handleslides of V past q using MCS moves 7–10, and if necessary, reorder the handleslide marks so that V is properly ordered. Move IV (sweeping V past a right cusp between strands i +1 and i assuming there are no handleslide marks in V beginning or ending on i +1 or i). The handleslides in V sweep past the right cusp using MCS moves 12 and 14. V stays properly ordered during this process. 6e2. The SR-form of an MCS. Given an MCS Ꮿ, the SR-algorithm results in an MCS with handleslide marks near switched crossings and some graded returns of NᏯ, and away from these crossings, the chain complexes are in simple form. The resulting MCS is called the SR-form of Ꮿ. The SR-form was inspired by discussions at the September 2008 AIM workshop, the work of Fuchs and Rutherford [2008] and the work of Ng and Sabloff [2006].

Deﬁnition 6.16. Given an MCS Ꮿ, we say a switch or return in NᏯ is simple if the ordered chain complexes of Ꮿ are in simple form before and after the switch or return, and in a neighborhood of the crossing the MCS is arranged as in Figures 26 or 27. A simple return is marked if it corresponds to one of the three arrangements in the top row of Figure 27. An MCS is in SR-form if all of its switched crossings and graded returns are simple, and besides the handleslides near simple switches and returns, no other handleslides appear in Ꮿ

Theorem 6.17. Every Ꮿ ∈ MCSb(6) is equivalent to an MCS in SR-form. CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 123

Figure 27. Simple returns.

Proof. We define an algorithm to sweep handleslide marks of Ꮿ from left to right in the front projection, beginning at the leftmost cusp and working to the right. As we sweep handleslides marks to the right, we ensure that the MCS we leave behind is in SR-form. By definition, the first chain complex of Ꮿ is in simple form and so the collection V begins empty. We consider each case of V encountering a handleslide mark, cusp, or crossing. In the case of a handleslide mark, we incorporate the mark into V using move II. In the case of a left cusp, we sweep V past the cusp using MCS moves 11 and 13. In either of these cases, if the MCS is in SR-form to the left of V before the handleslide mark or left cusp, then it is also in SR-form after we sweep past. Moves III and IV ensure that in certain situations we may sweep V past a crossing or right cusp so that the resulting MCS is in SR-form to the left of the singularity. Suppose we arrive at a crossing q between strands i + 1 and i and vi+1,i = 1. We use the following algorithm to ensure that q is a simple switch or simple return after we push V past q. We sweep the handleslide vi+1,i to the left of V using move I. We let h denote this handleslide. Now vi+1,i = 0 and so we sweep V past q using move III. If we suppose the ordered chain complex of Ꮿ is in simple form just before h, then around h, Ꮿ looks like one of the 6 cases in Figure 28. The top three cases indicate that q is a switch and the bottom three cases indicate that q is a return.

Switches: If q is a switch, we use MCS move 1 to introduce two handleslides just to the right of the crossing between strands i + 1 and i. If q is a switch of type (2) or (3), we also introduce two handleslides between the companion strands of strands i + 1 and i. Finally, we move the righthand mark of each new pair of marks into V using move I. The resulting MCS now has a simple switch at q; see Figure 26. 124 MICHAEL B. HENRY

Figure 28. Sweeping V past a crossing with a handleslide immediately to the left. The top row will be switches and the bottom will be marked returns. In each row, we label the cases (1), (2) and (3) from left to right.

Returns: If q is a return of type (1), then q is a simple return as in Figure 27. If q is a return of type (2) or (3), we use MCS move 1 to introduce two handleslides just to the right of the crossing between the companion strands of strands i + 1 and i and use move I to sweep the right-hand mark into V . The resulting MCS now has a simple return at q; see Figure 27. Suppose V arrives at a right cusp between strands i + 1 and i and there are handleslide marks in V beginning or ending on i + 1 or i. Suppose also that the MCS is in SR-form to the left of V . By applying move I possibly many times, we sweep the marks that end on strand i, begin on strand i + 1, begin on strand i, or end on strand i +1 out of V . The order in which these moves occur corresponds to the order given in the previous sentence. The Maslov potentials on strands i +1 and i differ by one; thus vi+1,i = 0. As a consequence, these moves do not introduce new handleslides ending or beginning on i +1 or i. Next we use move IV to sweep V past the right cusp. Now we remove the marks that have accumulated at the right cusp. We remove the marks ending on i + 1 and beginning on i using MCS moves 15 and 16. Let h1,..., hn denote the handleslides ending on strand i, ordered from left to right, and let g1,..., gm denote the handleslides beginning on strand i + 1. By assumption, the MCS is simple just to the left of h1. Thus just before and after h1, the pairing and MCS must look like one of the three cases in Figure 29(a). If h1 is of type 1 or 2, we can eliminate it using MCS move 17. Suppose h1 is of type 3, h1 begins on strand l, and generator l is paired with generator k in the ordered chain complex just before h. Introduce a new handleslide, denoted h0, between + 0 strands k and i 1 using MCS move 15. Move h1 past each of g1,..., gm using CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 125

(a) (b) Figure 29. Left: the 3 possible local neighborhoods of the handleslide h1. Right: MCS move 17 at a right cusp after introducing a new handleslide mark. In both (a) and (b), the two dark lines correspond to the strands entering the right cusp.

MCS move 6. Each time we perform such a move, we create a new handleslide mark which is then swept past the cusp using MCS move 12 and incorporated into 0 V using move I. Commute h1 past h2,..., hm using MCS moves 2–4. Now h1 0 and h1 look like Figure 29(b) and we can remove both using MCS move 17. Using this procedure, we eliminate all of h1,..., hn. The argument to eliminate g1,..., gm is essentially identical. Either we can eliminate g1 using MCS move 17 or we can push on a handleslide using MCS move 15 and then eliminate the pair with MCS move 17. After eliminating h1,..., hn and g1,..., gm, the MCS is in simple form just before and just after the right cusp. Hence, the resulting MCS is in SR-form to the left of V . As we carry out this algorithm from left to right, each time we encounter a handleslide, crossing or cusp we are able to sweep V past so that the MCS we leave behind is in SR-form. Corollary 6.18. Let N be a graded normal ruling on 6 with switched crossings q1,..., qn and graded returns p1,..., pm. Then N is the graded normal ruling of 2m MCSs in SR-form. Hence, N is the graded normal ruling of at most 2m MCS classes in MCS[ b(6). Proof. By Lemmas 3.14 and 3.15, each MCS equivalence class has an associated graded normal ruling, and by Theorem 6.17, each MCS equivalence class has at least one representative in SR-form. Thus, it is sufﬁcient to show there are exactly 2m MCSs in SR-form with graded normal ruling N. The set of marked returns of any MCS in SR-form with graded normal ruling N gives a subset of {p1,..., pm}. Conversely, by adding handleslide marks to 6 as dictated by N, Figure 26, and Figure 27, any R ⊂ {p1,..., pm} will determine an MCS in SR-form with graded normal ruling N and marked returns corresponding to R. 6e3. The A-form of an MCS. The A-algorithm is similar to the sweeping process in the SR-algorithm. However, we no longer introduce new handleslide marks after 126 MICHAEL B. HENRY crossings. The resulting MCS, called the A-form, has marks to the left of some graded crossings and nowhere else. The set of MCSs in A-form are in bijection with the augmentations on L6. In the algorithm, we still have to address the issue of mark accumulating at right cusps. If we begin with an MCS in SR-form, then this is easy to do.

Deﬁnition 6.19. An MCS Ꮿ ∈ MCSb(6) is in A-form if (1) outside a small neighborhood of the crossings of 6, Ꮿ has no handleslide marks; and (2) within a small neighborhood of a crossing q, either Ꮿ has no handleslide marks or it has a single handleslide mark to the left of q between the strands crossing at q. The left ﬁgure in Figure 21 is in A-form.

Theorem 6.20. Every Ꮿ ∈ MCSb(6) is equivalent to an MCS in A-form.

Proof. Let Ꮿ ∈ MCSb(6) and use the SR-algorithm to put Ꮿ in SR-form. The A-algorithm sweeps handleslide marks from left to right in the marked front of Ꮿ. As in the SR-algorithm, we will use V to keep track of handleslide marks and the moves I - IV to sweep V past handleslides, crossings and cusps. In the case of handleslides and left cusps, the A-algorithm works the same as in the SR- algorithm. Suppose V arrives at a crossing q between strands i + 1 and i. If vi+1,i = 0, then we sweep V past q as described in move III. If vi+1,i = 1, then we sweep the handleslide vi+1,i to the left of V and then sweep V past q using move III. Suppose V arrives at a right cusp q between strands i+1 and i. If no handleslides begin or end on i + 1 or i, then we sweep V past the right cusp using move IV. Otherwise, we sweep the handleslides that end on strand i, begin on strand i + 1, begin on strand i, or end on strand i + 1 out of V to the right using move II. The order in which these moves occur corresponds to the order given in the previous sentence. Let h1,..., hn denote the handleslides ending on strand i, ordered from right to left, and let g1,..., gm denote the handleslides beginning on strand i + 1, also ordered from right to left. Since the MCS is in SR-form to the right of h1, we know that the chain complex between h1 and the right cusp is simple. Thus just before and after h1, the pairing and MCS must look like one of the three cases in Figure 30(a). If h1 is of type (1) or (2) in Figure 30(a), we can eliminate it using MCS move 17. Suppose h1 is of type (3), h1 begins on strand l, and in the ordered chain complex just after h1, generator l is paired with generator k. Use MCS move 17 to introduce two new handleslides to the left of h1; see Figure 30(b). One new handleslide is between strands l and i. We remove h1 and this new handleslide using MCS move 1. The second handleslide is between strands k and i +1 and can CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 127

(a) (b) Figure 30. Left: the 3 possible local neighborhoods of the handleslide h1. Right: MCS move 17 at a right cusp introducing two new handleslides. The two dark lines correspond to the strands entering the right cusp.

be removed using MCS move 15. We eliminate all of h1,..., hn using this process. The argument to eliminate g1,..., gm is essentially identical. After eliminating h1,..., hn and g1,..., gm, we use MCS move 15 and 16 to remove all of the handleslides ending on i + 1 and beginning on i. We sweep V past the right cusp using move IV and continue to the right. As we progress from left to right in 6, marks remain to the immediate left of some graded crossings. Hence, after sweeping V past the rightmost cusp, we are left with an MCS in A-form.

The construction in the proof of Lemma 6.10 assigns an MCS Ꮿ to an augmentation ∈ Aug(L6). The MCS Ꮿ is in A-form. In fact, the crossings of Ꮿ with handleslide marks to their immediate left correspond to the resolved crossings in L6 that are augmented by . This process is invertible. Suppose Ꮿ is in A-form with handleslide marks to the immediate left of crossings p1,..., pk. From Lemma 6.4, we have an associated augmentation Ꮿ in a sufﬁciently dipped diagram with dips D1,..., Dm. We may undip D1,..., Dm, beginning with Dm and working to the left, so that the resulting augmentation on L6 only augments the crossings corresponding to p1,..., pk. As a result, we have the following corollary. Corollary 6.21. The set of MCSs in A-form are in bijection with the augmentations on L6, and if Ꮿ ∈ MCSb(6) is in A-form, then 9(Ꮿ) = [], where (p) = 1 if and only if p is a marked crossing in Ꮿ. The following corollary uses the SR-form and the A-algorithm to reprove the many-to-one relationship between augmentations and graded normal ruling ﬁrst noted in [Ng and Sabloff 2006]. Corollary 6.22. Let N be a graded normal ruling on 6 with switched crossings m q1,..., qn and graded returns p1,..., pm. Then the 2 MCSs in SR-form with m graded normal ruling N correspond to 2 different augmentations on L6. 128 MICHAEL B. HENRY

Proof. By Corollary 6.18, N corresponds to 2m MCSs in SR-form. In fact, two MCSs Ꮿ1 and Ꮿ2 in SR-form corresponding to N differ only by handleslide marks around graded returns. If Ꮿ1 and Ꮿ2 differ at the return p, then after applying the 0 0 A-algorithm, the resulting MCSs Ꮿ1 and Ꮿ2 differ at p as well. Thus the augmentations on L6 corresponding to Ꮿ1 and Ꮿ2 will differ on the resolved crossing corresponding to p.

7. Two-bridge Legendrian knots

In this section, we prove that in the case of front projections with two left cusps, ch the map 9b : MCS[ b(6) → Aug (L6) is bijective. Definition 7.1. A front 6 of a Legendrian knot K with exactly 2 left cusps is called a 2-bridge front projection. In [2001], Ng proves that every smooth knot admitting a 2-bridge knot projection is smoothly isotopic to a Legendrian knot admitting a 2-bridge front projection. Thus, the following results apply to an infinite collection of Legendrian knots. Definition 7.2. Given a graded normal ruling N on a front 6, we say two crossings qi < q j of 6, ordered by the x-axis, form a departure-return pair (qi , q j ) if N has a departure at qi and a return at q j and the two ruling disks that depart at qi are the same disks that return at q j . A departure-return pair (qi , q j ) is graded if both crossings have grading 0. There are five possible arrangements of qi and q j in a graded departure-return pair; see, for example, Figure 31(a). We let ν(N) denote the number of graded departure- return pairs of N. For a given fixed graded normal ruling N on any front projection 6, each unswitched crossing is either a departure or a return and thus part of a departure- return pair. In the case of a 2-bridge front projection, we can say more. Proposition 7.3. Suppose 6 is a 2-bridge front projection with graded normal ruling N. For each departure-return pair (qi , q j ) of N, no crossings or cusps of 6 may appear between qi and q j . In terms of the ordering of crossings by the x-axis, this says j = i + 1. Proof. Since 6 is a 2-bridge front projection, there are only two ruling disks for N. A switch or right cusp cannot appear after a departure since the two ruling disks overlap. Thus a return must immediately follow a departure.

Deﬁnition 7.4. An MCS Ꮿ ∈ MCSb(6) is in SRg-form if Ꮿ is in SR-form and each marked return is part of a graded departure-return pair.

Lemma 7.5. If 6 is a 2-bridge front projection, every MCS class in MCS[ b(6) has a representative in SRg-form. CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 129

(a) (b) Figure 31. Left: a graded departure-return pair. Right: unmark- ing a graded return paired with an ungraded departure.

Proof. Let [Ꮿ]∈MCS[ b(6), and let Ꮿ be an SR-form representative of [Ꮿ]. Suppose (qi , qi+1) is a departure-return pair of NᏯ such that qi+1 is a marked graded return with an ungraded departure qi . We can push the handleslide mark(s) at qi+1 to the left, past the ungraded departure qi ; see Figure 31(b). Using MCS move 17, we may remove these handleslide mark(s). In this manner, we eliminate all of the handleslide marks at graded returns that are paired with ungraded departures. Thus, Ꮿ is equivalent to an MCS in SRg-form.

In fact, the SRg-form found in the previous proof is unique.

Lemma 7.6. If 6 is a 2-bridge front projection, every MCS class in MCS[ b(6) has P ν(N) a unique representative in SRg-form. Thus |MCS[ b(6)| = N∈N(6) 2 .

Proof. Let Ꮿ1 and Ꮿ2 be two representatives of [Ꮿ] in SRg-form. Since Ꮿ1 ∼ Ꮿ2, Ꮿ1 and Ꮿ2 induce the same graded normal ruling on 6, which we will denote N. Thus, Ꮿ1 and Ꮿ2 have the same handleslide marks around switches and only differ on their marked returns. Suppose for contradiction that Ꮿ1 and Ꮿ2 differ at the graded departure-return pair (qi , qi+1). We assume qi+1 is a marked return in Ꮿ1 and is unmarked in Ꮿ2. We will prove that 9(Ꮿ1) 6= 9(Ꮿ2). Thus, by Lemma 6.15 we will have the desired contradiction.

Recall that 9(Ꮿ1) and 9(Ꮿ2) are computed by mapping the augmentations Ꮿ1 and Ꮿ2 constructed in Lemma 6.4 to augmentations in Aug(L6) using a dipping/undipping path. The augmentations Ꮿ1 and Ꮿ2 occur on sufficiently dipped d1 d2 d1 d2 diagrams L6 and L6 . The dipped diagrams L6 and L6 differ by one or two dips d2 between the resolved crossings qi and qi+1. Add these dips to L6 and extend Ꮿ2 ˜ d1 by 0 using Lemma 5.8. We let Ꮿ2 denote the resulting augmentation on L6 . From Lemma 6.9, the definition of 9 is independent of dipping/undipping paths. = ˜ Thus 9(Ꮿ1) 9(Ꮿ2) if and only if Ꮿ1 and Ꮿ2 are chain homotopic as augmen- d1 tations on L6 . We will show that they are not chain homotopic. Suppose for : d1 → ˜ contradiction that H (Ꮽ(L6 ), ∂) ޚ2 is a chain homotopy between Ꮿ1 and Ꮿ2 . We will prove a contradiction exists for two of the five possible arrangements of a graded departure-return pair. The arguments for the remaining three cases are essentially identical. Case 1: Suppose the graded departure-return pair is arranged as in Figure 32(a). d1 The dotted lines there indicate the location of the dips in L6 . Let k + 1 and k 130 MICHAEL B. HENRY

j j j j (a) (b)

Figure 32. In both (a) and (b), the MCS Ꮿ1 is on the left and Ꮿ2 is on the right. denote the strands crossing at qi+1. The following calculations use the formulas from Lemma 5.5 and the fact that all of the chain complexes involved are simple in the sense of Deﬁnition 3.10 or are only one handleslide away from being simple. The chain homotopy H must satisfy k+1,k = = − ˜ = (i) H(a j ) H(∂qi+1) Ꮿ1 Ꮿ2 (qi+1) 0, k+1,k + k+1,k = k+1,k = − ˜ k+1,k = (ii) H(a j ) H(a j−1 ) H(∂b j ) Ꮿ1 Ꮿ2 (b j ) 1, and k+1,k = k+1,k = − ˜ k+1,k = (iii) H(a j−1 ) H(∂b j−1 ) Ꮿ1 Ꮿ2 (b j−1 ) 0. k+1,k = Combining (i) and (ii), we see that H(a j−1 ) 1, but this contradicts (iii). Thus ˜ Ꮿ1 and Ꮿ2 are not chain homotopic. Case 2: Suppose as in Case 1 that the graded departure-return pair is arranged as in Figure 32(b), with the dotted lines as before. Let k + 1 and k denote the strands crossing at qi , and let l + 1 and l denote the strands crossing at qi+1. The chain homotopy H must satisfy l+1,l = = − ˜ = (i) H(a j ) H(∂qi+1) Ꮿ1 Ꮿ2 (qi+1) 0, k+1,k = k+1,l = − ˜ k+1,l = (ii) H(a j ) H(∂a j ) Ꮿ1 Ꮿ2 (a j ) 0, k+1,k + k+1,k = k+1,k = − ˜ k+1,k = (iii) H(a j ) H(a j−1 ) H(∂b j ) Ꮿ1 Ꮿ2 (b j ) 0, k+1,k + k+1,k = k+1,k = − ˜ k+1,k = (iv) H(a j−1 ) H(a j−2 ) H(∂b j−1 ) Ꮿ1 Ꮿ2 (b j−1 ) 1, and k+1,k = k+1,k = − ˜ k+1,k = (v) H(a j−2 ) H(∂b j−2 ) Ꮿ1 Ꮿ2 (b j−2 ) 0. l+1,l k+1,l The second equation follows from H(a j ) = 0 and the formula for ∂a j . k+1,k = By combining (ii) and (iii), we see that H(a j−1 ) 0. Combining (iv) with k+1,k = k+1,k = H(a j−1 ) 0, we see that H(a j−2 ) 1. This contradicts (v). Thus Ꮿ1 and ˜ Ꮿ2 are not chain homotopic. Using these two lemmas, we prove Theorem 1.4. ch Theorem 7.7. If 6 is a 2-bridge front projection, then 9b :MCS[ b(6)→Aug (L6) is a bijection. Proof. The surjectivity of 9b is the content of Theorem 6.12. We need only show it is injective. Suppose 9(b [Ꮿ1]) = 9(b [Ꮿ2]) and let Ꮿ1 and Ꮿ2 be SRg-representatives CONNECTIONSBETWEENLEGENDRIANKNOTINVARIANTS 131 of [Ꮿ1] and [Ꮿ1]. Since each MCS class has a unique SRg-representative by Lemma 7.6, we need to show Ꮿ1 = Ꮿ2. This is equivalent to showing that Ꮿ1 and Ꮿ2 induce the same graded normal ruling on 6 and have the same marked returns.

Recall that 9(Ꮿ1) and 9(Ꮿ2) are computed by mapping the augmentations Ꮿ1 and Ꮿ2 constructed in Lemma 6.4 to augmentations in Aug(L6) using a dipping/undipping path. The augmentations Ꮿ1 and Ꮿ2 occur on sufﬁciently dipped d1 d2 d1 d2 diagrams L6 and L6 . We may add dips to L6 and L6 and extend by 0 using ˜ ˜ Lemma 5.8 so that the resulting augmentations Ꮿ1 and Ꮿ2 occur on the same d [ ] = [ ] ˜ ˜ sufﬁciently dipped diagram L6. Then 9(b Ꮿ1 ) 9(b Ꮿ2 ) implies Ꮿ1 and Ꮿ2 are d chain homotopic as augmentations on L6. : d → ˜ ˜ ˜ Let H (Ꮽ(L6), ∂) ޚ2 be a chain homotopy between Ꮿ1 and Ꮿ2 . Since Ꮿ1 ˜ and Ꮿ2 are occ-simple, Corollary 6.14 gives ˜ = + ˜ + −1 Ꮿ1 (A j ) (I H(A j ))Ꮿ2 (I H(A j )) for all j. ˜ ˜ Recall that Ꮿ1 (A j ) and Ꮿ2 (A j ) encode the differential of a chain complex in ˜ ˜ Ꮿ1 and Ꮿ2, respectively. Since Ꮿ1 (A j ) and Ꮿ2 (A j ) are chain isomorphic by a ˜ lower triangular matrix, the pairing of the strands of 6 determined by Ꮿ1 (A j ) and ˜ Ꮿ2 (A j ) agree. Thus, Ꮿ1 and Ꮿ2 determine the same graded normal ruling on 6. In the proof of Lemma 7.6 we show two MCSs in SRg-form determining the same graded normal ruling on 6 are mapped to chain homotopic augmentations ˜ ˜ only if they have the same marked returns. Since Ꮿ1 (A j ) and Ꮿ2 (A j ) are chain homotopic, this implies Ꮿ1 and Ꮿ2 have the same marked returns and thus Ꮿ1 = Ꮿ2 as desired. Corollary 7.8. If 6 is a 2-bridge front projection, then

ch X ν(N) |Aug (L6)| = |MCS[ b(6)| = 2 , N∈N(6) where ν(N) denotes the number of graded departure-return pairs of N. Corollary 7.8 corresponds to Corollary 1.5 in Section 1.

Acknowledgments

My graduate advisors, Rachel Roberts and Joshua Sabloff, provided invaluable guidance and advice as I completed this work. In addition, I thank Sergei Chmutov, Dmitry Fuchs, Victor Goryunov, Paul Melvin, Dan Rutherford, and Lisa Traynor for many fruitful discussions and the American Institute of Mathematics (AIM), which held the September 2008 workshop conference where many of these discussions took place. 132 MICHAEL B. HENRY

It was at the AIM workshop that I first learned of the possibility of extending graded normal rulings by including handleslide data. Petya Pushkar had outlined such a program in emails with Dmitry Fuchs in 2000. After the AIM workshop, Fuchs gave me copies of these emails together with more recent emails from Pushkar in 2008. Along with the work done at AIM, these notes directly influenced the definition of a Morse complex sequence and the equivalence relation on the set of MCSs given in this article. The results claimed by Pushkar in 2008 also directed my efforts. It is my understanding that proofs of the claims in the Pushkar emails have not yet appeared.

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Received November 9, 2009. Revised September 1, 2010.

MICHAEL B.HENRY DEPARTMENT OF MATHEMATICS THE UNIVERSITYOF TEXAS AT AUSTIN 1 UNIVERSITY STATION, C1200 AUSTIN, TX 78712 UNITED STATES [email protected] http://www.mbhenry.com

PACIFIC JOURNAL OF MATHEMATICS Vol. 249, No. 1, 2011

A FUNCTIONAL CALCULUS FOR UNBOUNDED GENERALIZED SCALAR OPERATORS ON BANACH SPACES

DRAGOLJUB KECKIˇ CAND´ ÐORÐE KRTINIC´

We deﬁne weak and ultraweak functional calculus and construct using the Fourier transform technique an ultraweak functional calculus for an unbounded k-tuple of commuting generalized scalar operators Tj acting on a Banach space X. This functional calculus comprises the functions from the subset

k ˆ R α ˆ Fα = f : ޒ → ރ | f is a measure such that ޒk (1 + |t| )d| f | < ∞ , α of Sobolev space W∞(ޒ), where α > 0. It also contains some unbounded functions. We also give examples and related results.

1. Introduction

A functional calculus for an operator A is an (in some sense) continuous homomorphism 8 : Ꮽ → Ꮾ, where Ꮽ is a topological algebra of functions and Ꮾ is a topological algebra of operators, such that 8(1) = e is the unit of the algebra Ꮾ and 8(x) = A, where x denotes the identity function. Also, it is expected that the algebra Ꮽ is generated by 1 and x, that is, that polynomials are dense in Ꮽ (in the topology of Ꮽ). A familiar example of a functional calculus is the analytic functional calculus on Banach algebras. It is the homomorphism 8 : Ᏼ → Ꮿ, where Ꮿ is a Banach algebra, A ∈ Ꮿ and Ᏼ is an algebra of functions holomorphic in some neighborhood of σ (A) (in the topology generated by uniform convergence on compact sets), deﬁned by Z 1 − 8( f ) = f (λ)(λe − A) 1dλ = f (A), 2πi 0 where 0 ⊂ Dom( f ) and σ (A) b int 0. This mapping is continuous in the natural way.

MSC2000: primary 46H25, 46H30, 47A60, 47L60; secondary 47L10, 47D06. Keywords: functional calculus, Fourier transform, unbounded operators. Supported by MNZZS grant number 174034, Serbia.

135 136 DRAGOLJUB KECKIˇ C´ AND ÐORÐE KRTINIC´

Another example of a functional calculus is described by the spectral theorem. Let Ᏼ be a Hilbert space and A ∈ B(Ᏼ) be a normal operator. Then Z ∞ L (σ (A)) 3 f → f (λ)d Eλ = f (A) σ (A) ∞ (Eλ is the spectral measure) defines a homomorphism 8 : L → B(Ᏼ) that is continuous in the norm, because k f k∞ = k f (A)k and because the pointwise convergence of fn to f implies fn(A) → f (A), strongly (“continuity”). From these examples, we can see that if a functional calculus is defined for a wide class of operators, then the algebra of functions is very small. Analogously, if the algebra of functions is wide, then we must have stronger conditions on the operators. In this note we will use the Fourier transform technique, which is not new and has been applied in many papers. McIntosh and Pryde [1987] developed the functional calculus for commuting n-tuples of bounded generalized scalar operators 1 R s and functions that have Fourier transform in Ls = { f | | f (t)|(1+|t| )dt < +∞}. A more general result was obtained by Andersson and Berndtsson [2002]. The case of a single generator of a bounded strongly continuous group was treated in [Balabane et al. 1993]. For more details and references, see [deLaubenfels 1994]. A similar idea of using Laplace transform for generators of bounded or polynomially bounded semigroups was used in [deLaubenfels and Jazar 1999] and [deLaubenfels 1995]. Another direction of development of functional calculi uses the Cauchy–Green integral formula applied to the almost holomorphic extension of a test function. The initial work of Dynkin [1972] was followed by the articles [Helffer and Sjostrand¨ 1989; Andersson 2003; Andersson and Sjostrand¨ 2004; Andersson et al. 2006]. The results obtained by this approach are mostly consequences of our results. In particular, Andersson [2003] showed that the functional calculus can be extended via the Cauchy–Green formula to the algebra of C∞ functions with an additional condition at infinity if and only if the corresponding group is polynomially bounded, although a wider class of operators admits functional calculus for test functions. Andersson, Samuelsson and Sandberg [2006] observed that for any proper f ∈ C∞(ޒm, ޒk), the set {g ◦ f | g ∈ Ᏸ(ޒk)} is contained in D(ޒm); they then used this fact to construct the functional calculus (with test functions) for f (A), where f is a proper C∞ function and A is a Helffer–Sjostrand¨ operator, that is, an operator with real spectrum and rationally bounded resolvent on compacts. After basic definitions in Section 2, including those of weak and ultraweak functional calculus, we introduce in Section 3 the algebra Fα and in Section 4 the algebra Fα,loc. In Section 4, we construct the functional calculus for the n- tuple of commuting generalized scalar operators and functions from Fα,loc. In AFUNCTIONALCALCULUSFORUNBOUNDEDOPERATORS 137

Section 5, we prove that the support of such functional calculus coincides with the spectrum of some operator closely related to the initial n-tuple and acting on the Clifford algebra derived from the initial Banach space. In Section 6, we compare the (ultra) weak functional calculus with other axiomatic approaches and ﬁnd it to be more comprehensive. In other words, for very wide algebra of functions we obtain a more comprehensive theory. As shown in [Andersson 2003], this class of operators is exactly the one obtained by using the Cauchy–Green integral formula. In Section 7, we discuss relationship between some distinct deﬁnitions of commutativity of unbounded operators. In Section 8, we give some examples and comments.

2. Basic deﬁnitions

k In this paper ޒ 3 t = (t1,..., tk) and T = (T1,..., Tk) are k-tuples of numbers and operators,√ respectively. We define t ·T := t1T1 +· · ·+tk Tk, the “inner product”, | | := 2 + · · · + 2 and t t1 tk , the Euclidean norm. The next definition is a modification of definition given in [Vasilescu 1982].

Definition 2.1. Let T = (T1,..., Tk) be a k-tuple of closed densely defined operators on a Banach space X. We call the ordered couple (Ꮽ, 8) a weak functional calculus based on ޒk if (1) Ꮽ is a normed algebra of functions defined on ޒk with a topology τ, such that (a) τ is weaker than the norm topology; (b) the addition is continuous with respect to τ; the multiplication is separately continuous with respect to τ; (c) the test functions Ᏸ are included and dense in Ꮽ; (d) embedding Ᏸ ,→ Ꮽ is continuous; (2) 8 : Ꮽ → B(X) is homomorphism, which is norm continuous and continuous with respect to τ and the weak topology on B(X);

(3) for every polynomial p(t1,..., tk) of degree m and for every sequence of the test functions ϑn tending to 1 in the natural way (increasingly and uniformly on compacts) we have 3 (8(ϑn p)x) → 3(p(T1,..., Tk)x) for all m m ∗ x ∈ TP Ᏸ(T 1 ····· T k ) and all 3 ∈ X . m j =m 1 k Remark 2.2. In the condition (1), we expect continuity of multiplication only by coordinates. It would be to much to expect full continuity. Indeed, in that case from condition (2) it would follow that if An → A and Bn → B weakly, then An Bn → AB weakly, which is not true. To extend the functional calculus to a wide set of unbounded functions, we need another condition, included in the following deﬁnition. 138 DRAGOLJUB KECKIˇ C´ AND ÐORÐE KRTINIC´

Definition 2.3. Let T = (T1,..., Tk) be a k-tuple of closed densely defined operators on a Banach space X. We call the ordered couple (Ꮽ, 8) an ultraweak functional calculus based on ޒk if it is a weak functional calculus and if the algebra Ꮽ satisfies the following condition: For an arbitrary test function 0 6 ϕ 6 1 equalling 1 in some neighborhood of zero, and for any f ∈ Ꮽ, there holds ϕ(x/n) f (x) → f (x) as n → ∞, in the topology of Ꮽ. We will prove the existence of an ultraweak functional calculus when Ꮽ is the subspace n Z o = : k → ˆ + | |α | ˆ| ∞ Fα f ޒ ރ f is a measure such that (1 t )d f < ޒ α k Pk of the Sobolev space W∞(ޒ ), where α = j=1 α j is a positive real number. We assume that all operators Tj are closed and densely defined, σ (Tj ) ⊆ ޒ and

(2-1) keitTj k = O(|t|α j ) as t → ∞.

Also, we assume that Tj are mutually commuting, which means that the corresponding groups eitTj and eisTk mutually commute for any s, t ∈ ޒ and any j, k. Remark 2.4. We should specify what is meant by eitTj , because the set ∞ \ X kT n xk x ∈ X | x ∈ Ᏸ(T n) ∧ < ∞ n! n>1 n=1 (the so-called analytical vectors) need not be dense in X, according to Nelson’s theorem. In this paper, eitT is a one-parameter group C(t), the solution of the abstract Cauchy problem d (2-2) C(t) = iT C(t), C(0) = I. dt One can prove the existence of solution of Equation (2-2) under different kinds of conditions (for example, the Hille–Yoshida conditions). Also, it is possible to give an estimate of asymptotic behavior of that solution; see [deLaubenfels 1991; 1993b; 1993a].

Remark 2.5. The condition σ (Tj ) ⊆ ޒ is actually superﬂous. Namely, Andersson and Berndtsson [2002, Lemma 1.2] showed that the condition (2-1) implies that the joint spectrum of the n-tuple of commuting bounded operators is a subset of ޒn. The proof presented in their Remark 1 is still valid if we omit their boundedness. However, many conditions that ensure the existence of the solution of the abstract Cauchy problem include the reality of spectrum. AFUNCTIONALCALCULUSFORUNBOUNDEDOPERATORS 139

Remark 2.6. One can define commutativity of unbounded operators in various ways. For instance, one can assume there is a dense subspace on which all products Ti Tj are well-defined and Ti Tj = Tj Ti . In the literature, one also finds that the commutativity of the pair of unbounded operators is defined as the commutativity of their resolvents. In Section 7, we compare different commutativity conditions. Remark 2.7. From the Hille–Yoshida theorem (see [Dunford and Schwartz 1958]) we can conclude that for a closed, densely defined operator A, with real spectrum −1 m such that k(A − λI ) k 6 1/|λ| for nonreal λ, Equation (2-2) has a solution satisfying

(2-3) kC(t)k = o(eεt ) for all ε > 0.

A natural question arises: Can we derive a polynomial estimate of C(t) from (2-3)? The following example shows that this is not possible. Example 2.8. There is an operator for which (2-1) does not hold, and its asymptotic behavior is weaker than exponential (that is, eit A = o(eε|t|) as |t| → ∞ for every ε > 0).

Let E be the vector√ space of entire functions such that there exists C ∈ ޒ such |z| that | f (z)| 6 Ce . One can easily see that E is the Banach space if the norm k f k of a function f is deﬁned as the smallest√ constant C, for which the inequality |z| above holds (that is, k f k = supz∈ރ| f (z)|/e ). Since Z 0 1 f (ξ + z) f (z) = dξ, 2 2πi |ξ|=1 ξ it follows that operator A = −id/dz is a bounded linear operator on E. Hence

(eit A f )(z) = (etd/dz f )(z) = f (z + t).

Since √ it A |z+t| √ √ |(e f )(z)| | f (z + t)| e |z|+|t|− |z| √ = √ · √ 6 k f k · e for every f ∈ E, e z e |z+t| e |z| √ k it Ak = |t| | | → ∞ = P∞ n ! it follows that e O(e ) as t √. Let f0(z) 2 n=0 z /(2n) . From it A |t| the above it follows that ke f0k/k f0k 6 e . Since for x, t > 0, we have ∞ P + n ! √ it A 2 (x t) /(2n) (e f0)(x) n=0 2 ch( x + t) √ = √ = √ e x e x e x √ √ √ √ = e x+t− x + e− x+t− x , √ √ it A |t| it A |t| it follows that ke f0k ∼ e as t → ∞; therefore ke f0k ∼ e as t → ∞. 140 DRAGOLJUB KECKIˇ C´ AND ÐORÐE KRTINIC´

3. The algebra Fα

k Deﬁnition 3.1. Let α be a positive real number, and let ᏹα(ޒ ) be the space of k R + | |α | | ∞ k k = all Borel measures on ޒ such that ޒk (1 t )d µ < with the norm µ α R + | |α | | = { ∈ ∞ k | ˆ ∈ } ޒk (1 t )d µ . Let Fα f L (ޒ ) f ᏹα be the normed space with the k k = k ˆk norm f Fα f ᏹα . = ˇ k · k Since Fα ᏹα, it follows that Fα is the Banach space with the norm Fα ; that is Z 1 · 1 |µ( ˇ x)| = · eix t dµ(t) · kµk < ∞. k/2 6 k/2 ᏹα (2π) ޒk (2π) ∗ k Let τ be the topology on Fα generated by the weak topology on ᏹα(ޒ ) ⊆ k ∗ k (Cb,α(ޒ )) , where Cb,α(ޒ ) is the space of continuous functions such that f (t) | |α | | → ∞ k k = { | | | · + | |α } is O( t ) as t , with the norm f Cb,α inf M f (t) 6 M (1 t ) . It means that the subbase of the neighborhoods of zero are Z = ∈ ˆ ∈ k Ꮾ(g, ε) f Fα gd f < ε for g Cb,α(ޒ ) and ε > 0; ޒk see [Rudin 1973, Chapter 1].

Remark 3.2. Andersson and Berndtsson [2002] studied more general algebras Ꮽh, where h : ޒk → ޒ is a positive, continuous, subadditive function that is increasing on the rays from origin, h(0) = 0, and h(t) = o(|t|) as t → ∞. The algebra Fα is a α special case of Ꮽh, where h(x) = log(1+|x| ). The last function is not subadditive, but satisﬁes a somewhat weaker condition h(x + y) ≤ log 2α + h(x) + h(y).

Theorem 3.3. The space Fα is an algebra.

Proof. Fα is a linear space, obviously. Let us prove that f, g ∈ Fα implies f ·g ∈ Fα. The inequality

α α α α (3-1) 1 + |s + t| 6 2 (1 + |s| )(1 + |t| ) α α α α holds. Indeed, for |s| 6 |t| we have 1 + |s + t| 6 1 + (2|t|) 6 2 (1 + |t| ) 6 α α α ˆ 2 (1+|t| )(1+|s| ) and similarly for |s| > |t|. Therefore, if f , g ∈ Fα, then f and ˆ R +| |α | ˆ| +∞ R +| |α | ˆ| +∞ g are finite measures such that ޒk (1 t )d f < and ޒk (1 t )d g < . ˆ ˆ Also fd· g = f ∗g ˆ is well-defined and finite by the finiteness of f and gˆ, and we have Z Z Z α ˆ α ˆ (1 + |t| )d|f ∗g ˆ| 6 (1 + |s + t| )d|f |(s)d|g ˆ|(t) ޒk ޒk ޒk Z Z α α ˆ α 6 2 (1 + |s| )d|f |(s) (1 + |t| )d|g ˆ|(t) ޒk ޒk = αk k k k 2 f Fα g Fα . AFUNCTIONALCALCULUSFORUNBOUNDEDOPERATORS 141

Remark 3.4. Although such a space arises rarely in the literature, one can deﬁne α k the Sobolev space W∞ as the space of all functions f : ޒ → ޒ such that its fractional partial derivative (−1)α/2 f belongs to L∞. Here fractional derivative α/2 α (−1) is deﬁned via Fourier transform. However W∞ ⊃ Fα (that is, this inclusion is proper) since Fourier transforms of functions from Fα have always continuous fractional partial derivative.

Remark 3.5. One can, also, deﬁne the space F(α1,...,αn) consisting of all functions ˆ R + | |α1 · · · · · | |αn | ˆ| f whose Fourier transform f is a measure such that ޒk (1 t1 tn )d f ⊂ is ﬁnite. A simple application of the mean inequality shows that Fα F(α1,...,αn). From the following example we can see that this inclusion is strict.

Example 3.6. Let n = 2 and (α1, α2) = (1, 1). Then we have ={ |R +| | | ˆ| +∞} ={ |R + 2+ 2 ˆ +∞} F(1,1) f ޒ2 (1 xy )d f < and F2 f ޒ2 (1 x y )d f < . Consider the function ψ deﬁned as the inverse Fourier transform of the function 1 1 1 (x, y) 7→ · · . 1+|xy| 1+x2 +y2 log2(1+(x2 +y2)/2)

We claim that ψ ∈ F(1,1) \ F2. Indeed Z Z 1 1 (1 + |xy|)dψˆ = · dxdy < +∞, 2 2 2 2 2 ޒ2 ޒ2 1+x +y log (1+(x +y )/2) whereas Z Z 1 1 (1 + x2 + y2)dψˆ = · dxdy 2 2 2 ޒ2 ޒ2 1+|xy| log (1+(x +y )/2) Z 1 1 · dxdy > 1+|xy| 2 + 2 |x|>|y| log (1 x ) +∞ | | Z 1 Z x d|y| = 2 · dx 2 2 −∞ log (1+x ) 0 1+|xy| Z +∞ = dx = +∞ 2 2 . −∞ |x| log(1+x )

However, space F(α1,...,αn), although wider, need not be an algebra. 2 Example 3.7. Let n = 2 and (α1, α2) = (1, 1). The functions ϕ(x, y) = sin x and 2 ψ(x, y) = sin y are in F(1,1) since they are bounded and have bounded second order mixed derivatives ∂2ϕ/∂x∂y and ∂2ψ/∂x∂y. However, their product has no bounded second order mixed derivatives. Hence F(1,1) is not an algebra.

Theorem 3.8. The set Ᏸ is dense in the space Fα (with the topology τ) and the embedding Ᏸ ,→ Fα is continuous. 142 DRAGOLJUB KECKIˇ C´ AND ÐORÐE KRTINIC´

Proof. [−n n]k The sequence ( , )n>1 is an increasing sequence of sets and we have S k k k k n∈ގ[−n, n] = ޒ . Therefore |µ|([−n, n] ) → |µ|(ޒ ) as n → ∞ for µ ∈ ᏹα. k If g ∈ Cb,α(ޒ ) and M = supx∈ޒk |g(x)|, it follows that there is n ∈ ގ such that k k k |µ|(ޒ \[−n, n] ) < ε/M . Thus µˇ n ∈µ ˇ +Ꮾ(g, ε), where µn(E) = µ(E∩[−n, n] ) for every measurable set E. It follows that it is enough to prove the ﬁrst part of our statement for inverse images f ∈ Fα of functions with compact support. k Let µ be a measure with compact support. Let h j (x) = j h( jx) be an ap- k k proximative unit on ޒ (that is, a sequence (h j ) j∈ގ where h ∈ Ᏸ(ޒ ), h > 0 and R = ∗ ∈ ޒk h(x)dx 1; see [Rudin 1973, 6.31]). Then we have h j µ ᏿ (because α α D (h j ∗ µ) = (D h j ) ∗ µ, and since h j and µ have compact support, so does h j ∗ µ; here ᏿ is the Schwartz class). If g1(t) = g(−t), it follows that

(h j ∗ µ)(g) = ((h j ∗ µ) ∗ g1)(0)

= (h j ∗ (µ ∗ g1))(0) → δ ∗ (µ ∗ g1)(0) = µ ∗ g1(0) = µ(g) ∗ by [Rudin 1973, 6.30]. Therefore h j ∗ µ → µ as j → ∞ in the weak topology ∗ on ᏹα. Hence ᏿ is dense in the weak topology on ᏹα, so it is also dense in the ˆ space Fα with the topology τ, because S = S. Therefore, it is enough to prove that Ᏸ is dense in ᏿. Since Ᏸ is dense in ᏿ in the topology of the space ᏿,1 for ϕ ∈ ᏿ there exists → n → ∞ (ϕn)n>1 such that ϕn ϕ in ᏿ as . k 0 Let g ∈ Cb,α(ޒ ). Then gˆ ∈ ᏿ (see [Rudin 1973, 7.14(d)]) and therefore R ˆ − ˆ = R − ˆ → → → ∞ ޒk g(ϕn ϕ)dx ޒk (ϕn ϕ)dg 0. That means that ϕn ϕ as n in the topology τ. The last part of the statement is trivial for test functions with support in some ﬁxed compact set. Therefore by [Rudin 1973, Theorem 6.6], it is true for all Ᏸ.

Lemma 3.9. Pointwise addition and multiplication are continuous in (Fα, τ).

Proof. Let lim0 fγ = f and lim0 gγ = g in the topology of Fα, where 0 is a net. R ˆ → R ˆ R ˆ → R ˆ ∈ k Then ޒk h d fγ ޒk h d f and ޒk gdgγ ޒk h dg for arbitrary h Cb,α(ޒ ). Then we have Z Z Z Z Z Z ˆ ˆ h d( f\γ + gγ ) = h d fγ + h dgˆγ → h d f + h dgˆ = hd(\f + g) ޒk ޒk ޒk ޒk ޒk ޒk for γ ∈ 0. To prove the continuity of multiplication, it is enough to consider fγ → 0, and = { ∈ | |R ˆ| = } ﬁxed g. Let Ꮾh1,...,hn,ε1,...,εn ϕ Fα ޒk h j dϕ < ε j , j 1,..., n be an

1 Analogously to the procedure from the beginning of this proof, let (h j ) j∈ގ be an approximative unit, such that h(x) 6 1 and h ≡ 1 in some neighborhood of 0; then Ᏸ 3 h j ∗ (χ[−n,n]k · ϕ) → ϕ in the topology of ᏿ AFUNCTIONALCALCULUSFORUNBOUNDEDOPERATORS 143 arbitrary basic neighborhood of zero generated by h j ∈ Cb,α. Using inequality (3-1), it is easy to check that the functions Z ψ j (x) = h j (x + y)dgˆ(y) ޒk ∈ · ∈ also belong to Cb,α. If fγ Ꮾψ1,...,ψn,ε1,...,εn , then fγ g Ꮾh1,...,hn,ε1,...,εn . Indeed, Z ZZ Z ˆ ∗ ˆ = + ˆ ˆ = ˆ h j dfγ g h j (x y)dg(y)dfγ (x) ψ j dfγ < ε j . ޒk ޒk ×ޒk ޒk

In the general case, we have fγ · g − f g = ( fγ − f ) · g, and we can apply the special case.

From Theorems 3.3 and 3.8 and Lemma 3.9 it follows that the space (Fα, τ) (from Definition 3.1) satisfies condition (1) of Definition 2.1. Proposition 3.10. Let ϕ be a test function such that ϕ ≡ 1 on some neighborhood of zero, and such that 0 6 ϕ 6 1. Let χn(x) = ϕ(x/n). Then f χn → f in the topology of the space Fα (as n → ∞). R → R ˆ → ∞ ∈ k Proof. We need to prove ޒk gd df χn ޒk gd f as n for any g Cb,α(ޒ ). = R + ˆ ˆ → Let Fn(t, s) ޒk g(t s)dχn(s). Since χn(s) δ (the Dirac distribution) in 2 the topology of Fα as n → ∞, we have limn→∞ Fn(t, s) = g(t). Since Z Z Z Z Z ˆ ˆ ˆ gd df χn = gd(f ∗χ ˆn) = g(s + t)d f (t)dχˆn(s) = Fn(t, s)d f (t), ޒk ޒk ޒk ޒk ޒk k ˆ and χˆn(s) = n φ(ns) (see [Rudin 1973, Theorem 7.4]), our result follows from the dominated convergence theorem, because Z |Fn(t, s)| 6 |g(t + s)|d|χ( ˆ s)| ޒk Z k α ˆ 6 n (1 + |s + t| )dφ(ns) ޒk Z α α k α −α−n−1 6 2 (1 + |t| ) n (1 + |s| )(1 + |ns|) ds ޒk Z (1 + |u/n|α) = α( + |t|α) du 2 1 α+n+1 ޒk (1 + |u|) Z (1 + |u|α) α( + |t|α) du 6 2 1 α+n+1 ޒk (1 + |u|) α 6 const · (1 + |t| ).

2 χˆn(s) belongs to the Schwartz class ᏿ and tends to δ in the topology of ᏿; see Theorem 3.8. 144 DRAGOLJUB KECKIˇ C´ AND ÐORÐE KRTINIC´

k ˆ ˆ0 −m We used that g ∈ Cb,α(ޒ ), φ ∈ ᏿ (so |φ (s)| 6 const·(1+|s|) for any natural m), |u/n| 6 |u| for positive integer n, and inequality (3-1).

The preceding lemma asserts that the algebra Fα satisﬁes the condition from the deﬁnition of the ultraweak functional calculus.

4. The construction of the functional calculus

Following the technique from [McIntosh and Pryde 1987] we deﬁne the functional calculus using the Fourier transform, that is, by formula Z α ⊇ 3 7→ = = 1 · −it·T (4-1) W∞ Fα f 8( f ) f (T ) k/2 e dµfˆ, (2π) ޒk where the measure µfˆ is the Fourier transformation of the function f (namely, ˆ = −k/2 · R −its f (t) (2π) ޒk e f (s)ds can be divergent for some t, but this formula R ∞ + | |α | | ∞ deﬁnes a complex Borel measure µfˆ and we have −∞(1 t )d µfˆ < ). ˆ From now on, we shall abbreviate µfˆ to f .

Theorem 4.1. Suppose T = (T1,..., Tk) is a k-tuple of closed, densely deﬁned operators such that all eitTj exist as solutions of the corresponding abstract Cauchy problem (2-2), such that

keitTj k = O(|t|α j ) as t → ∞.

Also, we assume that Tj are mutually commuting, which means that the corresponding groups eitTj and eisTk mutually commute for any s, t ∈ ޒ and any j, k. P Let X be a reﬂexive Banach space and let α = j α j > 0. A homomorphism 8 : Fα → B(X) is deﬁned by formula (4-1). The integral in (4-1) exists as a weak integral and 8 is (τ, ω) continuous, where τ is the topology on Fα as the subspace of the dual space of Cb,α and ω is the weak topology on B(X). Proof. Let x ∈ X and 3 ∈ X ∗. Function t 7→ 3(eit·T x) is weakly continuous. Also, we have Z Z eit·T x d k k + |t|α kxkd| | k k · kxk · k f k ; 3( ) µfˆ 6 3 (1 ) µfˆ 6 3 α ޒk ޒk hence (4-1) exists as a weak integral. The mapping 8 is obviously linear; also, we have Z Z · = 1 · i(t+s)·T 8( f ) 8(g) k e dµfˆ(s)dµgˆ (t) (2π) ޒk ޒk Z Z = 1 · ir·T = 1 · ir·T = k/2 e dµfˆ∗g ˆ (r) k/2 e dµcf g(r) 8( f g), (2π) ޒk (2π) ޒk where r = s + t. AFUNCTIONALCALCULUSFORUNBOUNDEDOPERATORS 145

Finally, we need to prove the continuity. Let S3,x,ε = {A | |3(Ax)| < ε} be a subbase set in B(X), let f0 ∈ Fα, and let = −k/2 · R it·T ∈ ∈ A f (2π) ޒk e dµfˆ(t) B(X) for f Fα. Let Z 1 it·T V = f · 3(e x)dµ < ε . k/2 f[− f0 (2π) ޒk it·T Since t 7→ 3(e x) is continuous, V is a neighborhood in Fα. Since Z 1 · 3[(A − A )x] = · 3(eit T x)dµ (t) < ε, f f0 k/2 f[− f (2π) ޒk 0 + ⊆ + it follows that 8( f0 V ) A f0 S3,x,ε.

Theorem 4.2. Let p(t1, t2,..., tk) be a polynomial of degree m and let ϑn be a sequence of test functions tending to 1 increasingly and uniformly on compacts. m j w Then for all φ ∈ TP Dom Q T we have 8(ϑ p)φ −→ p(T , T ,..., T )φ m j =m j j n 1 2 k as n → ∞. Proof. Function t 7→ eit·T is strongly and weakly differentiable and satisﬁes (2-2). Let D j = −i∂/∂t j and let δ be the Dirac distribution. Since m ˆ m k/2 m m D j (1) = D j ((2π) δ) = (−1) tbj

(for all m ∈ ގ0) and ϑn has compact support, from the definition of the distribution and [Rudin 1973, 6.37], we have Z m1 ····· mk = 1 · m1 ····· mk it·T k/2 · T1 Tk φ k/2 T1 Tk e φ d((2π) δ) (2π) ޒk Z 1 m1 mk it·T = lim · D ····· D (e )φ dµ ˆ →∞ k/2 1 k (1∗ϑbn) n (2π) ޒk Z m1+···+mk 1 it·T = (−1) lim · e φ dµ m1 mk ˆ →∞ k/2 (D ·····D (1)∗ϑbn) n (2π) ޒk 1 k P Z m j 1 it·T = (−1) j lim · e φ dµ P →∞ k/2 − j m j m\1 ····· mk ∗ n (2π) ޒk ((( 1) t1 tk ) ϑbn) Z 1 · = lim · eit T φ dµ , →∞ k/2 m1 ·····\mk · n (2π) ޒk (t1 tk ϑn) = m1 ····· mk which proves our statement in the case where p(t1,..., tk) t1 tk is a monomial. In general case, the result easily follows by linearity of 8. The previous results can be summarized: Formula (4-1) defines an ultraweak functional calculus. Next, we shall see that the possibility of constructing the operator f (T ) does not depend on the behavior of the function f at infinity. In fact, it depends on the local properties of the function f . 146 DRAGOLJUB KECKIˇ C´ AND ÐORÐE KRTINIC´

Throughout the rest of this section, ϕn (and also ψn and χn) will always denote an increasing sequence of test functions with range contained in [0, 1], and such k that {t ∈ ޒ | ϕn(t) = 1} form an increasing sequence of sets whose union covers k all ޒ , unless ϕn is speciﬁed otherwise. Such sequences we shall call exhausting. S+∞ Proposition 4.3. Let Xn denote the linear space Im(χn(T )). Let E0 = n=1 Xn. Then Xn is invariant for all operators f (T ) and E = ᏸ(E0) is dense in X.

Proof. Let x ∈ Xn, that is, x = χn(T )y. Then

f (T )x = f (T )χn(T )y = χn(T ) f (T )y ∈ Xn.

Since χn → 1 in the topology of Fα, we have x = w −lim xn = w −lim χn(A)x for any x ∈ X. Therefore the set E0 is weakly dense, and its linear span E is also weakly dense. For linear spaces weak and strong closures coincide.

Deﬁnition 4.4. We say that f ∈ Fα,loc if f ϕ ∈ Fα for any ϕ ∈ Ᏸ.

Proposition 4.5. Fα ⊆ Fα,loc and Fα,loc is an algebra.

Proof. The ﬁrst assertion follows from Ᏸ · Fα ⊆ Fα · Fα ⊆ Fα. Let f, g ∈ Fα,loc and let ϕ ∈ Ᏸ be arbitrary. Choose another ψ ∈ Ᏸ such that ϕψ = ϕ, that is, a function equals 1 in the neighborhood of the support of ϕ. Then f ϕ, gψ ∈ Fα and f gϕ = f ϕ · gψ ∈ Fα.

Proposition 4.6. Let f ∈ Fα,loc, let χn be some exhausting sequence and let x ∈ E. The sequence ( f · χn)(T )x converges strongly and its limit does not depend on the choice of an exhausting sequence.

Proof. Let x ∈ Xk. Then x = ψk(A)y for some y ∈ X. Pick a positive integer n0 = · − · = such that χn0 1 on supp ψk. For n > n0 we have ( f χn)(T )x ( f χn0 )(T )x · − · = · (( f (χn χn0 ) ψk(T ))y 0, implying that ( f χn)(T )x is constant for n > n0. Therefore, this sequence is constant also for x ∈ E. To prove the independence, choose another exhausting ψn, and set

y = lim ( f χn)(T ) and z = lim ( f ψn)(T ). n→+∞ n→+∞ For an arbitrary θ ∈ Ᏸ, we have

θ(T )y = lim θ(T )( f χn)(T )x = (θ f )(T )x, n→+∞ and analogously θ(T )z = (θ f )(T )x. Thus for any θ ∈ Ᏸ, we have θ(T )(y −z) = 0. If θ runs through θn(x) = ϕ(x/n), then θn(T ) → I weakly, implying y − z = 0.

Definition 4.7. Let f ∈ Fα,loc and let x ∈ E. We define f (T )x := lim( f ·χn)(T )x. Thus, we defined the map f 7→ f (T ) Proposition 4.8. The map defined in Definition 4.7 is linear and multiplicative. AFUNCTIONALCALCULUSFORUNBOUNDEDOPERATORS 147

Proof. The linearity is obvious. Further, let us note that the sequence ( f χn)(T )x is not only convergent but constant for n > n0. Let χn be an arbitrary exhausting sequence and let x = ψk(T )y ∈ Xk. Choose another exhausting sequence ϕn such that χnϕn = χn. Then we have

(4-2) ( f g)(T )x = lim ( f χn)(T )(gϕn)(T )ψk(T )y. n→+∞

Since all the operators ( f χn)(T ), (gϕn)(T ) and ψk(T ) commute, we see that both (gϕn)(T )y and ( f χn)(T )y belong to Xk. Therefore (gϕn)(T )y and g(T )y (as its limit) are in the domain of f (T ). In other words, the operator f (T )g(T ) is well- deﬁned on Xn. By changing roles of f and g, we conclude that g(T ) f (T ) is also well-deﬁned on Xn. From (4-2), we get ( f g)(T ) = f (T )g(T ) = g(T ) f (T ).

Proposition 4.9. For any f ∈ Fα,loc, the operator f (T ) is closable.

Proof. Let xn → 0, and let f (T )xn → y. We have

y = w − lim χn(T ) = w − lim lim χn(T ) f (T )xm = 0 n→+∞ n→+∞ m→+∞ since both χn(T ) and (χn f )(T ) are bounded.

5. The support of the functional calculus

Deﬁnition 5.1. The support of the functional calculus is the smallest closed set supp 8 ⊆ ޒk such that if f ≡ g on supp 8, then 8( f ) = 8(g).

First we need some deﬁnitions and facts concerning Clifford algebras.

Definition 5.2. Suppose F is an m-dimensional real vector space F with a basis {e j | 1 6 j 6 m}. We define the Clifford algebra Ꮿl(F) as a vector space with basis { | ⊆ { }} = eS S 1, 2,..., m , with multiplication arising from identifications e∅ 1, e{ j} = e j and rules

2 (5-1) e j = −1 and e j ek = −eke j if k 6= j.

Remark 5.3. The rules (5-1) can be easily extended to the basis of Ꮿl(F) if we ··· = write e j1 e j2 e js for eS e{ j1, j2,..., js }, and further by linearity to whole Ꮿl(F).

Remark 5.4. For an m-dimensional real vector space F, we will write Ꮿl(m) for Ꮿl(F). It is well known that Ꮿl(1) is isomorphic to the ﬁeld of complex numbers, and that Ꮿl(2) to the quaternions. For further details on Clifford algebras, see [Brackx et al. 1982; Ryan 2003]. 148 DRAGOLJUB KECKIˇ C´ AND ÐORÐE KRTINIC´

Deﬁnition 5.5. Let X be a Banach space. The corresponding Banach module over P Ꮿl(m) is simply X(m) = X ⊗ޒ Ꮿl(m) = S⊆{1,2,...,m} uSeS | uS ∈ X , with the norm 1/2 X = X k k2 uSeS uS . S⊆{1,2,...,m} S⊆{1,2,...,m}

Definition 5.6. Let T0, T1,..., Tm be an (m +1)-tuple of operators acting on a Pm Banach space X (bounded or unbounded). The operator Tcl = j=0Tj e j is defined P Pm P by Tcl( S uSeS) = j=0 S Tj uSe j eS. The norm of Tcl is kTclk = sup kTcluk. kuk61 Also we can define the resolvent set and the spectrum of Tcl by m+1 m+1 ρ(Tcl) = {λ ∈ ޒ | λI − Tcl is invertible} and σ (Tcl) = ޒ \ ρ(Tcl). Proposition 5.7. (a) We have kuλk 2m/2kukkλk for all u ∈ X and λ ∈ Ꮿl(m). √ 6 m m+1 (b) σ (Tcl) ⊆ {λ ∈ ޒ | |λ| 6 m + 1 · kTclk}. Proof. Direct calculation.

Proposition 5.8. Let T0, T1,..., Tm be an (m +1)-tuple of commuting unbounded operators. By commutativity we mean that there is a dense set D ⊆ X such that Pm Tj Tk x = Tk Tj x for all x ∈ D and j, k. Also, let Tcl = j=0 Tj e j . (a) The following conditions are mutually equivalent:

(i) Tcl is invertible. P 2 (ii) ( j Tj )e0 is invertible. P 2 (iii) j Tj is invertible in B(X). m P 2 (b) σ (Tcl) = {λ ∈ ޒ | j (Tj − λ j ) is not invertible}. P 2 −1 = P 2 −1 · − P Proof. If j Tj is invertible in B(X), then Tcl ( j Tj ) (T0 j>0 Tj e j ), so (iii) implies (i). P P P |S| Since (T0 − j>0 Tj e j )U = UTcl, where U( j uSeS) = j (−1) uSeS, it fol- P P 2 P lows that (T0 − j>0 Tj e j ) is invertible; hence ( j Tj )e0 = Tcl(T0 − j>0 Tj e j ) is invertible, that is, (i) implies (ii). That (ii) is equivalent to (iii) is obvious. Part (b) follows from the equivalence of (i) and (iii). Theorem 5.9. The support of the weak functional calculus deﬁned in Deﬁnition 2.1 Pm is equal to σ ( i=1 Ti ei ). Pm Proof. Let n be an odd integer, let n > max{m, 2}, and let T = i=1 Ti ei . By Proposition 5.8, and since supp 8 ⊆ ޒm, it is enough to prove that σ (T ) = supp 8. P P If f = S fSeS, with 8(n)( f ) = S 8( fS)eS, we can extend 8 to a homomorphism 8(n) : Ꮽ ⊗ޒ ޒ(n) → B(X)(n), which is continuous with respect to τ and the weak topology on B(X)(n) and supp 8(n) = supp 8. AFUNCTIONALCALCULUSFORUNBOUNDEDOPERATORS 149

Let λ 6∈ supp 8 and let ϑi ∈ Ᏸ such that ϑi (λ) = 0 and supp ϑi → supp 8 (increasingly and uniformly on compacts). Then the function |x − λ|−n−1(x − λ) is well-defined (with value 0 for x = λ) and the function (x − λ)|x − λ|n−1 is polynomial (since n is odd). Therefore we have n−1 −n−1 (T − λI )|T − λI | · lim 8(n) ϑi · |x − λ| (x − λ) i→∞ n−1 −n−1 = lim 8(n) ϑi · (x − λ)|x − λ| 8(n) ϑi · |x − λ| (x − λ) i→∞ 2 = lim 8(n)(ϑi ) = I. i→∞ −n−1 n−1 Analogously, limi→∞ 8(n)(ϑi · |x − λ| (x − λ)) · (T − λI )|T − λI | = I , implying that T − λI is invertible, that is, λ 6∈ σ (T ). For the opposite direction, suppose f is in Ᏸ and satisfies supp f ∩ σ (T ) = ∅. Let (T − x I )−1|T − x I |−n+18 ( f ) for x 6∈ σ (T ) = (n) F(x) −n−1 8(n) f (t) · |t − x| (t − x) for x 6∈ supp f. Since supp f ∩ σ (T ) = ∅, it follows that the function F is defined for all x. If x 6∈ σ (T ), x 6∈ supp f , and ϑi is an element of Ᏸ such that ϑi ≡ 1 on supp f and ϑi tends to 1 increasingly and uniformly on compacts, then n−1 −n−1 (T − x I )|T − x I | · 8(n) f (t) · |t − x| (t − x) n−1 −n−1 = lim 8(n) ϑi · (t − x)|t − x| · 8(n) f (t) · |t − x| (t − x) i→∞

= lim 8(n)(ϑi f ) = 8(n)( f ), i→∞ implying F is well-deﬁned. Taking into account [McIntosh and Pryde 1987, Examples 5.3 and 5.4], we see −n−1 the function F is an entire monogenic function. Since f (t)·|t −x| (t − x) ⇒ 0 as |x| → ∞, it follows that

−n−1 w 8(n) f (t) · |t − x| (t − x) −→ 0 in B(X)(n), so the range of 8(n) is weakly bounded. But in locally convex space every weakly bounded set is originally bounded (see [Rudin 1973, Theorem 3.18] — actually, this theorem is consequence of the Banach–Alaoglu theorem, the Hahn–Banach −n−1 theorem and the Banach–Steinhaus theorem). So the 8(n)( f (t)·|t −x| (t − x)) are norm bounded. By Liouville’s theorem for monogenic functions (see [McIntosh and Pryde 1987, Theorem 5.1]), F is a constant function. However, a constant sequence of operators tending weakly to 0 must be identically equal to zero, that is, F ≡ 0. Hence 8(n)( f ) = 0, that is, supp 8(n) ⊆ σ (T ). Remark 5.10. McIntosh, Pryde and Ricker [McIntosh et al. 1988] proved that for the k-tuple of commuting bounded generalized scalar operators, the spectrum 150 DRAGOLJUB KECKIˇ C´ AND ÐORÐE KRTINIC´

σ (T ) coincides with many known joint spectra of the (T1,..., Tk), for instance Harte and Taylor spectrum. An interesting question for further research is to ﬁnd whether this statement holds for a k-tuple of unbounded operators.

6. Comparison of different deﬁnitions of functional calculi

We defined the notion of weak and ultraweak functional calculus. There are at least two other definitions of functional calculi. Definition 6.1 [deLaubenfels 1995]. Let T be an unbounded operator on a Banach space X and let Ᏺ be an algebra of functions. Also, let f1(x) = x be the identical −1 function, let f0(x) ≡ 1, and let gλ(z) denote the mapping z 7→ (z − λ) . The mapping Ᏺ 3 f 7→ f (T ) ∈ B(X) is a Ᏺ-functional calculus if it is a homomorphism and if the following conditions are satisfied:

(i) f (T )T ⊆ T f (T ) = ( f · f1)(T ) whenever f and f · f1 belong to Ᏺ.

(ii) f0(T ) = IX whenever f0 ∈ Ᏺ. k −k k (iii) T − λI is injective and (gλ)(T ) = (T − λI ) whenever gλ ∈ Ᏺ. Definition 6.2 [Andersson et al. 2006]. We call a continuous multiplicative map- n n ping A : Ᏸ(ޒ ) → B(X) is a hyperoperator on ޒ , writing A ∈ HᏰ(ޒn)(X), if S (i) DA = Im A(φ) is dense in X, and (ii) N = T Ker A(φ) = {0}. Definition 6.3 [Andersson et al. 2006]. Let c =(c, D) be a linear operator mapping the dense subspace D of X into itself. Moreover, assume that c is closable and that there is a linear and multiplicative mapping Ᏹ(ޒn) → Ł(D) that extends the trivial n one on polynomials, and such that hk(c)x → h(c)x for x ∈ D if hk → h in Ᏹ(ޒ ). Then we say that c, or rather (c, D), is a weak hyperoperator. In the preceding definition, Ᏹ(ޒn) is the algebra of all differentiable functions n P∞ −k on ޒ , which possess an asymptotic expansion f (x)∼ k=0 ak x (when x →∞), with ak ∈ ރ, in the sense that for every N ∈ ގ

N X −k −N−1 f (x) = ak x + x rN+1(x) for |x| > 1, k=0 where rN+1(x) is bounded with all its derivatives (for more details see [Andersson and Sjostrand¨ 2004]) and Ł(D) denotes the set (not the space!) of all closable operators on D. Proposition 6.4. Any ultraweak functional calculus is an Ꮽ-functional calculus in the sense of Deﬁnition 6.1. AFUNCTIONALCALCULUSFORUNBOUNDEDOPERATORS 151

Proof. (i) Let x ∈ Ᏸ(T ) and let χn = ϕ(x/n), where φ is a test function equal to 1 in some neighborhood of zero and satisﬁes 0 6 ϕ 6 1. By the deﬁnition of the ultraweak functional calculus, we have Ᏸ 3 f f1χn → f f1 in the topology of Ꮽ. Then ( f f1χn)(T ) → f f1(T ) weakly. So, for x in the domain of T we have

f (T )T x = w − lim f (T )( f1χn)(T )x = w − lim( f f1χn)(T )x = ( f f1)(T )x, implying f (T )T ⊆( f f1)(T ). On the other hand, for any x such that f (T )x belongs to the domain of T , we have

( f f1)(T )x = w − lim( f1χn)(T ) f (T )x = T f (T )x. The set of such vectors x contains the set E from Proposition 4.3, and therefore it is dense in X. By continuity, we get T f (T ) = ( f f1)(T ).

(ii) Let f0 ∈ Ꮽ. Choose a sequence χn ∈ Ᏸ tending to f0 in the topology of Ꮽ. (Note that the definition of the weak functional calculus contains the condition that Ᏸ is dense in Ꮽ in the topology of Ꮽ.) Then f0(T )x = w −lim χn(T )x = x by the definition of the ultraweak functional calculus. k k (iii) Let gλ ∈ Ꮽ. We have (z−λ) gλ(z) = 1. Let χn ∈ Ᏸ be an exhausting sequence. By the definition of the ultraweak functional calculus and by the condition (3) of Definition 2.1 (applied to the polynomial t 7→ t), we have

k k k k k k x = w − lim(χn( f1 − λ) gλ)(T )x = (T − λI ) gλ(T ) x = gλ(T )(T − λI ) x. Proposition 6.5. The restriction of any weak functional calculus to Ᏸ is a hyperoperator. Proof. We only have to prove the conditions (i) and (ii) from Definition 6.2. But this easily follows from condition (3) of Definition 2.1 applied to the polynomial f0(x) ≡ 1. Remark 6.6. We think that an interesting question for further investigation is to find the relationship between an ultraweak functional calculus and the notion of weak hyperoperator.

7. On commutativity conditions

For an n-tuple of unbounded operators, it is difficult to find a commutativity condition that ensures a reasonable theory. Any of the following four properties can be used to define of commutativity of two unbounded operators on a Banach space: (P1) There exists a dense space Ᏸ ⊆ X such that T Sx = ST x for any x ∈ Ᏸ. (P2) Suppose that there exists a Ᏸ(ޒ) functional calculus for both T and S. For any f, g ∈ Ᏸ(ޒ), we have f (T )g(S) = g(S) f (T ). 152 DRAGOLJUB KECKIˇ C´ AND ÐORÐE KRTINIC´

(P3) Suppose that T and S are generators of strongly continuous groups eitT and eit S, respectively. For any t, s ∈ ޒ, we have eitT eisS = eisSeitT . (P4) For any λ ∈ ρ(T ) and µ ∈ ρ(S), we have (T − λI )−1(S − µI )−1 = (S − µI )−1(T − λI )−1. In the general case all of these properties except (P1) might be meaningless, since an unbounded operator need not have a Ᏸ(ޒ) functional calculus, a solution of the corresponding abstract Cauchy problem, or a nonempty resolvent set. However, if S and T are generalized scalar operators, (P1)–(P4) make sense and are mutually equivalent. Proposition 7.1. (P1) implies (P4).

Proof. Obvious. Proposition 7.2. (P3) implies (P1) if S and T are generalized scalar operators. Proof. For such S and T , we constructed the ultraweak functional calculus and extended it to Fα,loc. The function f (x, y) = xy belongs to Fα,loc, so the operator f (T, S) is deﬁned on the linear span of E (deﬁned in Proposition 4.3). Examining the proof of Theorem 4.2, we see that on the set E both products TS and ST are equal to f (T, S). Proposition 7.3. (P3) implies (P2) if S and T are generalized scalar operators. Proof. For the pair (S, T ), we constructed a functional calculus for a wider algebra 2 than Ᏸ(ޒ ), and choosing a function of the form f (x)g(y), we obtain the result. Proposition 7.4. (P4) implies (P3) if S and T are generalized scalar operators. Proof. S and T are generalized scalar operators that are generators of strongly continuous groups, so this statement can be derived following the standard proof of the Hille–Yoshida theorem; see [Dunford and Schwartz 1958, Theorem VIII.1.13]. The spectrum of iT is on the imaginary line, so λ ∈ ρ(iT ) for all λ > 0. It follows −1 that Bλ =−iλ[I −λ(λI −iT ) ] (for λ>0) are bounded. Also, iT x =limλ→∞ Bλx t B for all x ∈ Ᏸ(iT ) = Ᏸ(T ) and limλ→∞ e λ x exists for all x ∈ Ᏸ(iT ) = Ᏸ(T ) (of itT −1 course, the value of this limit is e x). Analogously, if Cλ =−iλ[I −λ(λI −iC) ] sC (for λ>0), we have i Sx =limλ→∞ Cλx for all x ∈Ᏸ(i S)=Ᏸ(S) and limλ→∞ e λ x exists for all x ∈ Ᏸ(i S) = Ᏸ(S). Finally, from (P4) (the commutativity of resolvents) we have the commutativity of Bλ and Cλ (for all λ > 0), and, passing to λ → ∞, we get (P3). Proposition 7.5. (P2) implies (P4) if S and T are generalized scalar operators.

Proof. The function gλ(x) = 1/(x − λ) belongs to Fα for all α and for Im λ 6= 0. Indeed, it is easy to check that gλ is the inverse Fourier transform of the function AFUNCTIONALCALCULUSFORUNBOUNDEDOPERATORS 153 √ −iλx F(x) = i 2πe H(−x) for Im√λ > 0. Similarly, it is the inverse Fourier transform of the function F(x) = −i 2πe−iλx H(x) for Im λ < 0. Here H(x) is the Heaviside√ function, equal to zero if x < 0 and to 1 otherwise. Hence gλ ∈ Fα, and k k = | | + + | |α gλ Fα 2π(1/ Im λ 0(α 1)/ Im λ ). If ϕn(x) = ϕ(x/n), where ϕ is a test function valued in [0, 1] and equal to 1 in a neighborhood of zero, then the sequence of test functions ψn,λ = ϕn · gλ → gλ as n → +∞ in the topology τ deﬁned in Deﬁnition 3.1, and therefore ψn,λ(S) → −1 −1 gλ(S) = (S −λI ) weakly, by Proposition 6.4. Similarly ψm,µ(T ) → (T −µI ) weakly. From property (P2) we have ψn,λ(S)ψm,µ(T ) = ψm,µ(T )ψn,λ(S). Taking a limit as n → +∞ and m → +∞, we derive (P4).

8. Examples and remarks

Example 8.1. Example 2.8 can be generalized using examples from [Gorin and Karahanjan 1977] and techniques from [Evgrafov 1979]. There is an operator for which (2-1) does not hold, whose asymptotic behavior ρ satisﬁes eit A ∼ ρe|t| as |t| → ∞ for every ρ > 1/4. Let E be a vector space of entire functions, such that there exists C ∈ ޒ such σ|z|ρ that | f (z)| 6 Ce . Like in Example 2.8, E is a Banach space if the norm k f k of a function f is deﬁned as the smallest constant C for which the inequality above holds. Since Z 0 1 f (ξ + z) f (z) = dξ, 2 2πi |ξ|=1 ξ it follows that the operator A = −id/dz is a bounded linear operator on E; see [Gorin and Karahanjan 1977]. Hence

(eit A f )(z) = (etd/dz f )(z) = f (z + t).

ρ It is easy to see that keit Ak = O(e|t| ) as |t| → ∞. To see exact asymptotic behavior, we can use a function

∞ X zn f (z) = 0 0(n/ρ + 1) n=0

xρ (0 is the gamma function). Using that f0(x) ∼ ρe when x → ∞ for ρ > 1/4 (see [Evgrafov 1979, Example 1.5]), we derive our statement.

Example 8.2. Consider the operator A = −id/dx acting on the dense subspace of L p(ޒ) consisting of all absolutely continuous functions whose derivative also belongs to L p(ޒ). It is easy to see that eit A f = f (x + t), and we can conclude it A ke k = 1. Hence, for the operator A, we have the F0 functional calculus. 154 DRAGOLJUB KECKIˇ C´ AND ÐORÐE KRTINIC´

Example 8.3. Let H be a Hilbert space and let A be an unbounded selfadjoint operator. Also, let Q be a nilpotent operator of order n, that is, Qn = 0, and Qn−1 6= 0. Let Q commute with A. Such a Q is given by the matrix

0 1 0 ··· 0 0 0 1 ··· 0   . . . .  Q = ...... 0   ......  . . . . 1 0 0 ······ 0 with respect to the root subspace of some eigenvalue of A. Then we have

keit(A+Q)k = keit Aeit Qk = keit Qk = O(|t|n−1).

Therefore, for the operator A + Q, we have the Fn−1 functional calculus. Example 8.4. The advection operator

f (x, t) 7→ a · ∇ f (x, t)

k (here ∇ = (∂/∂x1, . . . , ∂/∂xn), x ∈ ޒ and t ∈ ޒ) is well known in the mechanics of ﬂuids. It is a generalized scalar operator in which f (x, t) solves the abstract Cauchy problem ∂ f (x, t)/∂t = a · ∇ f (x, t). This unbounded operator commutes with the Laplacian. In fact, it commute with all ∂/∂x j . So we can construct an ultraweak functional calculus for these operators and ﬁnally we can get any linear (even some nonlinear) combinations of the advection operator and the Laplacian.

Finally, we shall see that the functional calculus we established contains, in a sense, the widest algebra of functions. We deal with the one dimensional case.

Theorem 8.5. The operator A has the Fα functional calculus if and only if

it A α ke k 6 M(1 + |t| ) for some M ∈ ޒ.

it A α Proof. If there is M ∈ ޒ such that ke k 6 M(1 + |t| ), then by Section 1, A has the Fα functional calculus. In the opposite direction, since dαeitx /dxα = (it)αeitx , it follows that dα keitx k = |eitx | + eitx + |t|α, α sup sup α 6 1 x∈ޒ x∈ޒ dx and, from the existence of the functional calculus, it follows that

itx itx α k8(e )k 6 Mke k 6 M(1 + |t| ). AFUNCTIONALCALCULUSFORUNBOUNDEDOPERATORS 155

Acknowledgements

We are grateful to Professor Milosˇ Arsenovic´ for help and many useful suggestions during the work on this paper. We also thank the referee for useful suggestions during the review process.

References

[Andersson 2003] M. Andersson, “(Ultra)differentiable functional calculus and current extension of the resolvent mapping”, Ann. Inst. Fourier (Grenoble) 53:3 (2003), 903–926. MR 2004g:47010 Zbl 1052.47009 [Andersson and Berndtsson 2002] M. Andersson and B. Berndtsson, “Non-holomorphic functional calculus for commuting operators with real spectrum”, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1:4 (2002), 925–955. MR 2004d:47035 Zbl 1099.47505 [Andersson and Sjöstrand 2004] M. Andersson and J. Sjöstrand, “Functional calculus for non- commuting operators with real spectra via an iterated Cauchy formula”, J. Funct. Anal. 210:2 (2004), 341–375. MR 2005b:47010 Zbl 1070.47009 [Andersson et al. 2006] M. Andersson, H. Samuelsson, and S. Sandberg, “Operators with smooth functional calculi”, J. Anal. Math. 98 (2006), 221–247. MR 2007g:47008 Zbl 1152.47010 [Balabane et al. 1993] M. Balabane, H. Emamirad, and M. Jazar, “Spectral distributions and generalization of Stone’s theorem”, Acta Appl. Math. 31 (1993), 275–295. MR 94f:47038 Zbl 0802.47013 [Brackx et al. 1982] F. Brackx, R. Delanghe, and F. Sommen, Clifford analysis, Research Notes in Mathematics 76, Pitman, Boston, MA, 1982. MR 85j:30103 Zbl 0529.30001 [deLaubenfels 1991] R. deLaubenfels, “Existence and uniqueness families for the abstract Cauchy problem”, J. London Math. Soc. (2) 44:2 (1991), 310–338. MR 92k:34075 Zbl 0766.47011 [deLaubenfels 1993a] R. deLaubenfels, “Automatic well-posedness with the abstract Cauchy problem on a Fréchet space”, J. London Math. Soc. (2) 48:3 (1993), 526–536. MR 94i:34119 Zbl 0792. 47042 [deLaubenfels 1993b] R. deLaubenfels, “Unbounded holomorphic functional calculus and abstract Cauchy problems for operators with polynomially bounded resolvents”, J. Funct. Anal. 114 (1993), 348–394. MR 94h:47029 Zbl 0785.47018 [deLaubenfels 1994] R. deLaubenfels, Existence families, functional calculi and evolution equations, Lecture Notes in Mathematics 1570, Springer, Berlin, 1994. MR 96b:47047 Zbl 0811.47034 [deLaubenfels 1995] R. deLaubenfels, “Automatic extensions of functional calculi”, Studia Math. 114:3 (1995), 237–259. MR 96f:47029 Zbl 0834.47012 [deLaubenfels and Jazar 1999] R. deLaubenfels and M. Jazar, “Functional calculi, regularized semigroups and integrated semigroups”, Studia Math. 132:2 (1999), 151–172. MR 99k:47036 [Dunford and Schwartz 1958] N. Dunford and J. T. Schwartz, Linear Operators, I: General Theory, Wiley, New York, 1958. MR 1009162 (90g:47001a) Zbl 0084.10402 [Dynkin 1972] E. M. Dyn’kin, “An operator calculus based on the Cauchy–Green formula, III: Investigations on linear operators and the theory of functions”, Zap. Nauˇcn.Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 30 (1972), 33–39. In Russian. MR 48 #6982 [Evgrafov 1979] M. A. Evgrafov, Asimptotiqeskie ocenki i celye funkcii, 3rd ed., Nauka, Moscow, 1979. MR 81b:30048 Zbl 0447.30016 156 DRAGOLJUB KECKIˇ C´ AND ÐORÐE KRTINIC´

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Received October 19, 2009. Revised June 21, 2010.

DRAGOLJUB KECKIˇ C´ MATEMATICKIˇ FAKULTET UNIVERZITETU BEOGRADU STUDENTSKITRG 16 11000 BEOGRAD SERBIA [email protected] http://poincare.matf.bg.ac.rs/~keckic/

ÐORÐE KRTINIC´ MATEMATICKIˇ FAKULTET UNIVERZITETU BEOGRADU STUDENTSKITRG 16 11000 BEOGRAD SERBIA [email protected] http://poincare.matf.bg.ac.rs/~georg/ PACIFIC JOURNAL OF MATHEMATICS Vol. 249, No. 1, 2011

GEOMETRIC FORMALITY OF HOMOGENEOUS SPACES AND OF BIQUOTIENTS

D.KOTSCHICKAND S.TERZIC´

We provide examples of homogeneous spaces that are neither symmetric spaces nor real cohomology spheres, yet have the property that every invariant metric is geometrically formal. We also extend the known obstructions to geometric formality to some new classes of homogeneous spaces and of biquotients, and to certain sphere bundles.

1. Introduction

The notion of geometric formality was implicitly considered by Sullivan in the 1970s (see [Deligne et al. 1975; Sullivan 1975]), but the systematic study of this concept began only comparatively recently [Kotschick 2001]. A smooth manifold is geometrically formal if it admits a Riemannian metric for which all exterior products of harmonic forms are harmonic. Such a metric is then also called formal. Geometric formality clearly implies formality in the sense of Sullivan, and is even more restrictive. Since compact symmetric spaces are the classical examples of geometrically formal manifolds, it is natural to explore this notion in the context of homogeneous spaces, or, more generally, of manifolds with large symmetry groups. Trying to come up with generalizations of symmetric spaces, one might think ﬁrst of isotropy irreducible spaces. These are the homogeneous spaces G/H for which the isotropy representation of H on TeH (G/H) is irreducible. Such a space is strongly isotropy irreducible if the restriction of the isotropy representation to the identity component of H is also irreducible. These manifolds were originally classiﬁed by Manturov, and were further studied by Wolf and others, cf. [Besse 1987]. They share many properties of symmetric spaces, and indeed irreducible symmetric spaces are isotropy irreducible. A conceptual relationship between symmetric spaces and isotropy irreducible ones is explained in [Wang and Ziller 1993].

MSC2000: primary 53C25, 53C30; secondary 57T15, 57T20, 58A10. Keywords: formality, harmonic forms, homogeneous space, Stiefel manifold, biquotient. The work of the ﬁrst author was supported in part by the DFG Priority Program in Global Differential Geometry and by The Bell Companies Fellowship at the Institute for Advanced Study in Princeton. Copyright c 2009 D. Kotschick and S. Terzic.´

157 158 D. KOTSCHICK AND S.TERZIC´

However, the similarities between symmetric spaces and isotropy irreducible ones do not extend to (geometric) formality. Indeed, there are a number of strongly isotropy irreducible spaces which, by the results of [Kuin’ 1968], are not of Cartan type and therefore [Kotschick and Terzic´ 2003] are not formal in the sense of Sullivan. A fortiori, they cannot be geometrically formal. Example 1. The compact homogeneous spaces SU(pq)/(SU(p) × SU(q)) for 2 p, q ≥ 3, SO(78)/E6, and SO(n −1)/ SU(n) for n ≥ 3 are strongly isotropy irreducible, but are not formal in the sense of Sullivan. Another class of homogeneous spaces generalizing the symmetric ones consists of the so-called generalized symmetric spaces, sometimes called k-symmetric spaces. These are defined by replacing the involution in the definition of symmetric spaces by a symmetry of order k; see [Wolf and Gray 1968a; 1968b; Terzich 2001]. In [Kotschick and Terzic´ 2003] we proved that all generalized symmetric spaces of compact simple Lie groups are formal in the sense of Sullivan, and that many of them are not geometrically formal. The main purpose of this paper is to prove that, in spite of all these negative results, there are homogeneous spaces that are neither (homotopy equivalent to) symmetric spaces nor products of real homology spheres (which are trivially geometrically formal [Kotschick 2001]), yet are geometrically formal. We shall prove the following: Theorem 2. All homogeneous metrics on the following homogeneous spaces are geometrically formal: 2n+1 (1) the real Stiefel manifolds V4(ޒ ) = SO(2n + 1)/ SO(2n − 3) for n ≥ 3, 2n (2) the real Stiefel manifolds V3(ޒ ) = SO(2n)/ SO(2n − 3) for n ≥ 3, n (3) the complex Stiefel manifolds V2(ރ ) = SU(n)/ SU(n − 2) for n ≥ 5, n (4) the quaternionic Stiefel manifolds V2(ވ ) = Sp(n)/ Sp(n − 2) for n ≥ 3, 2 (5) the octonionic Stiefel manifold V2(ޏ ) = Spin(9)/G2, and (6) the space Spin(10)/ Spin(7). Moreover, none of these spaces is homotopy equivalent to a symmetric space. They are not homotopy equivalent to products of real cohomology spheres, except pos- 2n sibly for V3(ޒ ) with n even. The space Spin(10)/ Spin(7) above corresponds to a nonstandard embedding of 10 Spin(7) in Spin(10), so that the quotient is not V3(ޒ ) but a manifold with the same real homology and different integral homology. The homology of this space, 10 unlike that of V3(ޒ ), is torsion-free. In Section 2, we exhibit a very simple mechanism to prove that G-invariant metrics on certain homogeneous spaces G/H with simple cohomology rings are GEOMETRIC FORMALITY OF HOMOGENEOUS SPACES AND OF BIQUOTIENTS 159 geometrically formal. This mechanism in fact gives a new proof that certain symmetric spaces are geometrically formal, without using the symmetric space structure, but only the description of the cohomology ring. The argument applies to all homogeneous spaces that have the real cohomology of a product of odd- dimensional spheres. The examples (3)–(6) in Theorem 2 are all the homogeneous spaces that have the integral cohomology of such a product, but are not obviously diffeomorphic to products or to symmetric spaces. There are many more examples with the real cohomology of such a product, but different integral cohomology. We discuss the two infinite sequences of real Stiefel manifolds occurring in (1) and (2), leaving aside the other, sporadic, examples. Also in Section 2, by considering the Aloff-Wallach spaces [Aloff and Wallach 1975], we show that our results do not extend to all homogeneous spaces with the cohomology algebra of a product of spheres, if one does not insist that the spheres be odd-dimensional. In Section 3 we show that the homogeneous spaces listed in Theorem 2 are not homotopy equivalent to symmetric spaces or to nontrivial products, thereby completing the proof of the theorem. In the final two sections of this paper we add to the negative results of [Kotschick 2001; Kotschick and Terzic´ 2003; 2009] by providing further examples of manifolds that, though formal in the sense of Sullivan, are not geometrically formal. All the examples we give here are simply connected and of dimension six. In Section 4 we consider certain classes of biquotients in the sense of Eschenburg [1992b; 1992a] and in Section 5 we consider two-sphere bundles over ރP2. Many of these two-sphere bundles are known to carry special metrics of cohomogeneity one by the results of [Grove and Ziller 2008]. Both these collections of examples generalize the discussion of the flag manifold SU(3)/T 2 carried out in [Kotschick and Terzic´ 2003]. A different generalization, to certain partial flag manifolds of higher dimensions, is contained in [Kotschick and Terzic´ 2009].

2. Examples of geometrically formal homogeneous spaces

In this section, we describe a class of homogeneous spaces for which any homogeneous metric is formal. First, we recall the well-known fact that, on a compact homogeneous space G/H, the harmonic forms of any homogeneous metric are invariant. To check this, let h be a harmonic form with respect to some homogeneous metric on G/H. Then ∗ = + ∗ = ∗ + = R ∗ h and h can be written as h hi dα, and h ( h)i dβ, where hi G g h ∗ = R ∗ ∗ and ( h)i G g ( h) with respect to Haar measure with total volume 1. Since hi and (∗h)i are invariant forms, we have Z Z Z ∗ ∗ ∗ ∗hi = ∗ g h = ∗(g h) = g (∗h) = (∗h)i . G G G 160 D. KOTSCHICK AND S.TERZIC´

It follows that d(∗hi ) = d(∗h)i = 0 and thus hi is harmonic. This means that h = hi , that is, h is an invariant form. As a ﬁrst application of this fact, we have the following: Proposition 3. If G is a compact connected Lie group and H is a closed connected subgroup with the property that G/H is of even dimension 2k, and all its real cohomology is in degrees 0, k and 2k, then any homogeneous metric on G/H is formal. Proof. Because of the cohomology structure of G/H, to prove that it is geometrically formal with respect to some metric, it is enough to prove that x ∧ y is a harmonic form for any two harmonic k-forms x and y. If the metric g is homogeneous, then according to the previous observation, x ∧ y is an invariant form and being of top degree, we have x ∧ y = c · dvol, where c is constant. Thus, x ∧ y is harmonic and g is formal. Example 4. The complex projective plane ރP2 = SU(3)/S(U(2) × U(1)), the quaternionic projective plane ވP2 = Sp(3)/(Sp(2) × Sp(1)), the Cayley plane 2 ޏP = F4/ Spin(9), and G2/ SO(4) are examples to which the proposition applies; cf. [Borel and Hirzebruch 1958]. These spaces are all symmetric, but the argument proving geometric formality does not use the symmetric space structure. Unfortunately, there are no nonsymmetric homogeneous spaces to which we could apply Proposition 3. Indeed, if the number k is even, then such spaces belong to the class of rank one homogeneous spaces in the terminology of Onishchik [1995], while for odd k they belong to the class of rank two homogeneous spaces. Examining the classiﬁcation of homogeneous spaces of rank one and two given by Onishchik [1995], one sees that there are no examples other than the symmetric spaces mentioned in Example 4. To get new examples, we need the following slight variation of Proposition 3: Proposition 5. If G is a compact connected Lie group and H is a closed connected subgroup with the property that H ∗(G/H; ޒ) = V(x, y), where x and y are of odd degrees, then any homogeneous metric on G/H is formal. The spaces listed in Theorem 2 all have cohomology rings of this form; cf. [Onishchik 1963; Kramer 2002]. Thus Proposition 5 shows that those spaces are geometrically formal. Proof. Because of the cohomology structure of G/H, to prove that it is geometrically formal with respect to some metric, it is enough to prove that x ∧ y is a harmonic form, where x and y are the harmonic representatives of the cohomology generators. If the metric g is homogeneous, then as before, x ∧ y is an invariant form and, being of top degree, we have x ∧ y = c · dvol, where c is constant. It follows that x ∧ y is harmonic and g is formal. GEOMETRIC FORMALITY OF HOMOGENEOUS SPACES AND OF BIQUOTIENTS 161

Remark 6. At this point it is useful to recall that if G is simple and we endow G/H with the submersion metric g of a biinvariant metric on G, then the Riemannian homogeneous space (G/H, g) is irreducible as a Riemannian manifold [Kobayashi and Nomizu 1963]. This means that such a g cannot be a product metric. These are examples of normal homogeneous metrics.

As a simple application of our discussion so far, we can prove the following:

Proposition 7. The group SU(4) acts transitively on S5 × S7. All SU(4)-homogeneous metrics for this action are formal. Furthermore, the normal homogeneous metrics are not symmetric.

Proof. First, it is clear that SU(4) acts transitively on S7, with isotropy group SU(3). Second, SU(4) acts transitively on S5 via the double covering SU(4)→SO(6). The isotropy group of this action is the preimage of SO(5) under the covering, which can be identiﬁed with Sp(2). Now take the product action on S5 × S7. This is still transitive, for example because the restriction of the action of SU(4) on S5 to SU(3) is still transitive. It follows that S5 × S7 is a homogeneous space of SU(4) with isotropy group Sp(2) ∩ SU(3) = SU(2). Proposition 5 implies that all SU(4)-homogeneous metrics on S5×S7 are formal. By Remark 6, it follows that we cannot get the normal homogeneous metrics as product metrics. In particular, the metric on S5 × S7 that is the product of the symmetric space metrics on the factors, though also formal, is not normal homogeneous for the SU(4)-action. To end this section, we want to show that Proposition 5 is sharp in the sense that it does not extend to arbitrary homogeneous spaces with a cohomology algebra of the form V(x, y) with x of even degree and y of odd degree. A convenient class of examples to consider for this purpose are the so-called Aloff–Wallach spaces. These are homogeneous spaces of the form

1 Nk,l = SU(3)/T , where T 1 is embedded as the diagonal matrices D(zk, zl , z−k−l ), with k and l coprime integers with kl(k + l) 6= 0. Obviously, they are all homogeneous spaces of Cartan type, and are therefore [Kotschick and Terzic´ 2003] formal in the sense of Sullivan. It is also easy to see that they all have the real cohomology of S2×S5. The name derives from [Aloff and Wallach 1975], where it proved that these spaces have homogeneous metrics of positive sectional curvature. The numerical conditions on k and l are there to make sure that T 1 acts on ރ3 without nonzero ﬁxed points. Since SU(3) endowed with a biinvariant metric is geometrically formal, it is natural to consider the submersion metrics on SU(3)/T 1 which we get from the 162 D. KOTSCHICK AND S.TERZIC´ principal circle ﬁbration

1 π (1) T → SU(3) −→ Nk,l . Such a metric is often called a normal homogeneous metric. Theorem 8. The normal homogeneous metrics on Aloff–Wallach spaces are not formal. Proof. Assume that M = SU(3)/T 1 is geometrically formal for some embedding T 1 ⊂ SU(3). Since M has the real cohomology of S2 × S5, it carries two harmonic 2 = ∧ forms ω2 and ω5 such that ω2 0 and ω2 ω5 is a volume form. Since SU(3) is simply connected, the Euler class e of the principal bundle (1) is not zero, so that e = λ[ω2] with λ 6= 0. We now normalize our form so that λ = 1. In the following calculation we also ignore nonzero constants. There exists a connection form α on the principal bundle (1) such that

∗ (2) π (ω2) = dα, where π is the projection in (1). ∗ ∗ Let η5 = π (ω5) and η2 = π (ω2). Then η2 and η5 are closed and

2 (3) η2 = 0. Also,

∗ ∗ (4) ∗η5 = ∗π (ω5) = α ∧ π (∗ω5) = α ∧ η2. Then (2) gives

2 (5) d(∗η5) = dα ∧ η2 = η2 = 0.

This implies that η3 = ∗η5 is a harmonic form on SU(3) with the biinvariant metric. Since the harmonic forms on SU(3) coincide with the biinvariant ones, we get that η3 has the form

η3(X, Y, Z) = hX, [Y, Z]i for X, Y, Z ∈ su(3). On the other hand, we have a natural direct sum decomposition

(6) su(3) = t1 ⊕ (su(3)/t1), and (3) implies that there exists a 5-dimensional subspace ᏷ in su(3)/t1 such that iv(η3) = 0 for any vector v ∈ ᏷. Let H1, H2, E1, E2, F1, F2 be canonical (Chevalley) generators of the Lie algebra su(3). Consider the subspace ᏸ spanned by the vectors H1, H2, E1, F1. We are going to show that iX (η3) 6= 0 for any X ∈ ᏸ. Any X ∈ ᏸ can be written in the GEOMETRIC FORMALITY OF HOMOGENEOUS SPACES AND OF BIQUOTIENTS 163 form X = aH1 +bH2 +cE1 +d F1. To prove our claim, we consider the following cases: Case 1: If d 6= 0, then using the well-known relation between canonical generators of a simple Lie algebra and the root spaces related to the Killing form, we get η3(E1, H1, X) = −2dhE1, F1i 6= 0.

Case 2: If d = 0 and c 6= 0, then we have η3(F1, H1, X) = 2chF1, E1i 6= 0. Case 3: If c = d = 0, then we have

η3(F1, X, E1) = (2a − b)hF1, E1i and η3(F2, X, E2) = (2b − a)hF2, E2i, so there are always Ei and Fi for which η3(Fi , X, Ei ) 6= 0. This implies that ᏸ ∩ ᏷ = 0, which is impossible for dimension reasons. Remark 9. There are exactly two fibrations with fiber S2 over S5; cf. Section 5. The trivial bundle S2 ×S5 is of course geometrically formal with respect to product metrics. The nontrivial bundle is the 3-symmetric space SU(3)/T 1, where T 1 is embedded inside an SU(2) ⊂ SU(3). In the above notation, this is the case k = −l, which is excluded in the definition of Aloff–Wallach spaces. Nevertheless, the argument above applies to show that a normal homogeneous metric is not geometrically formal.

Remark 10. The Aloff–Wallach spaces have interesting homogeneous metrics that are not normal homogeneous. These include metrics of positive sectional curvature [Aloff and Wallach 1975] and Einstein metrics, some of which admit Killing spinors. The latter metrics are not geometrically formal because of the following result, communicated to us by U. Semmelmann: A metrically formal Riemannian spin manifold M of dimension ≥ 5 admitting a nontrivial Killing spinor must have vanishing second Betti number. It is known that a metric admitting a nontrivial Killing spinor must be Einstein, and thus is very special.

3. Some algebraic topology of Stiefel manifolds

In this section we prove that the homogeneous spaces listed in Theorem 2 are not homotopy equivalent to symmetric spaces or to products of real homology 2n spheres, except for the second property in the case of V3(ޒ ) with n even. To- gether with Proposition 5 proved in the previous section, this completes the proof of Theorem 2. n Let us consider ﬁrst the complex Stiefel manifolds V2(ރ ) = SU(n)/ SU(n −2) consisting of orthonormal pairs of vectors for the standard Hermitian inner product on ރn, with n ≥ 3. Projecting such a pair to its ﬁrst entry, we obtain a smooth 164 D. KOTSCHICK AND S.TERZIC´

n 2n−1 2n−3 fibration of V2(ރ ) over S with fiber S . It is a classical problem in homotopy theory to determine when the total space of such a fiber bundle is homotopy equivalent to the product of base and fiber. In the case at hand, for even n the action of the quaternions on ރn = ވn/2 defines a section of the fibration, which n splits the long exact homotopy sequence and makes V2(ރ ) indistinguishable from S2n−1 × S2n−3 at the level of homotopy groups. However, the following result was proved by James and Whitehead [1954] modulo Adams’s solution of the Hopf invariant-one problem: Theorem 11 [James and Whitehead 1954; Adams 1960]. For n ≥ 3, the Stiefel n 2n−1 2n−3 manifold V2(ރ ) = SU(n)/ SU(n−2) is not homotopy equivalent to S ×S , unless possibly if n = 4.

n In fact, James and Whitehead proved that if V2(ރ ) = SU(n)/ SU(n − 2) is homo- 2n−1 2n−3 2n topy equivalent to S × S , then π4n−1(S ) contains an element of Hopf invariant one. By the result of Adams, it follows that n is in {1, 2, 4}. This combination of results is also mentioned in [James 1971, Theorem 1.7], but the exceptional case is misstated there. In the notation of [James 1971], the exceptional case should be denoted n = 4 and k = 2, not n = k = 2. Remark 12. Some years after the results of [James and Whitehead 1954] and n [Adams 1960], Gilmore [1967] showed that π2n−2(V2(ރ )) is trivial for odd n. 2n−1 2n−3 2n−3 Since π2n−2(S × S ) = π2n−2(S ) = ޚ2, this gives another proof of Theorem 11 for odd n. As we remarked earlier, there is no such proof in the case of even n. The arguments of James and Whitehead only show that the existence of an n element of Hopf invariant one is necessary for V2(ރ ) = SU(n)/ SU(n − 2) to be homotopy equivalent to S2n−1 × S2n−3. It turns out that this condition is in fact sufﬁcient not just for homotopy equivalence, but for diffeomorphism. This following result completes the proof of case (3) in Theorem 2.

n Theorem 13. If a complex Stiefel manifold V2(ރ ) = SU(n)/ SU(n−2) with n ≥ 3 is homotopy equivalent to a symmetric space or to a product of real homology 3 spheres, then n = 3 or n = 4. In the ﬁrst case, V2(ރ ) = SU(3) is a symmetric space not homotopy equivalent to a product of real homology spheres. In the second case, 4 5 7 V2(ރ ) = SU(4)/ SU(2) is diffeomorphic to S × S . n Proof. Suppose that V2(ރ ) is homotopy equivalent to a product X1 × X2 of real n homology spheres. Then because V2(ރ ) is simply connected, so are both the Xi . n Moreover, because V2(ރ ) has the integral homology of a product of spheres, it follows that each Xi is an integral homology sphere. Thus the Xi are homotopy spheres. GEOMETRIC FORMALITY OF HOMOGENEOUS SPACES AND OF BIQUOTIENTS 165

n Now, if V2(ރ ) is homotopy equivalent to a product of (homotopy) spheres, then 4 5 7 Theorem 11 gives n = 4. Conversely, V2(ރ ) is an S bundle over S with structure group SU(4). The corresponding clutching map gives an element of π6(SU(4)), 4 which is trivial [Borel and Serre 1953]. Thus V2(ރ ) is diffeomorphic to the total space of the trivial bundle S5 × S7; compare with Proposition 7. n Next suppose that V2(ރ ) is homotopy equivalent to a symmetric space. On- ishchik [1963] and Kramer [2002] have classified the homogeneous spaces with the cohomology of a product of odd-dimensional spheres. If we assume that G/H has the integral cohomology of such a product, then apart from products of spheres and a handful of symmetric spaces of dimension ≤ 14, the remaining examples are exactly the spaces listed as (3)–(6) in Theorem 2 and the symmetric space E6/F4 9 17 n with the cohomology of S × S . If n ≥ 5, then the dimension of V2(ރ ) is ≥ 16, so we do not have to consider the sporadic irreducible symmetric spaces of dimension ≤ 14 arising in the classification of Onishchik and Kramer. Similarly, n n V2(ރ ) cannot be homotopy equivalent to E6/F4 for dimension reasons. If V2(ރ ) were homotopy equivalent to a reducible symmetric space, then each factor would be a homotopy sphere, and we would be in the case excluded already. 3 Finally, V2(ރ ) = SU(3) is of course a symmetric space, but it is not homotopy equivalent to a product of spheres. For if it were, then the spheres would have dimensions 3 and 5, leading to a contradiction with π6(SU(3)) = ޚ6 (cf. [Borel 3 and Serre 1953]), since π6(S ) = ޚ12. In a similar way one proves the corresponding result for the quaternionic Stiefel manifolds. The statement is cleaner since an exceptional case not excluded by the Hopf invariant does in fact not occur. This following result completes the proof of case (4) in Theorem 2. n Theorem 14. No quaternionic Stiefel manifold V2(ވ ) = Sp(n)/ Sp(n − 2) with n ≥ 2 is homotopy equivalent to a product of real homology spheres. If it is homotopy equivalent to a symmetric space, then n = 2. n Proof. If V2(ވ ) is homotopy equivalent to a nontrivial product, then as in the n previous proof, we may assume that each factor is a sphere. Assuming that V2(ވ ) is homotopy equivalent to S4n−5 × S4n−1, James and Whitehead proved in [1954, 4n Theorem 1.21] that π8n−1(S ) contains an element of Hopf invariant one. The result of Adams [1960] then implies that n = 2. However, for n = 2, we have 2 V2(ވ ) = Sp(2), and this is not homotopy equivalent to a product of spheres, for example because π6(Sp(2)) is trivial as first proved by Borel and Serre [1953]. n Next suppose that V2(ވ ) is homotopy equivalent to a symmetric space. If n ≥3, n then the dimension of V2(ވ ) is ≥ 18, so that, again, we do not have to consider the sporadic irreducible symmetric spaces of dimension ≤ 14 arising in the classification due to Onishchik [1963] and Kramer [2002]. For dimension reasons, E6/F4 166 D. KOTSCHICK AND S.TERZIC´

n cannot occur either. In the reducible case V2(ވ ) would be homotopy equivalent to a product of spheres, and this we have excluded already. Finally, for n = 2 we 2 have the symmetric space V2(ވ ) = Sp(2). Remark 15. Instead of using the results of James–Whitehead [1954] and Adams [1960], we could, in most cases, appeal to the calculations of Oguchi [1969]. On 4n−1 4n−5 4n−5 the one hand we have π4n−2(S × S ) = π4n−2(S ) = ޚ24 for n ≥ 3. On n the other, by [Oguchi¯ 1969], π4n−2(V2(ވ )) = ޚd , where d = gcd{n, 24}. Thus, n whenever n is not divisible by 24, one concludes that V2(ވ ) is not homotopy equivalent to a product of spheres. To complete the proof of case (5) in Theorem 2, we prove the following: 2 Theorem 16. The octonionic Stiefel manifold V2(ޏ ) = Spin(9)/G2 is not homotopy equivalent to a symmetric space, or to a product of real homology spheres. Proof. This manifold has the integral homology of S7 × S15. As in the previous 2 proofs, if V2(ޏ ) were homotopy equivalent to a nontrivial product, then this prod- 7 15 2 7 15 uct would have to be S × S . We now prove that V2(ޏ ) and S × S are in fact not homotopy equivalent, distinguished by their homotopy groups. On the one hand, we have 7 15 7 π14(S × S ) = π14(S ) = ޚ120. 2 On the other hand, we will show that π14(V2(ޏ )) has order at most 16. For this 2 we consider the principal G2 bundle with total space Spin(9) over V2(ޏ ), and the following piece of its exact homotopy sequence: 2 · · · → π14(Spin(9)) → π14(V2(ޏ )) → π13(G2) → · · ·

The group on the right is trivial and the group on the left is ޚ8 ⊕ޚ2, by calculations of [Mimura 1967]. 2 The proof that V2(ޏ ) is not homotopy equivalent to a symmetric space is the same as for the complex and quaternionic Stiefel manifolds. First of all, reducible 2 symmetric spaces cannot arise because V2(ޏ ) is not homotopy equivalent to a product, as we just proved. Second of all, there is no irreducible symmetric space with the correct cohomology. Next we deal with the space Spin(10)/ Spin(7), case (6) in Theorem 2. Theorem 17. The quotient Spin(10)/ Spin(7) is not homotopy equivalent to a symmetric space, or to a product of real homology spheres. Proof. Consider the principal Spin(9) bundle Spin(10) → Spin(10)/ Spin(9) = S9. The spin representation of Spin(9) on ޒ16 associates to this principal bundle a real vector bundle V of rank 16. Our homogeneous space X = Spin(10)/ Spin(7) is the unit sphere bundle in V , with ﬁber Spin(9)/ Spin(7) = S15. GEOMETRIC FORMALITY OF HOMOGENEOUS SPACES AND OF BIQUOTIENTS 167

The manifold X has the integral homology of S9 × S15. As in the previous proofs, if X were homotopy equivalent to a nontrivial product, then this product would have to be Y = S9 × S15. We will show below that in fact X and Y are not homotopy equivalent. The proof that X is not homotopy equivalent to a symmetric space is then the same as for the complex and quaternionic Stiefel manifolds. First, reducible symmetric spaces cannot arise because X is not homotopy equivalent to a product. Second, there is no irreducible symmetric space with the correct cohomology. Now we begin the proof that X and Y = S9 × S15 are not homotopy equivalent. 9 The principal bundle Spin(10) → S corresponds to an element of π8(Spin(9)), whose image in π8(SO(16)) classifies V . Since V is nontrivial, this element is the nontrivial element of π8(SO(16)) = ޚ2. Recall that the fibration X −→ S9 has a section for dimension reasons. Now [James and Whitehead 1954, Corollary 1.9] tells us that X is homotopy equivalent to Y = S9 × S15 if and only if a certain invariant λ(X) vanishes. This invariant is 15 an element in the group 39,15 = Im J/ Im P, where J : π8(SO(15)) → π23(S ) 15 15 is the classical J-homomorphism and P : π9(S ) → π23(S ) is, in our case, the zero map, as its domain is zero. Thus, in our case, λ(X) ∈ J(π8(SO(15)), and we only have to determine whether this is zero. Unravelling the definition of λ given in [James and Whitehead 1954], we find the following: if we identify π8(SO(15)) with π8(SO(16)), then λ(X) is just the image under the classical J-homomorphism of the classifying element of V in π8(SO(16)). Now π8(SO(15)) has order 2, and we know that the image of the J-homomorphism in this degree is the group J(S9), also of order 2, cf. [Bott 1969; Husemoller 1975]. Thus the J-homomorphism is an isomorphism, and λ(X) 6= 0 because the classifying map of V represents the nonzero element of π8(SO(16)). π Remark 18. The existence of a section to the fibration S15 → X −→ S9 implies that X and Y = S9 × S15 have isomorphic homotopy groups. Thus there can be no easy argument to prove that they are not homotopy equivalent. It remains to discuss the homotopy types of the real Stiefel manifolds occurring in cases (1) and (2) of Theorem 2. They are geometrically formal by Proposition 5, and they are not homotopy equivalent to products of spheres because of the pres- ence of torsion in their integral homology. However, for these manifolds it is more difficult to exclude the homotopy equivalence to products of real homology spheres, because the factors in such a decomposition would not necessarily be homotopy spheres. The following result completes the proof of Theorem 2. 2n+1 2n Theorem 19. The real Stiefel manifolds V4(ޒ ) and V3(ޒ ) with n ≥ 3 are not homotopy equivalent to symmetric spaces. They are not homotopy equivalent 2n to nontrivial products, except possibly for V3(ޒ ) with n even. 168 D. KOTSCHICK AND S.TERZIC´

Proof. These manifolds have the real cohomology algebras of S4n−5 × S4n−1 and S2n−1 × S4n−5, respectively. Therefore, if one of them is homotopy equivalent to a nontrivial product, then the factors are real homology spheres of dimensions 4n−5 and 4n − 1, respectively 2n − 1 and 4n − 5. For the real Stiefel manifolds the cohomology with coefﬁcients in ޚ2 and the Steenrod operations on it were determined by Borel [1953a]. From the structure 2n+1 of the Steenrod operations, Hsiang and Su [1968] deduced that V4(ޒ ) and 2n V3(ޒ ) cannot be homotopy equivalent to nontrivial products, except possibly in the second case, when n is even. Suppose that one of these manifolds is homotopy equivalent to a symmetric space. Then this symmetric space is reducible, and we are in the exceptional case 2n where V3(ޒ ) could be homotopy equivalent to a product. To see this, consider the irreducible symmetric spaces. Computations of their real cohomology algebras [Borel 1953b; Takeuchi 1962] or of their rational homotopy groups [Terzic´ 2003] show that, among them, only

SU(5)/ SO(5), SU(6)/ Sp(3), E6/F4, Sp(2), SU(3), G2, Spin(4) have the real cohomology algebra of a product of odd-dimensional spheres, with the dimensions given by the pairs

(5, 9), (5, 9), (9, 17), (3, 7), (3, 5), (3, 11), (3, 3), respectively. None of these dimension pairs is of the form (4n−5, 4n−1) or 2n+1 2n (2n−1, 4n−5) with n ≥ 3. Therefore, V4(ޒ ) or V3(ޒ ) cannot be homotopy equivalent to any irreducible compact simply connected symmetric space. 2n Finally we have to consider the possibility that V3(ޒ ) with n even could be homotopy equivalent to a reducible symmetric space. Then each factor would be a compact symmetric space with the real cohomology of S2n−1, respectively 4n−5 2n S . Since V3(ޒ ) has torsion in its integral homology, at least one of the factors has to be a simply connected symmetric space having the real cohomology of a sphere, but different integral cohomology. According to the classification, due to Onishchik [1963; 1995], of homogeneous spaces whose real cohomology is that of a sphere, the only possibility is SU(3)/ SO(3), of dimension 5. For even n, neither 2n − 1 nor 4n − 1 can equal 5, and this contradiction completes the proof. Remark 20. Projection of a 3-frame onto its first entry defines a smooth fibration 2n 2n−1 2n−1 2n−1 of V3(ޒ ) over S , with fiber V2(ޒ ). Note that the fiber V2(ޒ ) is a real homology sphere, but not an integral one. As far as we know, it is still an open problem whether for some even n the total space could be homotopy equivalent, or even diffeomorphic, to the product of base and fiber; see the discussion in GEOMETRIC FORMALITY OF HOMOGENEOUS SPACES AND OF BIQUOTIENTS 169

[James 1971]. If this happens for some n, then by a result of James [1971], this n is not just even, but a power of 2.

Onishchik [1963] and Kramer [2002] have classiﬁed all the compact homogeneous spaces with the real cohomology of a product of odd-dimensional spheres. In addition to nontrivial products and the spaces we have discussed already in Theorem 2 and its proof, there is a large number of sporadic cases that have the real cohomology of a product of spheres but different integral cohomology. All these spaces are geometrically formal by Proposition 5. To end this section, we discuss the diffeomorphism type of one of these sporadic examples. Consider the composition of standard inclusions SO(3) ⊂ SU(3) ⊂ SU(4). The quotient X = SU(4)/ SO(3) has the real cohomology of S5 × S7, but different integral cohomology.

Proposition 21. The manifold X = SU(4)/ SO(3) is not diffeomorphic to a nontrivial product, nor to a symmetric space.

Proof. Suppose X were diffeomorphic to some nontrivial product. Then the factors would have to be simply connected real homology spheres of dimensions 5 and 7, respectively. Since we are assuming that X is diffeomorphic to the product, not just homotopy equivalent to it, the factors must in fact be homogeneous spaces. There- fore, all the candidates must occur in the classification of Onishchik and Kramer. In dimension 5, the candidates are S5 and SU(3)/ SO(3); in dimension 7, they 7 5 are S , V2(ޒ ) and Sp(2)/ Sp(1) with a nonstandard embedding of the subgroup. Looking at the third homotopy groups, we have π3(X) = π3(SU(3)/ SO(3)) = ޚ4, 5 but π3(V2(ޒ )) = ޚ2 and π3(Sp(2)/ Sp(1)) = ޚ10; cf. [Kramer 2002, page 65]. Thus, on the one hand, the only product of homogeneous spaces that has the same third homotopy group as X is S7 × SU(3)/ SO(3). On the other hand, X and S7×SU(3)/ SO(3) have different sixth homotopy groups. This is so because the exact homotopy sequence of the fibration SU(4) → X shows that π6(X) = π5(SO(3)), which is of order 2, whereas π6(SU(3)/ SO(3)) is known to be of order 4; see [Lundell 1992]. Finally, if X were diffeomorphic to a symmetric space, then by what we just proved, that symmetric space would have to be irreducible. However, by the classification of irreducible symmetric spaces, there is no such space with the real 5 7 cohomology of S × S .

Note that if we discuss only the homotopy type of X = SU(4)/ SO(3), then we have to consider products of real homology spheres that are not necessarily homogeneous, and the argument breaks down. 170 D. KOTSCHICK AND S.TERZIC´

4. Biquotients

Let G be a compact Lie group, and U a closed subgroup of G × G that acts freely · = −1 on G by (u1, u2) g u1gu2 . Then the orbit space is a smooth manifold denoted by G/U and is called a biquotient of G. A biinvariant metric on G descends to a metric on any biquotient. More generally, one can consider subgroups U ⊂ ISO(G) acting freely. In some cases ISO(G) is strictly larger than G ×G, so one gets more examples. Biquotients were ﬁrst studied systematically by Eschenburg [1992a; 1992b] as a source of examples of manifolds with positive sectional curvature. Among Es- chenburg’s biquotients in [Eschenburg 1992a] there are several examples for which we can prove easily that they are not geometrically formal. Example 22. There is a 6-dimensional example M = G/U whose real cohomology is very similar but not isomorphic to that of SU(3)/T 2, considered in [Kotschick and Terzic´ 2003]. Moreover, M and SU(3)/T 2 have the same integral homology groups and the same cohomology algebra mod 2. We can show that this biquotient is not geometrically formal by using almost the same argument as the one we used for SU(3)/T 2 in [Kotschick and Terzic´ 2003]. Let G = U(3) and

U ⊂ {(D(a, a, a¯), D(b, c, 1)); a, b, c ∈ S1} ⊂ G × G.

∗ Eschenburg [1992a] proved that H (G/U) is generated by two generators√ x and y 2 2 3 of degree 2 with the relations xy√= y −x and x = 0. Setting z = (1/ 5)(x −2y), we get z2 = x2 and z3 = −(2/ 5) xy2 6= 0. If G/U is geometrically formal, then it has two 2-forms x and z, such that x3 = 0, x2 = z2, and z3 is a volume form. If G/U were geometrically formal, then for a formal metric these relations would hold pointwise for the harmonic forms representing the cohomology classes x and z and their products. The form x would then have rank 4 everywhere. If v 2 2 were a vector in its kernel, the relation x = z would show that ivz ∧ z = 0, which would contradict the fact that z3 would have to be a volume form. The second example uses a different argument to obstruct geometric formality, related to symplectic structures deﬁned by harmonic forms of formal metrics. Example 23. Let G = SU(3) and

U = T 1 × T 1 = {D(ak, al , a−k−l ), D(bm, bn, b−m−n); a, b ∈ S1}.

Using the results of [Eschenburg 1992a] on the cohomology of biquotients, it is easy to compute H ∗(G/U). We get the following algebra structure: There are two linearly independent generators x and y in degree 2, subject to the relations x2 = y2 and x3 = y3. Then H 4 is spanned by xy and x2 = y2, and H 6 is spanned GEOMETRIC FORMALITY OF HOMOGENEOUS SPACES AND OF BIQUOTIENTS 171 by x3 = y3 = x2 y = xy2. If we assume that M is geometrically formal, then the harmonic form representing x + y is a symplectic form. It then follows from (x − y)(x + y) = 0 that x − y vanishes, which contradicts the linear independence of x and y, because on a symplectic six-manifold the wedge product with the symplectic form is an isomorphism between two-forms and four-forms. Thus M cannot be geometrically formal. Next we consider Totaro’s biquotients of S3×S3×S3, which he studied in [2003] as an example of a family of 6-manifolds with nonnegative sectional curvature, but with infinitely many distinct isomorphism classes of rational cohomology rings. For all these manifolds we prove that they cannot be geometrically formal because of the structure of their cohomology rings. Theorem 24. The biquotients studied in [Totaro 2003] are not geometrically formal. Proof. The rational cohomology ring of the biquotients that Totaro considers has three generators x1, x2 and x3 in degree 2, satisfying the relations 2 x1 = 0, 2 (7) ax1x2 + x2x3 + x2 = 0, 2 bx1x3 + 2x2x3 + x3 = 0, for some integers a and b; see [Totaro 2003]. If we assume that these manifolds are geometrically formal, then the same relations (7) hold at the level of their representative harmonic forms, which we denote by the same letters. In order to prove that geometric formality leads to a contradiction, we differentiate the following cases according to the values of the constants a and b. Case 1: We first consider the case when a·b 6=0. Then we can obviously normalize x1 so that a becomes 1. Consider the forms = + 3 = + 3 y1 x1 b x2 and y2 x1 2 x3. 3 = 3 = 2 2 2 Using the relations (7), we see that y1 y2 0, while x1 y1 , x1 y2 and y1 y2 are volume forms. Since the cubes of y1 and y2 vanish, we have dim Ker(y1) = dim Ker(y2) = 2 2 ∩ = and, since y1 y2 is a volume form on M, it follows that Ker(y1) Ker(y2) 0. If we rewrite the relations (7) in terms of x1, y1 and y2, we find, after some straightforward calculations, that 2 x1 = 0, 2 (1 − 2b)x1 y1 − 2x1 y2 + 2y1 y2 + by1 = 0, 2 −2bx1 y1 + (b − 4)x1 y2 + 2by1 y2 + 2y2 = 0. 172 D. KOTSCHICK AND S.TERZIC´

If we multiply the ﬁrst relation by b − 4, and add it to the second one multiplied by 2, we get

2 2 2 (8) (5b − 2b − 4)x1 y1 + (6b − 8) y1 y2 + b(b − 4) y1 + 4y2 = 0.

Let u1 ∈ Ker(y1) and u2 ∈ Ker(y2) be such that u2 ∈/ ޒ{u1} ⊕ Ker(x1) (such a u2 always exists, otherwise we would have Ker(x1) ∩ Ker(y2) 6= 0 which is a 2 contradiction with the fact that x1 y2 is a volume form). In other words, Tp M = {u1, u2}⊕Ker(x1). Then if we contract the equation (8) with u1 and u2, we get − 2 − ∧ + − ∧ = (9) (5b 2b 4)iu2 (y1 iu1 x1) (6b 8)iu2 y1 iu1 y2 0.

Look at the 1-forms iu2 y1 and iu1 y2. Since the dimension of the kernel of each of them is ≥ 5, it follows that the intersection L of their kernels has dimension ≥ 4. 2 = = ∩ ≥ Since x1 0, it follows that dim Ker(x1) 4, and thus dim(Ker(x1) L) 2. Let w ∈ Ker(x1) ∩ L. If we contract (9) with w, we get x1(u1, u2) iw y1 = 0, since the coefﬁcient 5b − 2b2 − 4 has no real zeros. 2 This is in contradiction with the fact that x1 y1 is a volume form. Namely, take w1, w2, w3 ∈ Ker(x1) such that together with u1, u2, w, they form a basis of Tp M. 2 = Then we have x1 y1 (u1, u2, w, w1, w2, w3) 0, which is impossible.

Case 2: If a = 0 and b 6= 0, we ﬁrst normalize x1 to get b = 1 and proceed as in the previous case, taking y1 = x1 + 3x2 and y2 = x1 + 6x3.

Case 3: If a 6= 0 and b = 0, we again ﬁrst normalize x1 to have a = 1, and take = = + 3 y1 x2 and y2 x1 2 x3. Case 4: For a = b = 0, we have the following relations in cohomology:

2 2 2 x1 = 0, x2x3 + x2 = 0, 2x2x3 + x3 = 0. = + = + 1 Take y1 x2 x3 and y2 x2 2 x3. The cohomology relations imply that 3 = 3 = 2 y1 y2 0, and that y1 y2 is a volume form on M. If we rewrite these relations in terms of y1 and y2, we obtain

2 2 y2 − y1 y2 = 0 and y1 − 2y1 y2 = 0. ∈ 2 = Now take u Ker(y1). The relations imply that iu y2 0, which contradicts that 2 y1 y2 is a volume form.

5. Two-sphere bundles over the complex projective plane

For our calculations in [Kotschick and Terzic´ 2003], the example SU(3)/T 2 was the crucial case, from which all others were derived by various generalizations. The inclusions T 2 ,→ U(2),→ SU(3) show that SU(3)/T 2 ﬁbers over ރP2 with GEOMETRIC FORMALITY OF HOMOGENEOUS SPACES AND OF BIQUOTIENTS 173

fiber S2. In fact, this fibration is well known as the twistor fibration of ރP2; compare with [Kotschick and Terzic´ 2009]. We now generalize to arbitrary 2-sphere fibrations over ރP2. Assume first that we have an arbitrary smooth oriented fibration with fiber S2. Because the orientation-preserving diffeomorphism group of S2 is homotopy equivalent to SO(3), we may assume that the structure group is SO(3). Then M is the unit sphere bundle in the associated rank-3 vector bundle V . As SO(3) coincides with the projective unitary group PU(2), every 2-sphere bundle is the projectivization of a complex rank-2 vector bundle E. One can recover V by passing to the adjoint bundle. We have

2 w2(V ) = c1(E) (mod 2) and p1(V ) = c1(E) − 4c2(E).

In many cases, for example when w2(V ) is nontrivial and p1(V ) is divisible by 8, it follows from [Grove and Ziller 2008, Theorem C] that M admits a cohomogeneity- one action of SO(3) × SO(3). Thus, these manifolds are highly symmetric, and are, in this sense, close to homogeneous. Theorem 25. Let M6 be the total space of an S2 bundle E over ރP2. Then M is geometrically formal if and only if it is the trivial bundle S2 × ރP2. Proof. Assume M6 is the total space of the projectivization of a complex rank-2 vector bundle E over ރP2. Then the cohomology of M is multiplicatively generated by two degree 2 classes x and y, where we use x for the generator pulled back from ރP2 and y for a class that restricts as a generator to every ﬁber. By the deﬁnition of Chern classes we choose y so that

2 2 (10) y + c1(E)xy + c2(E)x = 0, where, by an obvious abuse of notation, we use ci (E) for the Chern numbers i 3 hci (E), [ރP ]i. We also have x = 0. After replacing y by a linear combination of x and y we may assume that

(11) y2 + cx2 = 0, = − 1 where c 4 p1(V ) vanishes if and only if M is the trivial bundle. To pass from (10) to (11), we can twist E by a line bundle. This does not affect the projectivization, but we can kill the ﬁrst Chern class if w2(V ) = 0. If this is not the case, we can still make the required base change, which amounts to twisting by a virtual line bundle whose Chern class is half-integral. In this case the constant c is not integral. Now (11) together with x3 = 0 implies that xy2 = 0. We also have y3 +cx2 y = 0. If c 6= 0, we conclude that y3 6= 0, for otherwise there would be no degree 6 cohomology in M. Thus, if c 6= 0 and M is geometrically formal, then the harmonic 174 D. KOTSCHICK AND S.TERZIC´

2-form representing y is nondegenerate. However, since x3 = 0 the harmonic form representing x has a nontrivial kernel and, if we contract (11) with an element v in this kernel, we deduce 2 iv y ∧ y = 0, which contradicts the nondegeneracy of y. Thus we have proved that a nontrivial M cannot be geometrically formal. Con- versely, the trivial bundle S2×ރP2 is geometrically formal with respect to a product of Kahler¨ metrics; cf. [Kotschick 2001]. In fact, it is a symmetric space.

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Received September 17, 2009.

D.KOTSCHICK MATHEMATISCHES INSTITUT LMUMÜNCHEN THERESIENSTR. 39 80333 MÜNCHEN GERMANY [email protected]

S.TERZIC´ FACULTY OF SCIENCE UNIVERSITYOF MONTENEGRO DŽORDŽA VAŠINGTONABB 81000 PODGORICA MONTENEGRO [email protected] PACIFIC JOURNAL OF MATHEMATICS Vol. 249, No. 1, 2011

POSITIVE SOLUTIONS FOR A NONLINEAR THIRD ORDER MULTIPOINT BOUNDARY VALUE PROBLEM

YANG LIU,ZHANG WEIGUO,LIU XIPING,SHEN CHUNFANG AND CHEN HUA

By using the Avery–Peterson ﬁxed point theorem, we obtain the existence of three positive solutions for a third order multipoint boundary value problem. An example illustrates the main results.

1. Introduction

We study the existence of positive solutions for the third order m-point boundary value problem x000(t) + f (t, x(t), x0(t), x00(t)) = 0 for t ∈ [0, 1], m−2 m−2 X 0 X 0 (1-1) x(1) = βi x(ξi ), x (0) = αi x (ξi ), i=1 i=1 x00(0) = 0, where 0 < ξ1 < ξ2 < ··· < ξm−2 < 1,

0 ≤ αi < 1 and 0 ≤ βi < 1 for i = 1, 2,..., m − 2, m−2 m−2 X X αi < 1 and βi < 1 i=1 i=1 and f ∈ C([0, 1] × [0, +∞) × R2, [0, +∞)). Third order differential equations arise in various areas of applied mathematics and physics, such as the deﬂection of a curved beam having a constant or varying cross section, three layer beams, electromagnetic waves, gravity driven ﬂows, and so on [Gregusˇ 1987]. In recent years, much attention has focused on positive solutions of boundary value problems (BVPs for short) for third order ordinary

MSC2000: primary 34B10; secondary 34B15. Keywords: third order multipoint boundary value problem, positive solution, cone, ﬁxed point. Sponsored by the Shanghai Leading Academic Discipline Project (number S30501), the Shanghai Natural Science Foundation (number 10ZR1420800), the Natural Science Foundation of Anhui Ed- ucational Department (number Kj2010B163), and the Youth Research Fund of ChangZhou Campus (number XZX/08B001-05). Zhang is the corresponding author.

177 178 YANG LIU, ZHANG WEIGUO, LIU XIPING, SHEN CHUNFANG AND CHENHUA differential equations. For example, Anderson [1998] established the existence of at least three positive solutions to the problem

−x000(t) + f (x(t)) = 0 for t ∈ (0, 1), 0 00 x(0) = x (t2) = x (1) = 0 for t2 ∈ (0, 1), where f : R → [0, +∞) is continuous and 1/2 ≤ t2 < 1. Palamides and Smyrlis [2008] proved that there exists at least one positive solution for the third order three-point BVP

x000(t) = a(t) f (t, x(t)) for t ∈ (0, 1), x(0) = x(1) = 0, x00(η) = 0 for η ∈ (0, 1).

The results are based on the well-known Guo–Krasnoselski˘ı ﬁxed point theorem [Guo and Lakshmikantham 1988]. Guo, Sun and Zhao [Guo et al. 2008] studied the positive solutions of the third order three-point problem

x000(t) = a(t) f (x(t)) for t ∈ (0, 1), x(0) = x0(0) = 0, x0(1) = x0(η) for η ∈ (0, 1), and obtained the existence of such solutions by using the Guo–Krasnoselski˘ı fixed point theorem. For more existence results for third order boundary value problems, see [Hopkins and Kosmatov 2007; Chu and Zhou 2006; Graef and Kong 2009; Lin et al. 2008; Pei and Chang 2007; Li 2006; Yao 2004] and references therein. In these works they concentrate on the two- or three-point BVPs. Few papers deal with the existence of positive solutions to m-point BVPs for third order differential equations, and in those that do, first and second order derivatives are not involved in the nonlinear term. We do allow first and second order derivatives to appear explicitly in the nonlinear term of multipoint third order boundary value problems (1-1), and consider the positive solutions of these problems. By using the Avery–Peterson fixed point theorem [2001] and analysis techniques, we prove that there exist at least three concave positive solutions of problem (1-1). The results so established are more general those of previous papers. We illustrate our results with an example.

2. Background

In this section, we present the necessary deﬁnitions from cone theory in Banach spaces and a ﬁxed point theorem due to Avery and Peterson. POSITIVE SOLUTIONS FOR A MULTIPOINT BOUNDARY VALUE PROBLEM 179

Definition 2.1. Let E be a real Banach space over ޒ. A nonempty convex closed set P ⊂ E is said to be a cone if (1) au ∈ P for all u ∈ P and a ≥ 0, and (2) u, −u ∈ P implies u = 0. Definition 2.2. An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets. Definition 2.3. A map α is said to be a nonnegative continuous convex functional on a cone P of a real Banach space E if α : P → [0, +∞) is continuous and α(tx + (1 − t)y) ≤ tα(x) + (1 − t)α(y) for all x, y ∈ P, t ∈ [0, 1]. Definition 2.4. A map β is said to be a nonnegative continuous concave functional on a cone P of a real Banach space E if β : P → [0, +∞) is continuous and β(tx + (1 − t)y) ≥ tβ(x) + (1 − t)β(y) for all x, y ∈ P, t ∈ [0, 1]. Let γ and θ be nonnegative continuous convex functionals on P, let α be a nonnegative continuous concave functional on P and let ψ be a nonnegative continuous functional on P. Then for positive numbers a, b, c and d, we define the convex sets P(γ, d) = {x ∈ P | γ (x) < d}, P(γ, α, b, d) = {x ∈ P | b ≤ α(x), γ (x) ≤ d}, P(γ, θ, α, b, c, d) = {x ∈ P | b ≤ α(x), θ(x) ≤ c, γ (x) ≤ d}, and a closed set R(γ, ψ, a, d) = {x ∈ P | a ≤ ψ(x), γ (x) ≤ d}. Lemma 2.5. Let P be a cone in a Banach space E. Let γ and θ be nonnegative continuous convex functionals on P, let α be a nonnegative continuous concave functional on P, and let ψ be a nonnegative continuous functional on P satisfying (2-1) ψ(λx) ≤ λψ(x) for 0 ≤ λ ≤ 1, such that for some positive numbers l and d, (2-2) α(x) ≤ ψ(x) and kxk ≤ lγ (x) for all x ∈ P(γ, d). Suppose T : P(γ, d) → P(γ, d) is completely continuous and there exist positive numbers a, b, c with a < b such that

(S1) {x ∈ P(γ, θ, α, b, c, d) | α(x) > b} 6= ∅ and α(T x) > b for x ∈ P(γ, θ, α, b, c, d),

(S2) α(T x) > b for x ∈ P(γ, α, b, d) with θ(T x) > c, 180 YANG LIU, ZHANG WEIGUO, LIU XIPING, SHEN CHUNFANG AND CHENHUA

(S3)0 6∈ R(γ, ψ, a, d) and ψ(T x) < a for x ∈ R(γ, ψ, a, d) with ψ(x) = a.

Then T has at least three ﬁxed points x1, x2, x3 ∈ P(γ, d) such that

γ (xi ) ≤ d for i = 1, 2, 3, (2-3) b < α(x1), a < ψ(x2), α(x2) < b, ψ(x3) < a.

3. Main results

Consider the problem

(3-1a) x000(t) + y(t) = 0 for t ∈ [0, 1], m−2 m−2 00 0 X 0 X (3-1b) x (0) = 0, x (0) = αi x (ξi ), x(1) = βi x(ξi ) i=1 i=1 Lemma 3.1. Let

ξ0 = 0, ξm−1 = 1, α0 = αm−1 = β0 = βm−1 = 0, Pm−1 Pm−1 ρ = (1 − i=0 αi )(1 − i=0 βi ) > 0. For y(t) ∈ C[0, 1], the problem (3-1) has the unique solution Z 1 Z s x(t) = G(t, s) y(τ)dτds, 0 0 where Pm−1 α (s − t) + Pm−1 β (t − s) + Pi−1 β (t − ξ )  k=i k k=i k k=0 k k  Pi−1 Pm−1 1  + (1 − = αk) (1 − s) + = βk(s − ξk) if t ≤ s, G(t, s) = k 0 k i ρ − Pi−1 − t + Pi−1 t − s + Pm−1 t − (1 k=0 αk) (1 ) k=0 βk( ) k=i βk( ξk)  Pm−1 Pi−1 + k=i αk k=0 βk(s − ξk) if t ≥ s, for ξi−1 < s < ξi and i = 1, 2,..., m − 1. Proof. Integrating both sides of (3-1a) and considering the boundary condition x00(0) = 0, we have Z t (3-2) −x00(t) = y(s)ds. 0 Let G(t, s) be the Green function for the problem

(3-3a) −x00(t) = 0, m−1 m−1 0 X 0 X (3-3b) x (0) = αi x (ξi ), x(1) = βi x(ξi ), i=0 i=0 POSITIVE SOLUTIONS FOR A MULTIPOINT BOUNDARY VALUE PROBLEM 181

Then for ξi−1 < s < ξi with i = 1, 2 ..., m − 1, we can assume A + Bt if t ≤ s, G(t, s) = C + Dt if t ≥ s. From the deﬁnition and properties of the Green function together with (3-3b), we have

i−1 m−1 X X A + Bs = C + Ds, B = αk B + αk D, k=0 k=i i−1 m−1 X X B − D = 1, C + D = βk(A + Bξk) + βk(C + Dξk), k=0 k=i Hence,

m−1 m m−1 m−1 1 X X X X A = 1 − α β ξ − 1 + 1 − α 1 − s + β (s − ξ ) , ρ k k k k k k k=i k=0 k=0 k=i m−1 m−1 X . X B = − αk 1 − αk , k=i k=0 i−1 i−1 m−1 m−1 i−1 1 X X X X X C = − 1 − α β s + β ξ − 1 + α β (s − ξ ) , ρ k k k k k k k k=0 k=0 k=i k=i k=0 i−1 m−1 X . X D = αk − 1 1 − αk . k=0 k=0 This gives the Green function explicitly. Considering (3-2) together, we obtain that (3-1) has the unique solution stated. Lemma 3.2. The Green function above satisﬁes G(t, s) ≥ 0 for t, s ∈ [0, 1].

Proof. For ξi−1 ≤ s ≤ ξi , with i = 1, 2,..., m − 1, and t ≤ s,

i−1 m−1 X X (s − t) + βk(t − ξk) + βk(t − s) k=0 k=i m−1 i−1 m−1 i−1 X X X X ≥ βk(s − t) + βk(t − ξk) + βk(t − s) = βk(s − ξk) ≥ 0, k=0 k=0 k=i k=0 and m−1 m−1 X X (1 − s) + βk(s − ξk) ≥ βk(1 − ξk) ≥ 0. k=i k=i 182 YANG LIU, ZHANG WEIGUO, LIU XIPING, SHEN CHUNFANG AND CHENHUA

For ξi−1 ≤ s ≤ ξi , with i = 1, 2,..., m − 1, and t ≥ s, i−1 m−1 i−1 m−1 X X X X (1 − t) + βk(t − s) + βk(t − ξk) ≥ βk(1 − s) + βk(1 − ξk) ≥ 0. k=0 k=i k=0 k=i These give that G(t, s) ≥ 0 for t, s ∈ [0, 1]. Lemma 3.3. If y(t) ≥ 0 for t ∈ [0, 1] and u(t) is the solution of (3-1), then

(1) min0≤t≤1|x(t)| ≥ δ max0≤t≤1|x(t)| and 0 (2) max0≤t≤1|x(t)| ≤ γ1 max0≤t≤1|x (t)|, where m−2 m−2 m−2 m−2 X . X X . X δ = βi (1−ξi ) 1− βi ξi and γ1 = 1− βi ξi 1− βi i=1 i=1 i=1 i=1 are constants. Proof. (1) For x000(t) = −y(t) ≤ 0 with t ∈ [0, 1], we see that x00(t) is decreasing on [0, 1]. Considering x00(0) = 0, we have x00(t) ≤ 0 for t ∈ (0, 1). Next we claim that x0(0) ≤ 0. Otherwise, if x0(0) > 0, we have the contradiction

m−2 m−2 0 0 0 X 0 X 0 0 0 = x (0) − x (0) = x (0) − αi x (ξi ) > αi (x (0) − x (ξi )) ≥ 0. i=1 i=1

Thus, max0≤t≤1 x(t) = x(0) and min0≤t≤1 x(t) = x(1). From the concavity of x(t), we have ξi (x(1) − x(0)) ≤ x(ξi ) − x(0). Pm−2 Multiplying both sides by βi and considering x(1) = i=1 βi x(ξi ), we have m−2 m−2 X X (3-4) 1 − βi ξi x(1) ≥ βi (1 − ξi )x(0). i=1 i=1 (2) By the mean value theorem, we obtain 0 x(1) − x(ξi ) = (1 − ξi )x (ηi ) for η ∈ (ξi , 1). From the concavity of x, similar to what we did above, we see that

m−2 m−2 X X 0 (3-5) 1 − βi x(1) < βi (1 − ξi )|x (1)|. i=1 i=1 0 Considering (3-4) together with (3-5) we have x(0) ≤ γ1|x (1)|. This completes the proof of Lemma 3.3. Pm−1 Pm−1 Lemma 3.4. If y ∈ C[0, 1] for y ≥ 0 and γ2 = 1 + ( i=0 αi ξi )/(1 − i=0 αi ) is 0 00 a positive constant, then max0≤t≤1|x (t)| ≤ γ2 max0≤t≤1|x (t)|. POSITIVE SOLUTIONS FOR A MULTIPOINT BOUNDARY VALUE PROBLEM 183

0 = 0 + R t 00 Proof. Since x (t) x (0) 0 x (s)ds, we have

m−1 m−1 m−1 Z ξi X 0 X 00 X 00 1 − αi x (0) = αi x (s)ds ≥ αi ξi x (1). i=0 i=0 0 i=0 Thus m−1 m−1 X 0 X 00 1 − αi |x (0)| ≤ αi ξi |x (1)|. i=0 i=0 Considering the concavity of x(t) and x0(t), we have

max |x0(t)| = |x0(1)| and max |x00(t)| = |x00(1)|, 0≤t≤1 0≤t≤1 and x0(1) − x0(0) = x00(η) ≥ x00(1) which give that

Pm−1 α ξ |x0(1)| ≤ 1 + i=0 i i |x00(1)|. Pm−1 1 − i=0 αi Remark. Lemmas 3.3 and 3.4 ensure that

0 00 00 max{ max |x(t)|, max |x (t)|, max |x (t)|} ≤ γ3 max |x (t)|, 0≤t≤1 0≤t≤1 0≤t≤1 0≤t≤1 where γ3 = γ1γ2 > 1. Let the Banach space E = C2[0, 1] be endowed with the norm n o kxk = max max |x(t)|, max |x0(t)|, max |x00(t)| for x ∈ E. 0≤t≤1 0≤t≤1 0≤t≤1 We deﬁne the cone P ⊂ E by

− − n m 2 m 2 = ∈ ≥ 00 = 0 = X 0 = X P x E x(t) 0, x (0) 0, x (0) αi x (ξi ), x(1) βi x(ξi ), = = i 1 i 1 o x(t) is concave on [0, 1] .

Let the nonnegative continuous concave functional α, the nonnegative continuous convex functionals γ and θ, and the nonnegative continuous functional ψ be deﬁned on the cone by

γ (x) = max |x00(t)|, θ(x) = ψ(x) = max |x(t)|, α(x) = min |x(t)|. 0≤t≤1 0≤t≤1 0≤t≤1 By Lemmas 3.3 and 3.4, the functionals deﬁned above satisfy

δθ(x) ≤ α(x) ≤ θ(x) = ψ(x) for kxk ≤ γ3γ (x). 184 YANG LIU, ZHANG WEIGUO, LIU XIPING, SHEN CHUNFANG AND CHENHUA

Therefore condition (2-2) of Lemma 2.5 is satisﬁed. Let Z 1 Z 1 m = sG(1, s)ds, N = sG(0, s)ds, λ = min{m, δγ3}. 0 0 Assume that there exist constants 0 < a, b, d with a < b < λd such that

(A1) f (t, u, v, w) ≤ d for (t, u, v) ∈ [0, 1] × [0, γ3d] × [−γ2d, 0] × [−d, 0];

(A2) f (t, u, v, w) > b/m for (t, u, v) ∈ [0, 1] × [b, b/δ] × [−γ2d, 0] × [−d, 0];

(A3) f (t, u, v, w) < a/N for (t, u, v) ∈ [0, 1] × [0, a] × [−γ2d, 0] × [−d, 0].

Theorem 3.5. Under assumptions (A1)–(A3), problem (1-1) has at least three pos- 00 itive solutions x1, x2, x3 satisfying max0≤t≤1|xi (t)| ≤ d for i = 1, 2, 3, where

min |x2(t)| < b < min |x1(t)| and max |x3(t)| ≤ a < max |x2(t)|. 0≤t≤1 0≤t≤1 0≤t≤1 0≤t≤1 Proof. Problem (1-1) has a solution x = x(t) if and only if x solves the operator equation Z 1 Z s x(t) = G(t, s) f (τ, x(τ), x0(τ), x00(τ))dτds = (T x)(t). 0 0 By a simple computation, we have Z t (T x)00(t) = − f (s, x, x0, x00)ds. 0 00 For x ∈ P(γ, d), we have γ (x) = max0≤t≤1|x (t)| ≤ d. Considering Lemmas 3.3 and 3.4 and assumption (A1), we obtain f (t, x(t), x0(t), x00(τ)) ≤ d, Z 1 = | 00 | = − 0 00 ≤ γ (T x) (T x) (1) f (s, x, x , x )ds d. 0 Hence, T : P(γ, d) → P(γ, d) and clearly T is a completely continuous operator. The fact that the constant function x(t) = b/δ is in P(γ, θ, α, b, c, d) and that α(b/δ) > b implies that {x ∈ P(γ, θ, α, b, c, d | α(x) > b)} 6= ∅. This gives that condition (S1) of Lemma 2.5 holds. For x ∈ P(γ, θ, α, b, c, d), we have b ≤ x(t) ≤ b/δ and |x00(t)| < d for 0 ≤ t ≤ 1. 0 00 From assumption (A2), we have f (t, x, x , x ) > b/m. By deﬁnition of α and the cone P, we have Z 1 Z s α(T x) = (T x)(1) = G(1, s) f (τ, x, x0, x00)dτds 0 0 b Z 1 b ≥ sG(1, s)ds > m = b, m 0 m POSITIVE SOLUTIONS FOR A MULTIPOINT BOUNDARY VALUE PROBLEM 185 which means α(T x) > b for all x ∈ P(γ, θ, α, b, b/δ, d). Second, with (3-4) and b < λd, we have α(T x) ≥ δθ(T x) > δ(b/δ) = b for all x ∈ P(γ, α, b, d) with θ(T x) > b/δ. Thus, condition (S2) of Lemma 2.5 holds. Finally we show that (S3) also holds. We see that ψ(0) = 0 < a and 0 6∈ R(γ, ψ, a, d). Suppose that x ∈ R(γ, ψ, a, d) with ψ(x) = a. Then by assumption (A3), Z 1 Z s ψ(T x) = max |(T x)(t)| = G(0, s) f (τ, x, x0, x00)dτds 0≤t≤1 0 0 a Z 1 < sG(0, s)ds = a. N 0 Thus, all conditions of Lemma 2.5 are satisﬁed. Hence (1-1) has at least three positive concave solutions x1, x2, x3 satisfying the conditions of the theorem.

4. Example

Consider the third order four-point boundary value problem

x000(t) + f (t, x(t), x0(t), x00(t)) = 0 for t ∈ [0, 1], (4-1) 00 = 0 = 1 0 1 + 1 0 2 = 1 1 + 1 2 x (0) 0, x (0) 4 x ( 3 ) 2 x ( 3 ), x(1) 3 x( 3 ) 2 x( 3 ), where  u5  ≤ ≤ 1 1 w 4  if 0 u 10, f (t, u, v, w) = et + + 10π 60 60 1800 10000  if u ≥ 10, π By a simple computation, the function G(t, s) is given by  24(5/9 − t/8 − s/24) if 0 ≤ s ≤ 1/3, t ≤ s,   − ≤ ≤ ≥  24(5/9 t/6) if 0 s 1/3, t s,   24(4/9 − s/8 − t/12) if 1/3 ≤ s ≤ 2/3, t ≤ s, G(t, s) =  24(4/9 − t/8 − s/12) if 1/3 ≤ s ≤ 2/3, t ≥ s,   − ≤ ≤ ≤  6(1 s) if 2/3 s 1, t s,   6(1 − t/6 − 5s/6) if 2/3 ≤ s ≤ 1, t ≥ s, Choosing a = 1, b = 4 and d = 1800, we note that

80 7 Z 1 35 Z 1 γ = , δ = , m = sG(1, s)ds = , N = sG(0, s)ds < 10. 9 10 0 108 0 186 YANG LIU, ZHANG WEIGUO, LIU XIPING, SHEN CHUNFANG AND CHENHUA

We can check that F = f (t, u, v, w) satisﬁes F ≤ 1800 in the region [0, 1] × [0, 16000] × [−4800, 0] × [−1800, 0], F ≥ 432/35 in the region [0, 1] × [4, 40/7] × [−4800, 0] × [−1800, 0], F ≤ 1/10 in the region [0, 1] × [0, 1] × [−4800, 0] × [−1800, 0]. Then all assumptions of Theorem 3.5 are satisﬁed. Thus, problem (4-1) has at least three positive solutions x1, x2, x3 such that 0 max |xi (t)| ≤ 1800 for i = 1, 2, 3, 0≤t≤1

min x1(t) > 4, max x2(t) > 1, 0≤t≤1 0≤t≤1

min x2(t) < 4, max x3(t) < 1. 0≤t≤1 0≤t≤1

Remark. Problem (4-1) is a third order four-point BVP and the nonlinear term is involved in the ﬁrst and second order derivative explicitly. Earlier results for positive solutions, to the authors’ knowledge, do not apply to problem (4-1).

Acknowledgment

We are very grateful to the referee for valuable comments and suggestions.

References

[Anderson 1998] D. Anderson, “Multiple positive solutions for a three-point boundary value problem”, Math. Comput. Modelling 27:6 (1998), 49–57. MR 99b:34040 Zbl 0906.34014 [Avery and Peterson 2001] R. I. Avery and A. C. Peterson, “Three positive ﬁxed points of nonlinear operators on ordered Banach spaces”, Comput. Math. Appl. 42:3-5 (2001), 313–322. MR 2002i: 47070 Zbl 1005.47051 [Chu and Zhou 2006] J. Chu and Z. Zhou, “Positive solutions for singular non-linear third-order periodic boundary value problems”, Nonlinear Anal. 64:7 (2006), 1528–1542. MR 2006i:34058 [Graef and Kong 2009] J. R. Graef and L. Kong, “Positive solutions for third order semiposi- tone boundary value problems”, Appl. Math. Lett. 22:8 (2009), 1154–1160. MR 2010g:34134 Zbl 1173.34313 [Greguš 1987] M. Greguš, Third order linear differential equations, Mathematics and its Appli- cations (East European Series) 22, D. Reidel Publishing Co., Dordrecht, 1987. MR 88a:34002 Zbl 0602.34005 [Guo and Lakshmikantham 1988] D. J. Guo and V. Lakshmikantham, Nonlinear problems in abstract cones, Notes and Reports in Mathematics in Science and Engineering 5, Academic, Boston, MA, 1988. MR 89k:47084 Zbl 0661.47045 [Guo et al. 2008] L.-J. Guo, J.-P. Sun, and Y.-H. Zhao, “Existence of positive solutions for nonlinear third-order three-point boundary value problems”, Nonlinear Anal. 68:10 (2008), 3151–3158. MR 2009d:34055 Zbl 1141.34310 POSITIVE SOLUTIONS FOR A MULTIPOINT BOUNDARY VALUE PROBLEM 187

[Hopkins and Kosmatov 2007] B. Hopkins and N. Kosmatov, “Third-order boundary value problems with sign-changing solutions”, Nonlinear Anal. 67:1 (2007), 126–137. MR 2008b:34026 Zbl 1130.34010 [Li 2006] S. Li, “Positive solutions of nonlinear singular third-order two-point boundary value problem”, J. Math. Anal. Appl. 323:1 (2006), 413–425. MR 2007g:34045 Zbl 1107.34019 [Lin et al. 2008] X. Lin, Z. Du, and W. Liu, “Uniqueness and existence results for a third-order nonlinear multi-point boundary value problem”, Appl. Math. Comput. 205:1 (2008), 187–196. MR 2466622 Zbl 1166.34008 [Palamides and Smyrlis 2008] A. P. Palamides and G. Smyrlis, “Positive solutions to a singular third- order three-point boundary value problem with an indeﬁnitely signed Green’s function”, Nonlinear Anal. 68:7 (2008), 2104–2118. MR 2008k:34092 Zbl 1153.34016 [Pei and Chang 2007] M. Pei and S. K. Chang, “Existence and uniqueness of solutions for third- order nonlinear boundary value problems”, J. Math. Anal. Appl. 327:1 (2007), 23–35. MR 2007i: 34031 Zbl 1160.34312 [Yao 2004] Q. Yao, “Solution and positive solution for a semilinear third-order two-point boundary value problem”, Appl. Math. Lett. 17:10 (2004), 1171–1175. MR 2005i:34030 Zbl 1061.34012

Received October 18, 2009. Revised March 9, 2010.

YANG LIU UNIVERSITYOF SHANGHAIFOR SCIENCEAND TECHNOLOGY SHANGHAI 200093 CHINA and DEPARTMENT OF MATHEMATICS HEFEI NORMAL UNIVERSITY HEFEI 230601 CHINA [email protected]

ZHANG WEIGUO UNIVERSITYOF SHANGHAIFOR SCIENCEAND TECHNOLOGY SHANGHAI 200093 CHINA [email protected]

LIU XIPING UNIVERSITYOF SHANGHAIFOR SCIENCEAND TECHNOLOGY SHANGHAI 200093 CHINA [email protected]

SHEN CHUNFANG DEPARTMENT OF MATHEMATICS HEFEI NORMAL UNIVERSITY HEFEI 230601 CHINA [email protected] 188 YANG LIU, ZHANG WEIGUO, LIU XIPING, SHEN CHUNFANG AND CHENHUA

CHEN HUA DEPARTMENT OF MATHEMATICS HOHAI UNIVERSITY,CHANGZHOU CAMPUS CHANGZHOU 213022 CHINA [email protected] PACIFIC JOURNAL OF MATHEMATICS Vol. 249, No. 1, 2011

THE BRAID GROUP SURJECTS ONTO G2 TENSOR SPACE

SCOTT MORRISON

Let V be the 7-dimensional irreducible representation of the quantum group Uq (g2). For each n, there is a map from the braid group Ꮾn to the en- domorphism algebra of the n-th tensor power of V, given by ᏾ matrices. Extending linearly to the braid group algebra, we get a map

⊗n ᏭᏮn → EndUq (g2)(V ). Lehrer and Zhang have proved that map is surjective, as a special case of a more general result. Using Kuperberg’s spider for G2, we give an elementary diagrammatic proof of this result.

1. Kuperberg’s spider for G2 We recall just enough from [Kuperberg 1994; 1996] for our purposes. We ﬁx the ground ring Ꮽ = ރ(q). Kuperberg’s q is q2 here, which brings our conventions into agreement with those for quantum groups as presented in [Chari and Pressley 1995; Jantzen 1996; Sawin 2006]. 0 First consider the braided tensor category Rep (G2), with objects tensor powers of the 7-dimensional representation of Uq (g2) and morphisms linear maps commuting with the actions of Uq (g2). (The prime in the notation indicates this is just a full subcategory of the actual representation category; we don’t allow arbitrary representations, although every representation does appears as a subobject of some tensor power.) Second consider the category ᐀(G2), with objects natural numbers and morphisms planar trivalent graphs embedded in a rectangle, modulo certain relations. A morphism from n to m should be a graph with n boundary points along the bottom edge of the rectangle, and m boundary points along the top edge. Note that we allow embedded circles in the interior of the rectangle, called loops. The edges of the graph break the rectangle into faces; those that do not meet the boundary of the rectangle are called internal faces. Composition of morphisms is vertical stacking. The category becomes a tensor category by adding natural numbers at

MSC2000: primary 17B37; secondary 20G42, 17B10, 18D10. Keywords: braid group, spider, G2, tensor category, representation theory.

189 190 SCOTT MORRISON

= q10 + q8 + q2 + 1 + q−2 + q−8 + q−10

= 0

= −(q6 + q4 + q2 + q−2 + q−4 + q−6)

= (q4 + 1 + q−4)

= −(q2 + q−2) + + (q2 + 1 + q−2) +

= + + + +

− − − − −

Figure 1. Kuperberg’s [1994; 1996] relations for the G2 spider, omitting relations involving double edges that we don’t use. Note also that there is a sign error in [1996] but not in [1994]. the level of objects, and juxtaposing graphs side by side at the level of morphisms. The relations from [Kuperberg 1996] are shown in Figure 1; the category ᐀(G2) becomes a braided tensor category with Kuperberg’s formulas for a crossing:

1 1 1 1 (1) = + + + 1+q−2 1+q2 q2 +q4 q−2 +q−4

We now need two theorems from [Kuperberg 1996], one easy, one hard.

Theorem 1.1 (web bases for ᐀(G2)). Each Hom space in ᐀(G2) has a basis given by diagrams with no loops, in which each internal face has at least 6 edges.

0 Theorem 1.2 (isomorphism). The categories Rep (G2) and ᐀(G2) are equivalent as braided tensor categories.

2. Diagrammatic proof

Our goal is now to prove that an arbitrary trivalent graph in the rectangle (with equal numbers of boundary points along the top and bottom edges) can be written, modulo the relations above, as a Ꮽ-linear combination of diagrams coming from THEBRAIDGROUPSURJECTSONTO G2 TENSOR SPACE 191 braids via equation (1). Using Kuperberg’s Theorems 1.1 and 1.2, this would then give a combinatorial proof of Lehrer and Zhang’s theorem:

Theorem 2.1 (surjectivity). The map from the braid group algebra to endomor- phisms of tensor powers of the 7-dimensional irreducible representation of Uq (g2),

→ ⊗n ᏭᏮn EndUq (g2)(V ), is surjective.

Remark. In fact, their result holds for any “strong multiplicity free” representation of a quantum group.

Beginning over-optimistically, we might guess that any such diagram can in fact be written as a composition of factors

, and and then make use of the (presumably easy) special case of Theorem 2.1 for n = 2. Even though this is false, it’s the right direction and contains the essential idea. Small counterexamples to this guess are provided by

⊗3 and in EndUq (g2)(V ).

The correct argument will involve three steps. First, we’ll prove a little lemma allowing us to rearrange connected components of a graph. Second, we’ll prove that the image of the braid group does hit a certain ﬁnite list of small graphs (including the examples above). Third, we’ll use Euler measure to inductively rewrite a graph in terms of linear combinations of braids and a graph with fewer vertices, eventually getting down to graphs in our list.

Lemma 2.2. Suppose a graph D1 ⊗ D2 is a tensor product of two diagrams. Then D1 ⊗ D2 is in the image of the braid group if and only if D2 ⊗ D1 is.

Proof. If 192 SCOTT MORRISON is in the image of the braid group, so is

Since the G2 spider is a braided tensor category, this is exactly the same as

Lemma 2.3. The graphs

, , and

are in the image of ᏭᏮ2.

Proof. Since all four diagrams have no loops or internal faces, they are basis diagrams. Thus by Theorem 1.1 they are linearly independent. Also, it is easy to see that they are the only basis diagrams with 4 boundary points, and so they ⊗2 span EndUq (g2)(V ). Moreover, the braiding (hereafter written as σ in formulas) lives in the same space, and has four distinct eigenvalues (see [Morrison, Peters ⊗2 and Snyder 2010]), and so its powers also span EndUq (g2)(V ). Purely for reference, we’ll give the characteristic equation for the braiding:

σ 4 = −q−16σ 0 + (q−18 − q−16 − q−10 + q−4)σ 1 + (q−18 + q−12 − q−10 − q−6 + q−4 + q2)σ 2 + (q−12 − q−6 − 1 + q2)σ 3.

Explicit formulas for these four diagrams as polynomials in σ appear in the appendix, along with instructions for using the short computer program that produces them. (However, these formulas are not actually needed anywhere here.) Remark. Note that corresponds to q10 +q8 +q2 +1+q−2 +q−8 +q−10 (this is the value of a loop, from Figure 1) times the idempotent projecting V ⊗ V onto its trivial submodule. Similiarly, corresponds to −(q6+q4+q2+q−2+q−4+q−6) times the idempotent projecting onto V . THEBRAIDGROUPSURJECTSONTO G2 TENSOR SPACE 193

Lemma 2.4. The graphs

and are in the image of ᏭᏮ3. Proof. We just do the computations directly, with the help of a computer, obtaining the formulas that appear in the appendix. In particular, we ﬁnd 35 braids that ⊗3 provide a basis for EndUq (g2)(V ). Remark. See also [Lehrer and Zhang 2006, 5.2.2(2) and 5.2.2(4)]. Corollary 2.5. The graphs

, , , and are in the image of the appropriate braid groups. Proof. We’ll show that each of these graphs is actually in the braided tensor subcategory generated by the graphs appearing in the two previous lemmas. (In fact, without even needing to take linear combinations.)

= =

Proposition 2.6 (base case for the induction). Any ᐀(G2) basis diagram in which every connected component contains at most one vertex is in the image of the braid group. Proof. Any such diagram is just a tensor product of factors, each of which is one of the graphs

, , , , or .

By Lemma 2.2, we can take this tensor product in any order we like. We claim that any such tensor product is actually a tensor product of the graphs in Lemmas 2.3 194 SCOTT MORRISON and 2.4 and Corollary 2.5, and so the entire diagram is in the image of the braid group. The argument is a tedious but straightforward case-bash. Suppose we have the diagram

⊗e ⊗ f ⊗a ⊗b ⊗c ⊗d .

Note that 2a + 3c + e = 2b + 3d + f , by counting boundary points at top and bottom. By splitting off factors of , we can assume that at most one of a and b is nonzero. Similarly, by splitting off or , we see that at most one of c and d and at most one of e and f are nonzero. Let’s assume without loss of generality that b = 0. If a > 0, then either we can split off copies of , or f < 2. If f = 1, e must be zero, and we have 2a + 3c = 3d + 1, and so a =∼ 2 (mod 3). Thus d ≥ 1, and we can split off a copy of . Otherwise, if f = 0, we have 2a + 3c + e = 3d, so d ≥ 1 and c = 0. Now, since 2a +e = 3d ≥ 3, either a ≥ 3, e ≥ 3, or a, e ≥ 1. In each of these cases, we can split something off, either , or , respectively. On the other hand, if a = 0, let’s further assume without loss of generality that c = 0. We thus have e = 3d + f ; since e = 0 would imply we have the empty diagram, we have f = 0 instead, and the entire diagram is just a tensor power of . Proof of Theorem 2.1. Suppose now we have an arbitrary basis diagram. We will show that a connected component with at least two vertices has either a or a attached along its top or bottom edge. Since these two diagrams are Laurent polynomials in the positive crossing, we can rewrite the basis diagram as the product of something in the image of the braid group and another basis diagram with fewer vertices. Repeating this, we reduce to the case that no connected component contains more than one vertex, at which point we’re done by Proposition 2.6. The Euler measure argument is straightforward. Consider a connected component of the basis diagram with at least two vertices. Assign formal angles of 2π/3 around trivalent vertices, and of π/2 on either side of an edge meeting the boundary. (We don’t assign angles to the corners of the rectangle, so the total Euler measure will be the Euler measure of the disc, +1.) Since internal faces of the component have at least 6 edges, by Theorem 1.1, they have nonpositive Euler measure. Let bk be the number of faces adjacent to the boundary, and meeting k edges of the graph. These faces meet the boundary of the disc surrounding the component exactly once, since the component is connected. Thus the Euler measure of a face counted by bk is 1 + 1 + 1 − − 1 + + = 2 − 1 4 4 3 (k 1) 2 (k 1) 1 3 6 k. THEBRAIDGROUPSURJECTSONTO G2 TENSOR SPACE 195

The number b1 is zero (this could only occur if the component were a single strand, but it must have at least two vertices), and so we obtain X 2 − 1 ≥ bk( 3 6 k) 1, k≥2 1 + 1 ≥ + ≥ and so 3 b2 6 b3 1, which we soften to b2 b3 3. There are thus at least three faces touching either 2 or 3 edges of the component. At least one of these must be attached to the top or bottom of the rectangle, avoiding the sides. If that face touches 3 edges, we’re done, as there must be a adjacent to the boundary. If it only touches 2 edges, the hypothesis that the component is connected and has at least two vertices ensures that there’s a adjacent to the boundary.

3. Questions

We end with two questions relating Kuperberg’s spider and the category of tilting modules at a root of unity.

Question 3.1. We can specialize Kuperberg’s spider for G2 to any root q of unity. The braiding is still deﬁned as long as q+q−1 6=0, and the braiding is still surjective as long as t(q8 − 1)(q4 − q2 + 1)(q6 + q4 + q2 + 1) 6= 0. (See the explicit formulas in the appendix.) We can form idempotent completion, and quotient by negligibles to produce a semisimple category. Is this category equivalent to the semisimple quotient of the category of tilting modules for the integral form of Uq (G2)? More optimistically, we might hope for an afﬁrmative answer to this: Question 3.2. Are these categories equivalent even before we quotient by negligi- ble morphisms on either side? Results in [Kuperberg 1994] and [Morrison, Peters and Snyder 2010] ensure that there is a functor from the specialization of the spider to the category of tilting modules. I’d hoped to be able to leverage the surjectivity of the braiding map into a proof that this functor was an equivalence, but I still don’t see how to do this.

Appendix: Explicit formulas

The online supplement to this article is a Mathematica notebook. It relies on the QuantumGroups package, which you can download as part of the KnotTheory package from [The Knot Atlas 2007]. After setting the path in the ﬁrst line to point at your copy of the QuantumGroups package, you can run the remaining lines to produce the formulas appearing below. 196 SCOTT MORRISON

Formulas for Lemma 2.3.

− = (q2 −1)(q4 −q2 +1)(q6 +q4 +q2 +1) 1 ×q22σ 0 +(−q20 +q22 +q28)σ 1 +(−q20 −q26 −q28)σ 2 −q26σ 3, − = (q2 −1)(q4 −q2 +1)(q6 +q4 +q2 +1) 1 ×(−q8 −q12)σ 0 +(q6 −q8 +q10 −q12 +q20 +q24)σ 1 +(q6 +q10 −q18 +q20 −q22 +q24)σ 2 +(−q18 −q22)σ 3, − = (q2 −1)(q4 −q2 +1)(q6 +q4 +q2 +1) 1 ×(q2 −q4 +2q6 +q12 −q14 −q18)σ 0 +(−q−2 −2q4 +2q6 −q8 +q10 +q12 +q16 −2q18 −q22 −q24)σ 1 +(−q4 −q8 +2q16 −q18 +q20 −q24)σ 2 +(q16 +q20 +q22)σ 3. Formulas for Lemma 2.4. (q2−1)2(q2+1)(q4−q2+1)2

= −q2(q2−1)(q26−q24+q22+q20−2q18 +4q16−3q14+2q10−4q8+q6−q4−1)1 8 22 20 18 16 12 10 8 6 4 2 +q (−q +3q −5q +4q −3q +7q −6q +2q +q −4q +2)σ1 − 4 2− 18− 16+ 14+ 8− 6+ 4+ −1 q (q 1)(q q 2q 2q q q 1)σ1 2 2 22 20 18 12 10 8 6 4 2 +q (q −1)(q −q +q +3q −2q +3q +q −2q +2q −1)σ2 − 4 2− 14+ 10+ 8− 6+ 4+ −1 q (q 1)(2q q 3q q q 1)σ2 10 2 2 2 12 10 8 6 4 +q (q −1) (q +1)(q −q +3q −q +3q +1)σ1σ1 24 22 20 18 16 14 12 10 8 6 4 2 +(−q +3q −3q +q +q −5q +5q −4q +q +2q −2q +q )σ1σ2 − 10 10− 8+ 6− 4− 2+ −1+ 4 4− 2+ 8+ 2− −1 q (q 2q q q q 1)σ1σ2 q (q q 1)(q q 1)σ1 σ2 − 6 8− 6+ 4− 2+ −1 −1 q (q q q q 1)σ1 σ2 24 22 20 18 16 14 12 10 8 6 4 2 +(−q +3q −3q +q +q −5q +5q −4q +q +2q −2q +q )σ2σ1 + 4 4− 2+ 8+ 2− −1+ 22− 24 q (q q 1)(q q 1)σ2σ1 (q q )σ2σ2 − 10 10− 8+ 6− 4− 2+ −1 − 6 8− 6+ 4− 2+ −1 −1 q (q 2q q q q 1)σ2 σ1 q (q q q q 1)σ2 σ1 + + 16 2− −1 0σ1σ1σ2 q (q 1)σ1σ1σ2 22 20 18 16 14 12 10 8 4 2 +(3q −6q +4q −q −2q +5q −3q +3q −q +q )σ1σ2σ1 + 4 12− 10+ 8− 6− 4+ 2− −1 q (q 2q q q q q 1)σ1σ2σ1 + 20 2− − 12 −1 q (q 1)σ1σ2σ2 q σ1σ2 σ1 THEBRAIDGROUPSURJECTSONTO G2 TENSOR SPACE 197

+ −1 −1+ 4 12− 10+ 8− 6− 4+ 2− −1 0σ1σ2 σ1 q (q 2q q q q q 1)σ1 σ2σ1 + 10− 8+ 6 −1 −1+ −1 + −1 −1 (q q q )σ1 σ2σ1 0σ1 σ2σ2 0σ1 σ2 σ1 + 8 4− 2+ −1 −1 −1+ + 20 2− q (q q 1)σ1 σ2 σ1 0σ2σ1σ1 q (q 1)σ2σ2σ1 + −1+ 16 2− −1 + 18− 20 0σ2σ2σ1 q (q 1)σ2 σ1σ1 (q q )σ1σ1σ2σ1 18 20 18 20 +(q −q )σ1σ2σ1σ1+(q −q )σ1σ2σ2σ1+0σ2σ1σ1σ2,

(q2−1)2(q2+1)(q4+1)(q4−q2+1)2

= −q2(q2−1)(q4+1)(q24−q22+2q18−2q16+q12−2q10−q4+q2−1)1 −q2(q30−3q28+5q26−4q24+2q20−3q18+q16 14 12 10 6 4 2 +2q −q +2q −2q +2q −2q +1)σ1 + − 26+ 24− 22+ 20+ 18− 16+ 14− 10+ 8− 6+ 4 −1 ( q 2q 2q q q q q q q q q )σ1 2 26 24 22 18 16 14 12 8 6 4 2 +q (q −2q +2q −4q +8q −9q +5q −5q +6q −5q +3q −1)σ2 − 4 18− 16+ 12− 10+ 6− 4+ 2− −1 q (q q q 2q q q q 1)σ2 30 28 26 24 18 12 +(q −2q +3q −2q −q +q )σ1σ1 −q2(q24−3q22+3q20+q18−5q16+9q14 12 10 6 4 2 −7q +3q −3q +3q −2q +1)σ1σ2 + 18− 10+ 8− 6+ 4 −1 (q q q q q )σ1σ2 + 4 4− 2+ 14− 12+ 10− 6+ 4− 2+ −1 − 6 −1 −1 q (q q 1)(q q q q q q 1)σ1 σ2 q σ1 σ2 6 20 18 16 12 10 8 4 2 −q (q −2q +2q −q +3q −2q +q −2q +1)σ2σ1 + 6 4− 2+ 8+ 2− −1− 14 2− 4+ 6− 2+ q (q q 1)(q q 1)σ2σ1 q (q 1)(q 1)(q q 1)σ2σ2 − 12 10− 8+ 6− 4− 2+ −1 q (q 2q q q q 1)σ2 σ1 − 2 14− 10+ 8+ 6− 4+ 2− −1 −1+ 14 4− 2+ q (q q q q 2q 2q 1)σ2 σ1 q (q q 1)σ1σ1σ2 − 16 −1+ 6 18− 16+ 14+ 12− 10+ 8− 6+ 4− q σ1σ1σ2 q (2q 4q 2q q 3q 5q 2q q 1)σ1σ2σ1 + 6 4+ 8− 6+ 2− −1+ 14 10− 8+ 4− 2+ q (q 1)(q 2q q 1)σ1σ2σ1 q (q q q q 1)σ1σ2σ2 − 14 −1 + 2+ 5− 3+ 2 −1 −1− 8 2− 4+ −1 q σ1σ2 σ1 (q 1)(q q q) σ1σ2 σ1 q (q 1)(q 1)σ1 σ2σ1 + 8 4− 2+ −1 −1− 16 4− 2+ −1 + −1 −1 q (q q 1)σ1 σ2σ1 q (q q 1)σ1 σ2σ2 0σ1 σ2 σ1 + − 8+ 6− 4 −1 −1 −1+ + 22 2− ( q q q )σ1 σ2 σ1 0σ2σ1σ1 q (q 1)σ2σ2σ1 + −1+ 18 2− −1 + 12 2− 4+ 0σ2σ2σ1 q (q 1)σ2 σ1σ1 q (q 1)(q 1)σ1σ1σ2σ1 20 22 20 22 +(q −q )σ1σ2σ1σ1+(q −q )σ1σ2σ2σ1+0σ2σ1σ1σ2. 198 SCOTT MORRISON

References

[Chari and Pressley 1995] V. Chari and A. Pressley, A guide to quantum groups, Cambridge Univer- sity Press, 1995. MR 96h:17014 Zbl 0839.17010 [Jantzen 1996] J. C. Jantzen, Lectures on quantum groups, Graduate Studies in Mathematics 6, American Mathematical Society, Providence, RI, 1996. MR 96m:17029 Zbl 0842.17012

[Kuperberg 1994] G. Kuperberg, “The quantum G2 link invariant”, Internat. J. Math. 5:1 (1994), 61–85. MR 95g:57013 Zbl 0797.57008 [Kuperberg 1996] G. Kuperberg, “Spiders for rank 2 Lie algebras”, Comm. Math. Phys. 180:1 (1996), 109–151. MR 97f:17005 Zbl 0870.17005 [Lehrer and Zhang 2006] G. I. Lehrer and R. B. Zhang, “Strongly multiplicity free modules for Lie algebras and quantum groups”, J. Algebra 306 (2006), 138–174. MR 2007g:17008 Zbl 1169.17003 [Morrison, Peters and Snyder 2010] S. Morrison, E. Peters, and N. Snyder, “Knot polynomial identities and quantum group coincidences”, preprint, 2010. To appear in Quantum Topology. arXiv 1003.0022 [Sawin 2006] S. F. Sawin, “Quantum groups at roots of unity and modularity”, J. Knot Theory Ramiﬁcations 15:10 (2006), 1245–1277. MR 2008f:17032 Zbl 1117.17006 [The Knot Atlas 2007] D. Bar-Natan and S. Morrison (editors), “The Knot Atlas”, wiki, 2007, available at http://katlas.org/.

Received September 22, 2009. Revised January 4, 2010.

SCOTT MORRISON MICROSOFT STATION Q CNSIBUILDING UNIVERSITYOF CALIFORNIA SANTA BARBARA, CA 93106-6105 [email protected] http://tqft.net/ PACIFIC JOURNAL OF MATHEMATICS Vol. 249, No. 1, 2011

ANALOGUES OF THE WIENER TAUBERIAN AND SCHWARTZ THEOREMS FOR RADIAL FUNCTIONS ON SYMMETRIC SPACES

E.K.NARAYANAN AND ALLADI SITARAM

We prove a Wiener Tauberian theorem for the L1 spherical functions on a semisimple Lie group of arbitrary real rank. We also establish a Schwartz- type theorem for complex groups. As a corollary we obtain a Wiener Taube- rian type result for compactly supported distributions.

Introduction

Two celebrated theorems from classical analysis dealing with translation invariant subspaces are the Wiener Tauberian theorem and the Schwartz theorem. Let f ∈ L1(ޒ) and f˜ be its Fourier transform. Then the Wiener Tauberian theorem says that the ideal generated by f is dense in L1(ޒ) if and only if f˜ is a nowhere vanishing function on the real line. The result due to L. Schwartz says that every closed translation invariant subspace V of C∞(ޒ) is generated by the exponential polynomials in V . In particular, such a V contains the function x → eiλx for some λ ∈ ރ. Interestingly, this result fails for ޒn if n ≥ 2. Even though the exact analogue of the Schwartz theorem fails in this case, it follows from the well-known theorem of Brown, Schreiber and Taylor [Brown et al. 1973] that if V ⊂ C∞(ޒn) is a closed subspace that is translation and rotation invariant, then V contains ψs for some s ∈ ރ, where Z n/2−1 isx.w ψs(x) = CJn/2−1(s|x|)/(s|x|) = e dσ (w). Sn−1

Here Jn/2−1 is the Bessel function of the ﬁrst kind and of order n/2−1 and σ is the unique, normalized rotation invariant measure on the sphere Sn−1. The constant C is such that ψs(0) = 1. It also follows from the work in [Brown et al. 1973] that

MSC2000: primary 43A20, 43A90; secondary 43A80, 43A30. Keywords: Wiener Tauberian theorems, Schwartz theorems, maximal ideals, Schwartz space. Narayanan was supported in part by a grant from UGC via DSA-SAP. Sitaram was supported by IISc Mathematics Initiative, and by a Raja Ramanna Fellowship from the Department of Atomic Energy, India.

199 200 E. K. NARAYANAN AND ALLADI SITARAM

z.x = ∈ n 2 + 2 + V contains all the exponentials e if z (z1, z2,..., zn) ރ satisﬁes z1 z2 2 2 · · · + zn = s for nonzero s. For s vanishing, ψs is just the constant function one. Our aim in this paper is to prove analogues of these results in the context of noncompact semisimple Lie groups.

1. Notation and preliminaries

For any unexplained terminology we refer to [Helgason 1994]. Let G be a connected noncompact semisimple Lie group with finite center and K a fixed maximal compact subgroup of G. Fix an Iwasawa decomposition G = KAN and let a be ∗ ∗ the Lie algebra of A. Let a be the real dual of a and aރ its complexification. Let ρ be the half sum of positive roots for the adjoint action of a on g, the Lie algebra of G. The Killing form induces a positive definite form h · , · i on a∗ × a∗. Extend ∗ this form to a bilinear form on aރ. We will use the same notation for the extension as well. Let W be the Weyl group of the symmetric space G/K . Then there is a ∗ ∗ h · · i natural action of W on a, a and aރ, and , is invariant under this action. ∈ ∗ For each λ aރ, let ϕλ be the elementary spherical function associated with λ. Recall that ϕλ is given by the formula Z (iλ−ρ)(H(xk)) ϕλ(x) = e dk for x ∈ G. K

See [Helgason 1994] for more details. It is known that ϕλ = ϕλ0 if and only if λ0 = τλ for some τ ∈ W. Let ` be the dimension of a and F denote the set (in ރ`)

∗ F = a + iCρ where Cρ = convex hull of {sρ : s ∈ W}.

Then it is a well-known theorem of Helgason and Johnson that ϕλ is bounded if and only if λ ∈ F. Let I (G) be the set of all complex valued spherical functions on G, that is,

I (G) = { f : f (k1xk2) = f (x) for k1, k2 ∈ K, x ∈ G}.

1 Fix a Haar measure dx on G, and let I1(G) = I (G) ∩ L (G). Then it is well known that I1(G) is a commutative Banach algebra under convolution and that the maximal ideal space of I1(G) can be identiﬁed with F/W. ˆ For f ∈ I1(G), deﬁne its spherical Fourier transform f on F by Z ˆ f (λ) = f (x)ϕ−λ(x)dx. G

Then fˆ is a W-invariant bounded function on F that is holomorphic in the interior F0 of F and is continuous on F. Also [f ∗ g = fˆgˆ, where the convolution of f WIENER TAUBERIAN AND SCHWARTZ THEOREMS FOR RADIAL FUNCTIONS 201 and g is defined by Z ( f ∗ g)(x) = f (xy−1)g(y)dy. G Next, we define the L1-Schwartz space of K -biinvariant functions on G, which will be denoted by S(G). Let x ∈ G. Then x = k exp X for k ∈ K and X ∈ p, where g = k + p is the Cartan decomposition of the Lie algebra g of G. Put σ (x) = kXk, where k · k is the norm on p induced by the Killing form. For any left-invariant differential operator D on G and any integer r ≥ 0, we define for a smooth K - biinvariant function f

r −2 pD,r ( f ) = sup(1 + σ (x)) |ϕ0(x)| |D f (x)|, x∈G where ϕ0 is the elementary spherical function corresponding to λ = 0. Deﬁne

S(G) = { f : pD,r ( f ) < ∞ for all D, r}. Then S(G) becomes a Frechet´ space when equipped with the topology induced by the family of seminorms pD,r . = ∗ ∗ ∈ Let ᏼ ᏼ(aރ) be the symmetric algebra over aރ. Then each u ᏼ gives rise ∗ to a differential operator ∂(u) on aރ. Let Z(F) be the space of functions f on F such that (i) f is holomorphic in F0 (the interior of F) and continuous on F; (ii) if u ∈ P and m ≥ 0 is any integer, then

2 m qu,m( f ) = sup (1 + kλk ) |∂(u) f (λ)| < ∞; λ∈F0 (iii) f is W-invariant. Then Z(F) is an algebra under pointwise multiplication and a Frechet´ space when equipped with the topology induced by the seminorms qu,m. If a ∈ Z(F), we deﬁne the “wave packet” ψa on G by Z 1 −2 ψa(x) = a(λ)ϕλ(x)|c(λ)| dλ, |W| a∗ where c(λ) is the well-known Harish-Chandra c-function. By the Plancherel theorem of Harish-Chandra, we also know that the map f → fˆ extends to a unitary map from L2(K \G/K ) onto L2(a∗, |c(λ)|−2dλ). We can now state a result of Trombi and Varadarajan [1971]. Theorem 1.1. (i) If f ∈ S(G), then fˆ ∈ Z(F).

(ii) If a ∈ Z(F), then the integral deﬁning the “wave packet” ψa converges absolutely, and ψa ∈ S(G). Moreover, ψˆ a = a. 202 E. K. NARAYANAN AND ALLADI SITARAM

(iii) The map f → fˆ is a topological linear isomorphism of S(g) onto Z(F). The plan of this paper is as follows. In Section 2, we prove a Wiener Tauberian theorem for L1(K \G/K ) assuming more symmetry on the generating family of functions. In Section 3, we establish a Schwartz-type theorem for complex semisimple Lie groups. As a corollary we also obtain a Wiener Tauberian-type theorem for compactly supported distributions on G/K .

2. A Wiener Tauberian theorem for L1(K\G/K)

Ehrenpreis and Mautner [1955] observed that an exact analogue of the Wiener Tauberian theorem is not true for the commutative algebra of K -biinvariant functions on the semisimple Lie group SL(2, ޒ). Here K is the maximal compact subgroup SO(2). However, they did prove an analogue of the Wiener Tauberian theorem under an additional “not too rapidly decreasing condition” on the spherical Fourier transform: If f is a K -biinvariant integrable function on G = SL(2, ޒ) whose spherical Fourier transform fˆ does not vanish anywhere on the maximal ideal space (which can be identified with a certain strip on the complex plane), then f generates a dense subalgebra of L1(K \G/K ) provided fˆ does not vanish too fast at ∞. There have been a number of attempts to generalize these results to L1(K \G/K ) or L1(G/K ), where G is a noncompact connected semisimple Lie group with finite center. Almost complete results have been obtained when G is a real rank one group. See [Benyamini and Weit 1992; Ben Natan et al. 1996; Sarkar 1998; Sitaram 1988] for results on rank one case. See also [Sarkar 1997] for a result on the whole group SL(2, ޒ). Sitaram [1980] proved that under suitable conditions on the spherical Fourier transform of a single function f , an analogue of the Wiener Tauberian theorem holds for L1(K \G/K ) with no assumptions on the rank of G. Recently, Narayanan [2009] improved this result to include the case of a family of functions rather than just a single function. One difference between rank one results and those of higher rank has been the precise form of the “not too rapid decay condition”. In [Sitaram 1980; Narayanan 2009], this condition on the spherical Fourier transform of a function is assumed to be true on the whole maximal domain, while for rank one groups it suffices impose this condition on a∗; see [Benyamini and Weit 1992; Sarkar 1998]. (An important corollary of this is that in the rank one case one can get a Wiener Tauberian-type theorem for a wide class of functions purely in terms of the nonvanishing of the spherical Fourier transform in a certain domain, without having to check any decay conditions; see [Mohanty et al. 2004, Theorem 5.5].) In the first part of this paper we show that such a stronger result is true for the higher rank case as well, provided we assume more symmetry on the generating family WIENER TAUBERIAN AND SCHWARTZ THEOREMS FOR RADIAL FUNCTIONS 203 of functions, and again as a corollary we get a result of the type alluded to in the parenthesis above. ∗ = ∗ ` ∈ ∗ If dim a `, then aރ may be identified with ރ and a point λ aރ will be = 2 + 2 + · · ·+ 2 1/2 denoted λ (λ1, λ2, . . . λ`). Denote by r(λ) its radius (λ1 λ2 λ`) . Let ∗ BR denote the ball of radius R centered at the origin in a , and let FR denote the ∗ domain in aރ defined by ∗ FR = {λ ∈ aރ : kIm(λ)k < R}.

For a > 0, let Ia denote the strip in the complex plane deﬁned by

Ia = {z ∈ ރ : |Im z| < a}.

Now, suppose that f is a holomorphic function on FR and that f depends only on 2 2 r(λ). Then it is easy to see that g(s) = f (λ1, λ2, . . . λ`), where s = r(λ) deﬁnes an even holomorphic function on IR and vice versa. We will need some lemmas. Let A(Ia) denote the collection of functions g such that

(i) g is even, bounded and holomorphic on Ia, ¯ (ii) g is continuous on Ia, and

(iii) lim|s|→∞ g(s) = 0.

Then A(Ia) with the supremum norm is a Banach algebra under pointwise multiplication.

Lemma 2.1. Let {gα : α ∈ 3} be a collection of functions in A(Ia). Assume that ¯ there is no s ∈ Ia such that gα(s) = 0 for all α ∈ I . Further assume that there exists ∈ α0 I such that gα0 does not decay very rapidly on ޒ, that is, | | ke|s| lim sup gα0 (s) e > 0 for all k > 0. |s|→∞

Then the closed ideal generated by {gα : α ∈ I } is the whole of A(Ia).

Proof. Let ψ be a suitable biholomorphic map that maps the strip Ia onto the unit disc; see [Benyamini and Weit 1992]. Let hα(z) = gα(ψ(z)). Then hα ∈ A0(D), where A0(D) is the collection of even holomorphic functions h on the unit disc that are continuous up to the boundary and satisfy h(i) = h(−i) = 0. The not too rapid decay condition on ޒ is precisely what is needed to apply the Beurling– Rudin theorem to complete the proof. See the proofs of [Benyamini and Weit 1992, Theorem 1.1 and Lemma 1.2] for the details. −thλ,λi Let pt denote the K -biinvariant function deﬁned by pˆt (λ) = e . It is easy to see that pt ∈ S(G). 204 E. K. NARAYANAN AND ALLADI SITARAM

1 Lemma 2.2. Let J ⊂ L (K \G/K ) be a closed ideal. If pt ∈ J for some t > 0, then J = L1(K \G/K ).

Proof. Since pˆt has no zeros and does not decay too rapidly, this immediately follows from the main result in [Narayanan 2009] or [Sitaram 1980]. We say a function f ∈ L1(K \G/K ) is radial if the spherical Fourier transform fˆ(λ) is a function of r(λ). Notice that, if the group G is of real rank one, then the class of radial functions is precisely the class of K -biinvariant functions in L1(G). When the group G is complex, it is possible to describe the class of radial functions (see the next section). The following is our main theorem in this section: 1 Theorem 2.3. Let { fα : α ∈ I } be a collection of radial functions in L (K \G/K ). ˆ Assume that the spherical transform fα extends as a bounded holomorphic function ˆ to the bigger domain FR, where R > kρk with lim|λ|→∞ fα(λ) = 0 for all α and ˆ that there exists no λ ∈ FR such that fα(λ) = 0 for all α. Further assume that there ˆ ∗ exists an α0 such that fα0 does not decay too rapidly on a , that is, | ˆ | |λ| lim sup fα0 (λ) exp(ke ) > 0 for all k > 0. |λ|→∞

1 Then the closed ideal generated by { fα : α ∈ I } is all of L (K \G/K ). ˆ Proof. Since fα is radial, each fα gives rise to an even bounded holomorphic function gα(s) on the strip IR. If |ρ| < a < R, then the collection {gα(s) : α ∈ I } satisﬁes the hypotheses in Lemma 2.1 on the domain Ia. It follows that the family {gα} generates A(Ia). In particular, we have a sequence n + n + · · · + n → −s2/2 h1(s)gα1(n)(s) h2(s)gα2(n)(s) hk (s)gαk (n)(s) e ¯ n ∈ uniformly on Ia, where gα j (n) are in the given family and h j (s) A(Ia). n Each h j can be viewed as a holomorphic function on the domain Fa contained ∗ n | | in aރ that depends only on r(λ). Since the h j are bounded and ρ < a it can be −hλ,λi/2 n easily checked that e h j (λ) ∈ Z(F). Again, an application of the Cauchy integral formula says that −hλ,λi/2 n ˆ + −hλ,λi/2 n ˆ + · · · −hλ,λi/2 n ˆ e h1(λ) fα1(n)(λ) e h2(λ) fα2(n)(λ) e hk (λ) fαk (n)(λ) converges to e−hλ,λi in the topology of Z(F); see the proof of [Benyamini and Weit 1992, Theorem 1.1]. By Theorem 1.1, this simply means that the ideal generated 1 −hλ,λi by { fα : α ∈ I } in L (K \G/K ) contains the function p , where pˆ(λ) = e . We ﬁnish the proof by appealing to Lemma 2.2.

Corollary 2.4. Let { fα : α ∈ I } be a family of radial functions satisfying the hypotheses of Theorem 2.3. Then the closed subspace spanned by the left G- translates of the this family is all of L1(G/K ). WIENER TAUBERIAN AND SCHWARTZ THEOREMS FOR RADIAL FUNCTIONS 205

Proof. Let J be the closed subspace generated by the left translates of the given family. By Theorem 2.3, L1(K \G/K ) ⊂ J. Now, it is easy to see that J has to be 1 equal to L (G/K ). 1 Corollary 2.5. Let { fα : α ∈ I } be a family of L -radial functions. Assume that ˆ each fα extends to a bounded holomorphic function on the bigger domain FR for ˆ some R > kρk. Assume further that limkλk→∞ fα(λ) → 0. If there exists an α0 such that fα0 is not equal to a real analytic function almost everywhere, then the left G-translates of the family above span a dense subset of L1(G/K ). Proof. This follows exactly as in [Mohanty et al. 2004, Theorem 5.5].

3. Schwartz theorem for complex groups

When G is a connected noncompact semisimple Lie group of real rank one with finite center, a Schwartz-type theorem was proved by Bagchi and Sitaram [1979]. Let K be a maximal compact subgroup of G. Then their result states the following: Let V be a closed subspace of C∞(K \G/K ) with the property that f ∈ V implies w ∗ f ∈ V for every compactly supported K -biinvariant distribution w on G/K . ∈ ∗ Then V contains an elementary spherical function ϕλ for some λ aރ. This was proved by establishing a one-one correspondence between ideals in C∞(K \G/K ) ∞ and those of C (ޒ)even. This also proves that a similar result cannot hold for higher rank groups. Going back to ޒn, we notice that if f ∈ C∞(ޒn) is radial, then the translation invariant subspace V f generated by f is also rotation invariant. It follows from [Brown et al. 1973] that V f contains ψs for some s ∈ ރ, where ψs is the Bessel function defined in the introduction. Our aim in this section is to prove a similar result for the complex semisimple Lie groups. Our definition of radiality, taken from [Volchkov and Volchkov 2008], coincides with the definition in the previous section when the function is in L1(K \G/K ). Throughout this section we assume that G is a complex semisimple Lie group. Let Exp:p→ G/K denote the map P →(exp P)K . Then Exp is a diffeomorphism. If dx denotes the G-invariant measure on G/K , then Z Z (1) f (x)dx = f (Exp P)J(P)d P, G/K p where sinh ad P J(P) = det . ad P Since G is a complex group, the elementary spherical functions are given by a simple formula: Z −1/2 ihAλ,Ad(k)Pi (2) ϕλ(Exp P) = J(P) e dk for P ∈ p. K 206 E. K. NARAYANAN AND ALLADI SITARAM

Here Aλ is the unique element in aރ such that λ(H) = hA, Aλi for all H ∈ aރ. Let E(K \G/K ) be the strong dual of C∞(K \G/K ). Then E(K \G/K ) can be identiﬁed with the space of compactly supported K -biinvariant distributions on G/K . If w is such a distribution, then w(λ)ˆ = w(ϕλ) is well deﬁned and is called the spherical Fourier transform of w. By the Paley–Wiener theorem, we know that λ→w(λ) ˆ is an entire function of exponential type. Similarly, E(ޒ`) will denote the space of compactly supported distribution on ޒ` and E W (ޒ`) consists of the Weyl group invariant ones. From the work of Gangolli and others, as noted in [Bagchi and Sitaram 1979], we know that the Abel transform

S : E(K \G/K ) → E W (ޒ`) is an isomorphism and S](w)(λ) =w(λ) ˆ for w ∈ E(K \G/K ), where S](w)(λ) is the Euclidean Fourier transform of the distribution S(w).

Proposition 3.1 [Bagchi and Sitaram 1979]. There exists a linear topological isomorphism T from C∞(K \G/K ) onto C∞(ޒ`)W such that

S(w)(T ( f )) = w( f ) for all w ∈ E(K \G/K ) and f ∈ C∞(K \G/K ). We also have

S(w0) ∗ T (w ∗ f ) = T (w0 ∗ w ∗ f ) for all w, w0 ∈ E(K \G/K ) and f ∈ C∞(K \G/K ). Moreover, 1 X T (ϕ )(x) = exp(ihτ.λ, xi). λ |W| τ∈W A K -biinvariant function f is called radial if it is of the form

f (x) = J(Exp−1 x)−1/2u(d(0, x)), where d is the Riemannian distance induced by the Killing form on G/K and u is a function on [0, ∞). Then [Volchkov and Volchkov 2008, Theorem 4.6] shows that this definition of radiality coincides with the one in the previous section if the function is integrable. That is, f ∈ L1(K \G/K ) has the above form if and only if the spherical Fourier transform fˆ(λ) depends only on r(λ). We denote the class ∞ ∞ of smooth radial functions by C (K \G/K )rad, and Cc (K \G/K )rad will consist ∞ of compactly supported functions in C (K \G/K )rad. For f ∈ C∞(K \G/K ) define Z f #(Exp P) = J(P)−1/2 J(σ.P)1/2 f (σ.P)dσ, SO(p) WIENER TAUBERIAN AND SCHWARTZ THEOREMS FOR RADIAL FUNCTIONS 207 where SO(p) is the special orthogonal group on p and dσ is the Haar measure on SO(p). Here, by f (P) we mean f (Exp P). Clearly, f → f # is the projection ∞ ∞ from C (K \G/K ) onto C (K \G/K )rad. ∞ Proposition 3.2. (a) The space C (K \G/K )rad is reflexive. ∞ (b) The strong dual E(K \G/K )rad of C (K \G/K )rad is given by {w ∈ E(K \G/K ) :w(λ) ˆ is a function of r(λ)}. ∞ (c) C (K \G/K )rad is invariant under convolution by w ∈ E(K \G/K )rad. ∞ ∞ Proof. (a) The space C (K \G/K )rad is a closed subspace of C (K \G/K ), which is a reflexive Frechet´ space. # ∞ (b) Define Bλ = ϕλ, the projection of ϕλ into C (K \G/K )rad. A simple computation shows that Z −1/2 ihAλ,σ.Pi Bλ(Exp P) = J(P) e dσ. SO(p)

It is clear that Bλ as a function of λ depends only on r(λ). Now, let w ∈ E(K \G/K ). Deﬁne a distribution w# by w#( f ) = w( f #). It is easy to see that w# is a compactly # supported K -biinvariant distribution. Clearly, if w ∈ E(K \G/K )rad, then w = w . It follows that w(λ)ˆ = w(ϕλ) = w(Bλ). Consequently, w(λ)ˆ is a function of r(λ). It also follows that E(K \G/K )rad is reﬂexive. ∞ ∞ (c) If w ∈ E(K \G/K )rad and g ∈ Cc (K \G/K )rad, then w ∗ g ∈ Cc (K \G/K )rad. This follows from (b) above and [Volchkov and Volchkov 2008, Theorem 4.6]. ∞ Next, if g is arbitrary, we may approximate g with gn ∈ Cc (K \G/K )rad. We can now state our main result in this section. Let V be a closed subspace ∞ ∞ of C (K \G/K )rad. We say V is an ideal in C (K \G/K )rad if f ∈ V and w ∈ E(K \G/K )rad implies that w ∗ f ∈ V . ∞ Theorem 3.3. (a) If V is a nonzero ideal in C (K \G/K )rad, then there exists a ∈ ∗ ∈ λ aރ such that Bλ V. ∞ (b) If f ∈ C (K \G/K )rad, then the closed left G-invariant subspace generated ∞ ∈ ∗ by f in C (G/K ) contains ϕλ for some λ aރ. Proof. We closely follow the arguments in [Bagchi and Sitaram 1979]. (a) Notice that the map

` S : E(K \G/K )rad → E(ޒ )rad is a linear topological isomorphism. Using the reﬂexivity of the spaces involved and arguing as in [Bagchi and Sitaram 1979] we obtain that (as in Proposition 3.1)

∞ ∞ ` T : C (K \G/K )rad → C (ޒ )rad 208 E. K. NARAYANAN AND ALLADI SITARAM

∞ ` ∞ is a linear topological isomorphism, where C (ޒ )rad stands for the space of C radial functions on ޒ` and ∞ S(w)(T ( f )) = w( f ) for all w ∈ E(K \G/K )rad, f ∈ C (K \G/K )rad. Another application of Proposition 3.1 implies that we have a bijection between ∞ ∞ ` ∞ ` the ideals in C (K \G/K )rad and C (ޒ )rad. Here, an ideal in C (ޒ )rad is a closed subspace invariant under convolution by compactly supported radial distributions on ޒ`. From [Bagchi and Sitaram 1990] or [Brown et al. 1973], any ideal ∞ ` in C (ޒ )rad contains ψs (Bessel function) for some s ∈ ރ. To complete the proof it sufﬁces to show that under the topological isomorphism T the function Bλ is 2 2 mapped into ψs, where s = r(λ) . Now, we have S(w)(T (Bλ)) = w(Bλ). Since w ∈ E(K \G/K )rad, we know that w(Bλ) is nothing but w(ϕλ), which equals (]Sw)(λ). Since S is onto, this implies 2 2 that T (Bλ) = ψs, where s = r(λ) .

(b) From [Bagchi and Sitaram 1979] we know that T (ϕλ) = 8λ where 8λ(x) = −1 P |W| τ∈W exp(iτλ.x). Let V f denote the left G-invariant subspace generated by f . Then T (V f ) surely contains the space

VT ( f ) = {S(w) ∗ T ( f ) : w ∈ E(K \G/K )}. From Proposition 3.2, T ( f ) is a radial C∞ function on ޒ`. Hence, from [Brown ∞ ` et al. 1973], the translation invariant subspace XT ( f ) generated by T ( f ) in C (ޒ ) contains ψs for some s ∈ ރ. Consequently, if s 6= 0, the space XT ( f ) will contain iz.x 2 2 all the exponentials e , where z = (z1, z2,..., z`) satisfies r(z) = s . If s = 0, then XT ( f ) contains the constant functions. Now, it is easy to see that the map −1 P XT ( f ) → VT ( f ), x 7→ |W| τ∈W g(τ.x) is surjective. Hence, there exists a l λ ∈ ރ such that 8λ ∈ VT ( f ). Since T (ϕλ) = 8λ, this finishes the proof. Our next result is a Wiener Tauberian-type theorem for compactly supported distributions. Let E(G/K ) denote the space of compactly supported distributions on G/K . If g ∈ G and w ∈ E(G/K ), then the left g-translate of w is the compactly supported distribution gw defined by

−1 gw( f ) = w(g f ) for f ∈ C∞(G/K ), where x f (y) = f (x−1 y).

Theorem 3.4. Suppose {wα : α ∈ I } is a family of distributions contained in E(K \G/K )rad. Then the left G-translates of this family span a dense subset of ∈ ∗ ˆ = ∈ E(G/K ) if and only if there is no λ aރ such that wα(λ) 0 for all α I. Proof. We start with the “if” part of the theorem. Let J stand for the closed span of the left G-translates of the distributions wα in E(G/K ). It suffices to show that E(K \G/K ) ⊂ J. To see this, let f ∈ C∞(G/K ) be such that w( f ) = 0 for WIENER TAUBERIAN AND SCHWARTZ THEOREMS FOR RADIAL FUNCTIONS 209 all w ∈ E(K \G/K ). Since J is left G-invariant, we also have w( fg) = 0 for all ∈ = R g G, where fg is the K -biinvariant function defined by fg(x) K f (gkx)dk. It follows that fg ≡ 0 for all g ∈ G and consequently f ≡ 0. Next, we claim that E(K \G/K ) ⊂ J if E(K \G/K )rad ⊂ J. To prove this it is enough to show that ∞ {g ∗ w : w ∈ E(K \G/K )rad, g ∈ Cc (K \G/K )} is dense in E(K \G/K ). By Proposition 3.2, the map S from E(K \G/K ) onto ` W ` E(ޒ ) is a linear topological isomorphism mapping E(K \G/K )rad onto E(ޒ )rad ` isomorphically. Hence, it suffices to prove a similar statement for E(ޒ )rad and E(ޒ`)W — an easy exercise in distribution theory. So, to complete the proof of Theorem 3.4 we only need to show that ∞ {g ∗ wα : α ∈ I, g ∈ Cc (K \G/K )rad} is dense in E(K \G/K )rad. If not, consider ∞ ∞ Jrad = { f ∈ C (K \G/K )rad : (g ∗ wα)( f ) = 0 for all g ∈ Cc (K \G/K ), α ∈ I }. ∞ This set is clearly a closed subspace of C (K \G/K )rad that is invariant under ∞ convolution by Cc (K \G/K )rad. By Theorem 3.3 we have Bλ ∈ Jrad for some ∈ ∗ ˆ = ∈ λ aރ. It follows that wα(λ) 0 for all α I , which is a contradiction. This finishes the proof. ∞ For the “only if” part, it suffices to observe that if g ∈ Cc (G/K ) then ∗ = # ˆ # = R g wα(ϕλ) gb(λ)wα(λ), where g (x) K g(kx)dk.

Remark. A single distribution w ∈ E(K \G/K )rad cannot generate the whole of E(G/K ) unless w is the measure supported at the identity coset. This is because wˆ cannot have zeroes, and so by the Hadamard factorization theorem it has to be an exponential function, which in turn has to be a constant due to the Weyl group invariance. Remark. A similar theorem for all rank one groups (not necessarily complex) may be derived from the results in [Bagchi and Sitaram 1990].

References

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[Sarkar 1997] R. P. Sarkar, “Wiener Tauberian theorems for SL2(ޒ)”, Pacific J. Math. 177:2 (1997), 291–304. MR 98m:43003 Zbl 0868.43010 [Sarkar 1998] R. P. Sarkar, “Wiener Tauberian theorem for rank one symmetric spaces”, Pacific J. Math. 186:2 (1998), 349–358. MR 99k:43004 Zbl 0931.43009 [Sitaram 1980] A. Sitaram, “An analogue of the Wiener Tauberian theorem for spherical transforms on semisimple Lie groups”, Pacific J. Math. 89:2 (1980), 439–445. MR 82e:43008 [Sitaram 1988] A. Sitaram, “On an analogue of the Wiener Tauberian theorem for symmetric spaces of the noncompact type”, Pacific J. Math. 133:1 (1988), 197–208. MR 89e:43024 Zbl 0628.43011 [Trombi and Varadarajan 1971] P. C. Trombi and V. S. Varadarajan, “Spherical transforms of semisimple Lie groups”, Ann. of Math. (2) 94 (1971), 246–303. MR 44 #6913 Zbl 0203.12802 [Volchkov and Volchkov 2008] V. V. Volchkov and V. V. Volchkov, “Convolution equations and the local Pompeiu property on symmetric spaces and on phase space associated to the Heisenberg group”, J. Anal. Math. 105 (2008), 43–123. MR 2010f:43018 Zbl 1152.43006

Received November 10, 2009.

E.K.NARAYANAN INDIAN INSTITUTEOF SCIENCE DEPARTMENT OF MATHEMATICS BANGALORE 560012 INDIA [email protected] http://www.math.iisc.ernet.in/~naru

ALLADI SITARAM INDIAN INSTITUTEOF SCIENCE DEPARTMENT OF MATHEMATICS BANGALORE 560012 INDIA [email protected] PACIFIC JOURNAL OF MATHEMATICS Vol. 249, No. 1, 2011

SEMIDIRECT PRODUCTS OF REPRESENTATIONS UP TO HOMOTOPY

YUNHE SHENGAND CHENCHANG ZHU

We study the semidirect product of a Lie algebra with a representation up to homotopy and provide various examples coming from Courant algebroids, Lie 2-algebras of string type, and omni-Lie algebroids. In the end, we study the semidirect product of a Lie group with a representation up to homotopy and use it to give an integration of a certain Lie 2-algebra of string type.

1. Introduction

This paper is the ﬁrst part of our project to integrate representations up to homotopy of Lie algebras (algebroids). Our original motivation is to integrate the standard Courant algebroid TM ⊕T ∗ M, since it is this Courant algebroid that is much used in Hitchin and Gualtieri’s program of generalized complex geometry. Courant algebroids are Lie 2-algebroids in the sense of Roytenberg [2002] and Severaˇ [2005]. The general procedure to integrate Lie n-algebras (algebroids) is already described in [Getzler 2009; Henriques 2008; Severaˇ 2005]. We want to pursue some explicit formulas for the special case of the standard Courant algebroid. It turns out that the sections of the Courant algebroid TM ⊕ T ∗ M form a semidirect product of a Lie algebra with a representation up to homotopy. Abad and Crainic [2009] recently studied the representations up to homotopy of Lie algebras, Lie groups, and even Lie algebroids, Lie groupoids, in general. Just as one can form the semidirect product of a Lie algebra with a representation, one can form the semidirect product with representations up to homotopy too. In our case, the semidirect product coming from the standard Courant algebra is a Lie 2-algebra. But using the fact that it is also a semidirect product, the integration becomes easier. The integration result is related to the semidirect product of Lie groups with its representation up to homotopy, as will be discussed in Section 3. However it turns out that Abad

MSC2000: primary 17B65; secondary 18B40, 58H05. Keywords: representation up to homotopy, L∞-algebra, integration, Lie 2-algebras, Courant algebroids. Supported by the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)) through the Institutional Strategy of the University of Göttingen, NSF of China (10871007) and China Post- doctoral Science Foundation (20090451267).

211 212 YUNHE SHENG AND CHENCHANGZHU and Crainic’s concept of representations up to homotopy of Lie groups will not be general enough to cover all the integration results. This we will continue in a forthcoming paper [Sheng and Zhu 2010]. In this paper we focus on exhibiting more examples of representations up to homotopy and their semidirect products in order to demonstrate the importance of our integration procedure. The examples are all variations of Courant algebroids. One is Chen and Liu’s omni-Lie algebroids, which generalize Weinstein’s omni- Lie algebras. Hence we expect in [Sheng and Zhu 2010] to give an integration to Weinstein’s omni-Lie algebras via Lie 2-algebras. Another example comes from a more general string Lie 2-algebra, which we call the Lie 2-algebra of string type. It is essentially a Courant algebroid over a point (see Example II), namely a Lie algebra with an adjoint-invariant inner product. This sort of Lie algebra is usually called a quadratic Lie algebra. This concept also appears in the context of Manin triples and double Lie algebras. The example ޒ → g ⊕ g∗ that we consider in this paper is an analogue of the standard Courant algebroid, and is basically a special case taken from [Lu and Weinstein 1990].1 We give an integration of the Lie 2-algebra of string type ޒ → g ⊕ g∗ at the end. Usually people require the base Lie algebra of a string Lie 2-algebra to be semisimple and of compact type (see Remark 2.11). For such string Lie 2-algebras, Baez and Lauda [2004] have proved a no-go theorem: such string Lie 2-algebras cannot be integrated to finite-dimensional semistrict Lie 2-groups. Here a semistrict Lie 2-group is a group object in DiffCat, where DiffCat is the 2-category consisting of categories, functors, and natural transformations in the category of differential manifolds, or equivalently DiffCat is a 2-category with Lie groupoids as objects, strict morphisms of Lie groupoids as morphisms, and 2-morphisms of Lie groupoids as 2-morphisms. Our semistrict Lie 2-group is actually called a Lie 2-group by Baez and Lauda. However, we call it a semistrict Lie 2-group because compared to the Lie 2-group in the sense of Henriques [2008], or equivalently (the equivalency was proved in [Zhu 2009]) the stacky group in the sense of Blohmann [2008], it is stricter. Basically, their Lie 2-group is a group object in the 2-category with objects as Lie groupoids, morphisms as Hilsum–Skandalis bimod- ules (or generalized morphisms), 2-morphisms as 2-morphisms of Lie groupoids. Schommer-Pries [2010] realizes the string 2-group as such a Lie 2-group with a finite-dimensional model; the integration of a string Lie 2-algebra to such a model is work in progress [Schommer-Pries et al. ≥ 2011]. It is not needed in the definition of the string Lie 2-algebra for the base Lie algebra to be semisimple of compact type. One only needs a quadratic Lie algebra. As soon as we relax this condition, we find out that one can integrate ޒ → g ⊕ g∗

1private conversation with Jiang-Hua Lu SEMIDIRECT PRODUCTS OF REPRESENTATIONS UP TO HOMOTOPY 213 to a ﬁnite dimensional semistrict Lie 2-group in the sense of Baez and Lauda. The integrating object is actually a special Lie 2-group (very close to a strict Lie 2-group) in the sense of Baez and Lauda. Of course, as we relax the condition, we are in danger that the class corresponding to this Lie 2-algebra in H 3(g⊕g∗, ޒ) might be trivial, and consequently our Lie 2-algebra might be trivially strictiﬁed. Then what we have done would not have been a big surprise because a strict Lie 2-algebra corresponds to a crossed module of Lie algebras, and it easily integrates to a strict Lie 2-group by integrating the crossed module. However, we verify that when g itself (not g ⊕ g∗) is semisimple, this class is not trivial.

2. Representations up to homotopy of Lie algebras

Here, we ﬁrst consider the 2-term representation up to homotopy of Lie algebras. We give explicit formulas of the corresponding 2-term L∞-algebra, which is their semidirect product. Then we give several interesting examples including Courant algebroids and omni-Lie algebroids.

2a. Representations up to homotopy of Lie algebras and their semidirect products. L∞-algebras, sometimes called strongly homotopy Lie algebras, were introduced by Drinfeld and Stasheff [Stasheff 1992] as a model for “Lie algebras that satisfy Jacobi identity up to all higher homotopies”. The following convention of L∞-algebras has the same grading as in [Henriques 2008] and [Roytenberg and Weinstein 1998].

Deﬁnition 2.1. An L∞-algebra is a graded vector space L = L0⊕L1⊕· · · equipped Vk with a system {lk | 1 ≤ k < ∞} of linear maps lk : L → L with degree deg(lk) = k − 2, where the exterior powers are interpreted in the graded sense and the sum

X i( j−1) X (−1) sgn(σ ) Ksgn(σ )l j (li (xσ (1),..., xσ (i)), xσ (i+1),..., xσ (n)) i+ j=n+1 σ vanishes for all n ≥ 0, where Ksgn is the Koszul sign and the sum is taken over all (i, n − i)-unshufﬂes with i ≥ 1. Letting n = 1, we have

2 l1 = 0, l1 : Li+1 → Li , which means that L is a complex; we usually write d = l1. Letting n = 2, we have p dl2(x, y) = l2(dx, y) + (−1) l2(x, dy) for all x ∈ L p, y ∈ Lq , which means that d is a derivation with respect to l2. We usually view l2 as a bracket [· , ·]. However, it is not a Lie bracket: the obstruction of the Jacobi 214 YUNHE SHENG AND CHENCHANGZHU identity is controlled by l3:

(p+q)r qr+1 (1) l2(l2(x, y), z) + (−1) l2(l2(y, z), x) + (−1) l2(l2(x, z), y) pq (p+q)r = −dl3(x, y, z)−l3(dx, y, z)+(−1) l3(dy, x, z)−(−1) l3(dz, x, y), where x ∈ L p, y ∈ Lq , z ∈ Lq and l3 also satisﬁes higher coherence laws. In particular, if the k-th brackets are zero for all k > 2, we recover the usual notion of differential graded Lie algebras (DGLA). If L is concentrated in degrees less than n, then L is called an n-term L∞-algebra. In this paper, we mainly consider 2-term L∞-algebras, which are equivalent to the Lie 2-algebras given by John Baez and Alissa Crans [2004]. In this special case, l4 is always zero. Thus by restricting the coherence law satisﬁed by l3 on degree 0, we obtain

(2) l3(l2(x, y), z, w) + c.p. − (l2(l3(x, y, z), w) + c.p.) = 0

for all x, y, z, w ∈ L0, where c.p. stands for cyclic permutation. Lie 2-algebras are categorified versions of Lie algebras. In a Lie 2-algebra, the Jacobi identity is replaced by an isomorphism called the Jacobiator. The Jacobiator satisfies a certain law of its own. Given a d 2-term L∞-algebra L1 −→ L0, the underlying 2-vector space of the corresponding Lie 2-algebra is made up by L0 as the vector space of objects and L0 ⊕ L1 as the vector space of morphisms. See [Baez and Crans 2004, Theorem 4.3.6] for details. Recall from [Baez and Crans 2004] that a Lie 2-algebra is skeletal if isomorphic objects are equal. Viewing a Lie 2-algebra as a 2-term L∞-algebra, explicitly we have this: d Definition 2.2. A 2-term L∞-algebra L1 −→ L0 is called skeletal if d = 0. Theorem 2.3 [Baez and Crans 2004]. There is a bijection between 2-term skeletal d L∞-algebra L1 −→ L0 and quadruples (k1, k2, φ, θ), where k1 is a Lie algebra, k2 is a vector space, φ is a representation of k1 on k2, and θ is a 3-cocycle on k1 with values in k2. d Given a 2-term skeletal L∞-algebra L1 −→ L0, recall that k1 is L0, k2 is L1, the representation φ comes from l2, and the 3-cocycle θ is obtained from l3. See [Abad and Crainic 2009; Abad 2008] for the general theory of representation up to homotopy of Lie algebroids. In this paper we only consider the 2-term representations up to homotopy of Lie algebras. Definition 2.4 [Abad and Crainic 2009]. A 2-term representation up to homotopy of a Lie algebra g consists of the following: d (i) A 2-term complex of vector spaces V1 −→ V0. SEMIDIRECT PRODUCTS OF REPRESENTATIONS UP TO HOMOTOPY 215

(ii) Two linear maps µi : g → End(Vi ) that are compatible with d. That is, for any X ∈ g and ξ ∈ V1, we have

(3) d ◦ µ1(X)(ξ) = µ0(X) ◦ d(ξ). V2 (iii) A linear map ν : g → Hom(V0, V1) such that

(4) µ0[X1, X2] − [µ0(X1), µ0(X2)] = d ◦ ν(X1, X2),

(5) µ1[X1, X2] − [µ1(X1), µ1(X2)] = ν(X1, X2) ◦ d, and

(6) [µ0(X1) + µ1(X1), ν(X2, X3)] + c.p. = ν([X1, X2], X3) + c.p.

We usually write µ = µ0 + µ1 and denote a 2-term representation up to homotopy d of a Lie algebra g by (V1 −→ V0, µ, ν). In [2009, Example 3.25], Abad and Crainic proved that one can associate to any representation up to homotopy V• of a Lie algebra g a new L∞-algebra g n V•, which is their semidirect product. Here we make this construction explicit in the 2-term case. d Let (V1 −→ V0, µ, ν) be a 2-term representation up to homotopy of g. Then we can form a new 2-term complex

d (g n V•, d) : V1 −→ (g ⊕ V0). V2 Deﬁne l2 : (g n V•) → g n V• by setting

l2(X + ξ, Y + η) = [X, Y ] + µ0(X)(η) − µ0(Y )(ξ),

(7) l2(X + ξ, f ) = µ1(X)( f ),

l2( f, g) = 0 for any X + ξ, Y + η ∈ g ⊕ V0 and f, g ∈ V1. Note that l2 is not a Lie bracket, but instead

l2(l2(X + ξ, Y + η), Z + γ ) + c.p. = d(ν(X, Y )(γ )) + c.p. V3 Deﬁne l3 : (g n V•) → g n V• by setting

(8) l3(X + ξ, Y + η, Z + γ ) = −ν(X, Y )(γ ) + c.p. Then: d Proposition 2.5. With the notations above, if (V1 −→ V0, µ, ν) is a 2-term rep- d resentation up to homotopy of a Lie algebra g, then (V1 −→ (g ⊕ V0), l2, l3) is a 2-term L∞-algebra. 216 YUNHE SHENG AND CHENCHANGZHU

Example I (Courant algebroids TM ⊕ T ∗ M). Courant algebroids, introduced in [Liu et al. 1997] to study the double of Lie bialgebroids, each consist of a vector bundle E → M equipped with a nondegenerate symmetric bilinear form h · , · i on the bundle, an antisymmetric bracket [[· , · ]] on the section space 0(E) and a bundle map ρ : E → TM such that a set of axioms are satisﬁed. One can be viewed as a Lie 2-algebroid with a “degree-2 symplectic form” [Roytenberg 2002]. The ﬁrst example is the standard Courant algebroid (᐀ = TM ⊕T ∗ M, h · , · i, [[· , · ]], ρ) associated to a manifold M, where ρ : ᐀ → TM is the projection, the canonical pairing h · , · i is given by h + + i = 1 + ∈ ∈ 1 (9) X ξ, Y η 2 (ξ(Y ) η(X)) for all X, Y X(M), ξ, η (M), and the antisymmetric bracket [[· , · ]] is given by [[ + + ]] [ ] + − + 1 − (10) X ξ, Y η , X, Y L X η LY ξ 2 d(ξ(Y ) η(X)) for all X + ξ, Y + η ∈ 0(᐀). This is not a Lie bracket, but we have

(11) [[[[e1, e2]], e3]]+ c.p. = dT (e1, e2, e3) for all e1, e2, e3 ∈ 0(᐀), where T (e1, e2, e3) is given by = 1 h[[ ]] i + (12) T (e1, e2, e3) 3 ( e1, e2 , e3 c.p.). Now we realize the section space of ᐀ as the semidirect product of the Lie algebra X(M) of vector ﬁelds with the natural 2-term deRham complex

(13) C∞(M) −→d 1(M).

For this we need to define a representation up to homotopy of X(M) on this complex. For any X ∈ X(M), define linear actions µ0 and µ1 by [[ ]]= − 1 ∈ 1 (14) µ0(X)(ξ) , X, ξ L X ξ 2 d(ξ(X)) for all ξ (M), h i = 1 ∈ ∞ (15) µ1(X)( f ) , X, d f 2 X ( f ) for all f C (M). Define ν : V2 X(M) → Hom(1(M), C∞(M)) by

(16) ν(X, Y )(ξ) = T (X, Y, ξ) for all X, Y ∈ X(M), ξ ∈ 1(M).

∞ d 1 Proposition 2.6. With the above notations, (C (M) −→ (M), µ = µ0 +µ1, ν) is a representation up to homotopy of the Lie algebra X(M). Proof. For any f ∈ C∞(M), we have = − 1 = 1 µ0(X)(d f ) L X d f 2 d X ( f ) 2 d X( f ), SEMIDIRECT PRODUCTS OF REPRESENTATIONS UP TO HOMOTOPY 217 which implies µ0 ◦d = d ◦µ1, that is, the µi are compatible with the differential d. By straightforward computations, we have

µ0[X, Y ](ξ) − [µ0(X), µ0(Y )](ξ) = [[[[X, Y ]], ξ]]+ c.p. = dT (X, Y, ξ) = d(ν(X, Y )(ξ)), [ ] − [ ] = 1 [ ] − 1 − µ1 X, Y ( f ) µ1(X), µ1(Y ) ( f ) 2 X, Y ( f ) 4 (X (Y ( f )) Y (X ( f ))) = 1 [ ] 4 X, Y ( f ), = = 1 1 [ ] + 1 − ν(X, Y )(d f ) T (X, Y, d f ) 3 ( 2 X, Y ( f ) 4 (X (Y ( f )) Y (X( f ))) = 1 [ ] 4 X, Y ( f ), which implies (4) and (5). At last we need to prove (6), which is obviously equivalent to

(17) µ1(X)T (Y, Z, ξ) − T (Y, Z, µ0(X)(ξ)) + c.p.(X, Y, Z) = T ([X, Y ], Z, ξ) + c.p.(X, Y, Z).

Observe that since µ0(X)(ξ) = [[X, ξ]], we have

T (Y, Z, µ0(X)(ξ)) + c.p.(X, Y, Z) + T ([X, Y ], Z, ξ) + c.p.(X, Y, Z) = T ([X, Y ], Z, ξ) + c.p.(X, Y, Z, ξ).

∞ Furthermore, since µ1(X)( f ) = hX, d f i for any f ∈ C (M) and the cotangent bundle T ∗ M is isotropic under the pairing (9), we have

µ1(X)T (Y, Z, ξ) + c.p.(X, Y, Z) = hX, dT (Y, Z, ξ)i + c.p.(X, Y, Z, ξ).

Thus, (17) is equivalent to

hX, dT (Y, Z, ξ)i + c.p.(X, Y, Z, ξ) = T ([X, Y ], Z, ξ) + c.p.(X, Y, Z, ξ), which holds by [Roytenberg and Weinstein 1998, Lemma 4.5]. By Proposition 2.5, we have this:

∞ d=0⊕d 1 Corollary 2.7. (C (M) −−−−→ (X(M)⊕ (M)), l2, l3) is a 2-term L∞-algebra, where l2 and l3 are given by (7) and (8), in which µ0, µ1 and ν are given by (14), (15) and (16), respectively. Remark 2.8. Roytenberg and Weinstein [1998] have proved that the sections of a Courant algebroid (Ꮿ, h · , · i, [[· , · ]], ρ) form an L∞-algebra. In the case when Ꮿ = ᐀ the standard Courant algebroid, the 2-term L∞-algebra is given by

d=0⊕d C∞(M) −−−−→ X(M) ⊕ 1(M) = 0(᐀), 218 YUNHE SHENG AND CHENCHANGZHU with brackets given by

l2(e1, e2) = [[e1, e2]], l2(e1, f ) = he1, d f i, l3(e1, e2, e3) = −T (e1, e2, e3),

∞ and li≥4 = 0 for any e1, e2, e3 ∈ 0(᐀) and f ∈ C (M). Here T is deﬁned by (12). It is easy to verify that this is the same as our 2-term L∞-algebra in Corollary 2.7. We can also modify our complex (13) to 1(M) −−→Id 1(M). Following the same procedure, we get another representation up to homotopy of the Lie algebra X(M) and therefore obtain another 2-term L∞-algebra that is also totally determined by the Courant algebroid (᐀, h · , · i, [[· , · ]], ρ). More precisely, µ0 = µ1 is given by µ0(X)(ξ) , [[X, ξ]], and ν : X(M) → 2(X2(M), End(1(M), 1(M))) is given by

ν(X, Y )(ξ) , dT (X, Y, ξ) for any X, Y ∈ X(M) and ξ ∈ 1(M).

1 Id 1 Proposition 2.9. With the above notations, ( (M) −−→ (M), µ = µ0 = µ1, ν) is a representation up to homotopy of the Lie algebra X(M). Example II (Courant algebroids over a point and Lie 2-algebras of string type).A Courant algebroid over a point is literally a quadratic Lie algebra, namely a Lie algebra k together with nondegenerate inner product h · , · i that is invariant under the adjoint action. People often think of a Courant algebroid over a point as a string Lie 2-algebra.2 Here we justify this thinking. Deﬁnition 2.10. The Lie 2-algebra of string type associated to a quadratic Lie 0 algebra (k, h · , · i) is a 2-term L∞-algebra ޒ −→ k, whose degree-0 part is k and whose degree-1 part is ޒ, with l2, l3 given by

l2((e1, c1), (e2, c2)) = ([e1, e2], 0),

l3((e1, c1), (e2, c2), (e3, c3)) = (0, h[e1, e2], e3i), where e1, e2, e3 ∈ k and c1, c2, c3 ∈ ޒ. The representation φ of k on ޒ and the 3-cocycle θ : V3 k → ޒ in the corresponding quadruple in Theorem 2.3 is given by

ρ(e)(c) = l2(e, c),

(18) θ(e1, e2, e3) = h[e1, e2], e3i.

2Private conversation with John Baez and Urs Schreiber. SEMIDIRECT PRODUCTS OF REPRESENTATIONS UP TO HOMOTOPY 219

Remark 2.11. In the definition of string Lie 2-algebras [Baez and Rogers 2010; Henriques 2008], the base Lie algebra k is usually required to be semisimple and of compact type, such that the Jacobiator gives rise to the generator of H 3(k, ޚ) = ޚ. This is because Witten’s original motivation was to obtain a 3-connected cover of Spin(n), and so(n) is simple and of compact type. However, to write down the structure of the string Lie 2-algebra, we only need a quadratic Lie algebra. This is how we obtain the definition above on Lie 2-algebras of string type. Then H 3(k, ޚ) is not necessarily ޚ for a general quadratic Lie algebra k. For example, for the abelian Lie algebra ޒ, any inner product is adjoint-invariant, and H 3(ޒ, ޚ) = 0. We thus face the danger that sometimes a Lie 2-algebra of string type might be trivial, that is, the Jacobiator might correspond to the trivial element in H 3(k, ޚ). Then what we have can be trivially strictified to a strict Lie 2-algebra, which is a crossed module of Lie algebras. Then the integration of a crossed module of Lie algebras is simply a crossed module of Lie groups. However, we will verify that the example we consider is not such a case. The standard Courant algebroid motivates us to consider the case of the direct sum k = g ⊕ g∗ of a Lie algebra g and its dual with the semidirect product Lie algebra structure:

∗ ∗ [X + ξ, Y + η] = [X, Y ]g + adX η − adY ξ, where [· , ·]g is the Lie bracket of g. The nondegenerate invariant pairing h · , · i on g ⊕ g∗ is given by h + + i = 1 + + + ∈ ⊕ ∗ X ξ, Y η 2 (η(X) ξ(Y )) for all X ξ, Y η g g . With these deﬁnitions, (g ⊕ g∗, [· , ·], h · , · i) is a quadratic Lie algebra. In fact, we have ∗ ∗ h[X1 + ξ1, X2 + ξ2], X3 + ξ3i = h[X1, X2]g + adX ξ2 − adY ξ1, X3 + ξ3i

= h[X1, X2]g, ξ3i + c.p. Similarly, we have

hX2 + ξ2, [X1 + ξ1, X3 + ξ3]i = h[X1, X3]g, ξ2i + c.p., which implies hX2 + ξ2, [X1 + ξ1, X3 + ξ3]i + h[X1 + ξ1, X2 + ξ2], X3 + ξ3i = 0, that is, the nondegenerate inner product h · , · i is invariant under the adjoint action. This example is a special case of [Lu and Weinstein 1990, Theorem 1.12] with g∗ equipped with the 0 Lie bracket. Thus (g, g∗) forms a Lie bialgebra or equivalently (g ⊕ g∗, g, g∗) is a Manin triple. However, honestly we have not found other Lie bialgebras (Manin triples) giving rise to Lie 2-algebras of the form of semidirect products. 220 YUNHE SHENG AND CHENCHANGZHU

We denote by

(19) ޒ −→0 g ⊕ g∗. the corresponding Lie 2-algebra of string type of (g⊕g∗, [· , ·], h · , · i). We denote by ν˜ : V3(g ⊕ g∗) → ޒ the corresponding 3-cocycle (see (18))

ν(˜ X1 + ξ1, X2 + ξ2, X3 + ξ3) = h[X1 + ξ1, X2 + ξ2], X3 + ξ3i (20) = h[X1, X2]g, ξ3i + c.p. 0 ∗ ∗ Proposition 2.12. (ޒ −→ g , µ1 = 0, µ0 = ad , ν = [ · , ·]g) is a 2-term representation up to homotopy of the Lie algebra g. Moreover the Lie 2-algebra of string type ޒ −→0 g ⊕ g∗ is the semidirect product of g and the complex ޒ −→0 g∗.

Proof. Since d=0, we only need to verify that µ0 and µ1 are Lie algebra morphisms and (6). Both ad∗ : g → End(g∗) and 0 are Lie algebra morphisms, and (6) follows from Jacobi identity of [· , ·]g. Then it is not hard to see that the Lie 2-algebra of string type ޒ −→0 g ⊕ g∗ with formulas in Deﬁnition 2.10 is exactly the semidirect product of g with the complex 0 ∗ ∗ (ޒ −→ g , µ1 = 0, µ0 = ad , ν = [ · , ·]g) with the formulas (7) and (8). Proposition 2.13. If the Lie algebra g is semisimple, the Lie algebra 3-cocycle ν˜ given by (20) is not exact.

Proof. Let h · , · ik be the Killing form on g. The proof follows from the fact that the Cartan 3-form h[ · , ·], · ik on a semisimple Lie algebra is not exact. Since g is ∗ semisimple, the Killing form h · , · ik is nondegenerate. Identify g and g by using the Killing form h · , · ik and let ᏷ be the corresponding isomorphism,

h᏷(ξ), Xik = hξ, Xi. Assume that ν˜ = dφ for some φ : V2(g ⊕ g∗) → ޒ; deﬁne ϕ : V2(g ⊕ g) → ޒ by

φ(X + ξ, Y + η) = ϕ(X + ᏷(ξ), Y + ᏷(η)).

Then we have ν(˜ X, Y, ξ) = dφ(X, Y, ξ) ∗ ∗ = −φ([X, Y ], ξ) + φ(adX ξ, Y ) − φ(adY ξ, X) = −ϕ([X, Y ], ᏷(ξ)) + ϕ([X, ᏷(ξ)], Y ) − ϕ([Y, ᏷(ξ)], X) = dϕ(X, Y, ᏷(ξ)). On the other hand, we have

ν(˜ X, Y, ξ) = h[X, Y ], ξi = h[X, Y ], ᏷(ξ)ik, SEMIDIRECT PRODUCTS OF REPRESENTATIONS UP TO HOMOTOPY 221 which implies that the Cartan 3-form

h[X, Y ], ᏷(ξ)ik = dϕ(X, Y, ᏷(ξ)) is exact. This is a contradiction. In Section 4, we will give the integration of the Lie 2-algebra of string type ޒ −→0 g ⊕ g∗ by using the semidirect product of a Lie group with its 2-term representation up to homotopy. It turns out that this Lie 2-algebra of string type can be integrated to a special Lie 2-group with a ﬁnite-dimensional model. Example III (Omni-Lie algebroids DE ⊕ JE). Chen and Liu [2010] introduced omni-Lie algebroids to generalize Weinstein’s omni-Lie algebras. Just as Dirac structures of an omni-Lie algebra characterize Lie algebra structures on a vector space, Dirac structures of an omni-Lie algebroid characterize Lie algebroid structures on a vector bundle. See [Chen et al. 2008] for more details. In fact, the role of the omni-Lie algebroids DE ⊕ JE in E-Courant algebroids, which were introduced in [Chen et al. 2010], is the same as the role of standard Courant algebroids TM ⊕ T ∗ M in Courant algebroids. We brieﬂy recall the notion of omni-Lie algebroids. We will see that it gives rise to a 2-term L∞-algebra that is a semidirect product. In this subsection E is a vector bundle over a smooth manifold M, and 0(E) is the section space of E. Let DE be the covariant differential operator bundle of a vector bundle E. The associated Atiyah sequence is given by i a (21) 0 → gl(E) / DE / TM → 0.

We deﬁne the associated 1-jet vector bundle JE as follows. For any m ∈ M, we deﬁne (JE)m as a quotient of local sections of E. Two local sections u1 and u2 are equivalent (we denote this by u1 ∼ u2) if ∗ u1(m) = u2(m) and dhu1, ξim = dhu2, ξim for all ξ ∈ 0(E ).

So any µ ∈ (JE)m has a representative u ∈ 0(E) such that µ = [u]m. Let p be ∼ the projection that sends [u]m to u(m). Then Ker p = Hom(TM, E) and there is a short exact sequence e p (22) 0 → Hom(TM, E) / JE / E → 0, called the jet sequence of E. From this it is straightforward to see that JE is a ﬁnite dimensional vector bundle. Also, 0(JE) is isomorphic to 0(E)⊕0(T ∗ M ⊗ E) as an ޒ-vector space, and any u ∈0(E) has a lift du ∈0(JE) by taking its equivalence class, such that

(23) d( f u) = f du + d f ⊗ u for all f ∈ C∞(M). 222 YUNHE SHENG AND CHENCHANGZHU

Chen and Liu [2010] proved that ∼ JE = {ν ∈ Hom(DE, E) | ν(8) = 8 ◦ ν(IdE ) for all 8 ∈ gl(E)} . Therefore, there is an E-pairing between JE and DE obtained by setting

(24) hµ, diE , d(u) for all µ ∈ (JE)m, d ∈ (DE)m, where u ∈ 0(E) satisﬁes µ = [u]m. Particularly, one has p (25) hµ, 8iE = 8 ◦ (µ) for all 8 ∈ gl(E), µ ∈ JE;

(26) hy, diE = y ◦ a(d) for all y ∈ Hom(TM, E), d ∈ DE. Furthermore, we claim that 0(JE) is an invariant subspace of the Lie derivative Ld for any d ∈ 0(DE), which is deﬁned by the Leibniz rule as follows: 0 0 0 0 hLdµ, d iE , dhµ, d iE − hµ, [d, d ]DiE for all µ ∈ 0(JE), d ∈ 0(DE).

Deﬁne a nondegenerate symmetric E-valued 2-form ( · , · )E on Ᏹ ,DE⊕JE by + + 1 h i + h i ∈ ∈ (d µ, r ν)E , 2 ( d, ν E r, µ E ) for all d, r DE, µ, ν JE. Deﬁne an antisymmetric bracket [[· , · ]] on 0(Ᏹ) by [[ + + ]] [ ] + − + 1 dh i − dh i d µ, r ν , d, r D Ldν Lrµ 2 ( µ, r E ν, d E ).

Chen and Liu [2010] call the quadruple (Ᏹ, [[· , · ]],( · , · )E , ρε) the omni-Lie algebroid associated to the vector bundle E, where ρ is the projection of Ᏹ onto DE.3 Even though [[· , · ]] is antisymmetric, it is not a Lie bracket. More precisely, we have

[[[[X, Y ]], Z]]+ c.p. = dT (X, Y, Z) for any X, Y, Z ∈ 0(Ᏹ), where T : 0(V3 Ᏹ) → 0(E) is deﬁned by = 1 [[ ]] + (27) T (X, Y, Z) 3 (( X, Y , Z)E c.p.).

Let us construct a 2-term L∞-algebra from the omni-Lie algebroid Ᏹ. Obviously, 0(DE) is a Lie algebra and there is a natural 2-term complex d 0(E) −−−→0⊕ 0(JE).

For any d ∈ 0(DE), deﬁne linear actions µ0 and µ1 by d µ0(d)(µ) , [[d, µ]]= Ldµ − (µ, d)E for all µ ∈ 0(JE), (28) d = 1 ∈ µ1(d)(u) , (d, u)E 2 d(u) for all u 0(E).

3This is slightly different from the notion given in [Chen and Liu 2010], where the bracket is not skew symmetric. SEMIDIRECT PRODUCTS OF REPRESENTATIONS UP TO HOMOTOPY 223

Deﬁne ν : V2 0(DE) → Hom(0(JE), 0(E)) by (29) ν(d, t)(ξ) = T (d, t, µ) for all d, t ∈ 0(DE), µ ∈ 0(JE). Similarly to Proposition 2.6, we prove: 0⊕d Proposition 2.14. With the notations above, (0(E) −−−→ 0(JE), µ = µ0 +µ1, ν) is a representation up to homotopy of the Lie algebra 0(DE). 0⊕d Corollary 2.15. (0(E) −−−→ 0(DE) ⊕ 0(JE), l2, l3) is a 2-term L∞-algebra, where l2 and l3 are given by (7) and (8), and µ and ν are given by (28) and (29), respectively. Remark 2.16. If the base manifold M is a point, that is, E is a vector space, for which we use a new notation V , then DV = gl(V ) and JV = V , and we d recover the notion of omni-Lie algebras. The complex 0(V ) −−−→0⊕ 0(JV ) reduces Id to V −−→ V , which is a representation up to homotopy of gl(V ) with (µ0 = µ1, ν) given by = 1 = 1 [ ] ∈ ∈ (30) µ0(A)(u) 2 Au, ν(A, B) 4 A, B for all A, B gl(V ), u V. Hence even though an omni-Lie algebra gl(V ) ⊕ V is not a Lie algebra, we can extend it to a 2-term L∞-algebra, of which L0 = gl(V )⊕ V, L1 = V , l2 and l3 are given by (7) and (8), in which µ and ν are given by (30). This 2-term L∞-algebra is a semidirect product of gl(V ) with V −−→Id V . We will study the global object of the 2-term L∞-algebra associated to an omni- Lie algebra in the forthcoming paper [Sheng and Zhu 2010].

3. Representations up to homotopy of Lie groups and semidirect products

The representation up to homotopy of a Lie group was introduced in [Abad 2008]. In this section we define the semidirect product of a Lie group with a 2-term representation up to homotopy and prove that the semidirect product is a Lie 2-group. Thus we first recall some background on Lie 2-groups. A group is a monoid where every element has an inverse. A 2-group is a monoidal category where every object has a weak inverse and every morphism has an inverse. Denote the category of smooth manifolds and smooth maps by Diff, a semistrict4 Lie 2-group is a 2-group in DiffCat, where DiffCat is the 2-category consisting of categories, functors, and natural transformations in Diff. For more details, see [Baez and Lauda 2004]. Here we only recall the expanded definition: Definition 3.1 [Baez and Lauda 2004]. A semistrict Lie 2-group consists of an object C in DiffCat, that is, s C / C , 1 t 0

4See the introduction for the reason we call it a semistrict Lie 2-group. 224 YUNHE SHENG AND CHENCHANGZHU where C1 and C0 are objects in Diff, s and t are the source and target maps, and there is a vertical multiplication ·v : C × C → C, together with

• a functor (horizontal multiplication) ·h : C × C → C, • an identity object 1, • a contravariant functor inv : C → C and the following natural isomorphisms:

• the associator ax,y,z : (x ·h y) ·h z → x ·h (y ·h z),

• the left and right unit lx : 1 ·h x → x and rx : x ·h 1 → x,

• the unit and counit ix : 1 → x ·h inv(x) and ex : inv(x) ·h x → 1, which are such that the following diagrams commute.

• The pentagon identity for the associator:

(w ·h x) ·h (y ·h z) 3 a · a(w·hx),y,z w,x,y hz + ((w ·h x) ·h y) ·h z w ·h (x ·h (y ·h z)) 8

aw,x,y ·h1z 1w·hax,y,z % aw,x·h y,z (w ·h (x ·h y)) ·h z / w ·h ((x ·h y) ·h z)

• The triangle identity for the left and right unit lows:

ax,1,y (x ·h 1) ·h y / x ·h (1 ·h y) rx ·h1y 1 · l & x x h y x ·h y

• The ﬁrst zig-zag identity:

ax,inv(x),x (x ·h inv(x)) ·h x / x ·h (inv(x) ·h x) 7 ix ·h1x 1x ·hex ' · · 1 h x 3 x h 1 −1 lx rx

+ x SEMIDIRECT PRODUCTS OF REPRESENTATIONS UP TO HOMOTOPY 225

• The second zig-zag identity:

ainv(x),x,inv(x) inv(x) ·h (x ·h inv(x)) / (inv(x) ·h x) ·h inv(x) 7 1inv(x)·hix ex ·h1inv(x)

' inv(x) ·h 1 1 ·h inv(x) − − 3 rinv(x) linv(x) 1

+ inv(x)

In the special case where ax,y,z, lx , rx , ix and ex are all identity isomorphisms, we call such a Lie 2-group a strict Lie 2-group.5 Definition 3.2 [Baez and Lauda 2004].A special Lie 2-group is a Lie 2-group of which the source and target coincide and the left unit law l, the right unit law r, the unit i and the counit e are identity isomorphisms. For classification of special Lie 2-groups, we need the group cohomology with smooth cocycles, that is, we consider the cochain complex with smooth morphisms G×n → M with G a Lie group and M its module. The differential is defined as • usual for group cohomology. We denote this cohomology by Hsm(G, M). Theorem 3.3 [Baez and Lauda 2004, Theorem 8.3.7]. There is a one-to-one correspondence between special Lie 2-groups and quadruples (K1, K2, 8, 2) consisting a Lie group K1, an abelian group K2, an action 8 of K1 as automorphisms of : 3 → K2 and a normalized smooth 3-cocycle 2 K1 K2. Two special Lie 2-groups 6 are isomorphic if and only if they correspond to the same (K1, K2, 8) and the 3 corresponding 3-cocycles represent the same element in Hsm(K1, K2).

Remark 3.4. Given a quadruple (K1, K2, 8, 2), the corresponding semistrict Lie 2-group has the Lie group K1 as the space of objects and the semidirect product Lie group K1 n8 K2 as the space of morphisms. The associator is given by 2. Deﬁnition 3.5. A unital 2-term representation up to homotopy of a Lie group G consists of d (a) a 2-term complex of vector spaces V1 −→ V0;

(b) a nonassociative action F1 on V0 and V1 satisfying dF1 = F1d and F1(1G)=Id; and

(31) F1(g1) · F1(g2) − F1(g1 · g2) = [d, F2(g1, g2)]

5The notion of strict Lie 2-groups is the same as in [Baez and Lauda 2004]. 6up to isomorphisms of groups, of course 226 YUNHE SHENG AND CHENCHANGZHU

and

(32) F1(g1)◦F2(g2, g3)−F2(g1·g2, g3)+F2(g1, g2·g3)−F2(g1, g2)◦F1(g3) = 0. We denote this 2-term representation up to homotopy of the Lie group G by d (V1 −→ V0, F1, F2). One should be careful: Even if F1 is a usual associative action, (32) is not equivalent to F2 being a 2-cocycle. This is strangely different from the 3 Lie algebra case (see Section 4). Deﬁne Fe2 : (G n V0) → V1 by

(33) Fe2((g1, ξ1), (g2, ξ2), (g3, ξ3)) = F2(g1, g2)(ξ3).

If F1 is a usual associative action, we form the semidirect product G n V0. Then V1 is a G n V0-module with an associated action Fe1 of G n V0 on V1 given by

Fe1(g, ξ)(m) = F1(g)(m) for all m ∈ V1.

Proposition 3.6. If F1 is the usual associative action of the Lie group G on the d complex V1 −→ V0, then Fe2 deﬁned by (33) is a group 3-cocycle representing an 3 element in Hsm(G n V0, V1). Proof. By direct computation, we have d Fe2((g1, ξ1), (g2, ξ2), (g3, ξ3), (g4, ξ4))

= Fe1(g1, ξ1)Fe2((g2, ξ2), (g3, ξ3), (g4, ξ4))

−Fe2((g1, ξ1)·(g2, ξ2), (g3, ξ3), (g4, ξ4))+Fe2((g1, ξ1), (g2, ξ2)·(g3, ξ3), (g4, ξ4))

−Fe2((g1, ξ1), (g2, ξ2), (g3, ξ3)·(g4, ξ4))+Fe2((g1, ξ1), (g2, ξ2), (g3, ξ3))

= F1(g1)F2(g2, g3)(ξ4)−F2(g1·g2, g3)(ξ4)+F2(g1, g2·g3)(ξ4)

−F2(g1, g2)(ξ3+F1(g3)(ξ4))+F2(g1, g2)(ξ3) = F1(g1)◦F2(g2, g3)−F2(g1·g2, g3)+F2(g1, g2·g3)−F2(g1, g2)◦F1(g3) (ξ4).

By (32), Fe2 is a Lie group 3-cocycle. Just as we can associate to any representation of a Lie group a new Lie group that is their semidirect product, we can use a 2-term representation up to homotopy of a Lie group to form a Lie 2-group. d Theorem 3.7. Given a 2-term representation up to homotopy (V1 −→ V0, F1, F2) of a Lie group G, its semidirect product with G is deﬁned to be

G × V0 × V1

(34) s t G × V0. Then it is a Lie 2-group with the following structure maps: SEMIDIRECT PRODUCTS OF REPRESENTATIONS UP TO HOMOTOPY 227

The source and target are given by

(35) s(g, ξ, m) = (g, ξ) and t(g, ξ, m) = (g, ξ + dm).

The vertical multiplication ·v is given by

(h, η, n) ·v (g, ξ, m) = (g, ξ, m + n), where h = g, η = ξ + dm.

The horizontal multiplication ·h of objects is given by

(36) (g1, ξ) ·h (g2, η) = (g1 · g2, ξ + F1(g1)(η)).

The horizontal multiplication ·h of morphisms is given by

(37) (g1, ξ, m) ·h (g2, η, n) = (g1 · g2, ξ + F1(g1)(η), m + F1(g1)(n)).

The inverse map inv is given by

−1 −1 (38) inv(g, ξ) = (g , −F1(g )(ξ)).

The identity object is (1G, 0). The associator

: · · → · · a(g1,ξ),(g2,η),(g3,γ ) (g1, ξ) h (g2, η) h (g3, γ ) (g1, ξ) h (g2, η) h (g3, γ ) is given by

= · · + + · (39) a(g1,ξ),(g2,η),(g3,γ ) (g1 g2 g3, ξ F1(g1)(η) F1(g1 g2)(γ ), F2(g1, g2)(γ )).

The unit i(g,ξ) : (1G, 0) → (g, ξ) ·h inv(g, ξ) is given by

−1 (40) i(g,ξ) = (1G, 0, −F2(g, g )(ξ)).

All the other natural isomorphisms are identity isomorphisms.

Proof. By (35), (36) and (37), it is straightforward to see that s (g1, ξ, m) ·h (g2, η, n) = s(g1, ξ, m) ·h s(g2, η, n), t (g1, ξ, m) ·h (g2, η, n) = t(g1, ξ, m) ·h t(g2, η, n).

Thus the multiplication ·h respects the source and target map. Furthermore, it is not hard to check that the horizontal and vertical multiplications commute, that is,

0 0 (g, ξ + dm, n) ·h (g , η + dp, q) ·v (g, ξ, m) ·h (g , η, p) 0 0 = (g, ξ + dm, n) ·v (g, ξ, m) ·h (g , η + dp, q) ·v (g , η, p) 228 YUNHE SHENG AND CHENCHANGZHU or, in terms of a diagram,

(g, ξ) (g0, η)

m p Ö Õ 0 (41) ••o (g, ξ + dm) o (g , η + dp) •. Y Y n q (g, ξ + d(m + n))(g0, η + d(p + q))

It follows from (31) that the associator a(g1,ξ),(g2,η),(g3,γ ) deﬁned by (39) is indeed a morphism from ((g1, ξ)·h (g2, η))·h (g3, γ ) to (g1, ξ)·h ((g2, η)·h (g3, γ )). To see that it is natural, we need to show that

· · · (42) a(g1,ξ+dm),(g2,η+dn),(g3,γ +dk) h (((g1, ξ, m) h (g2, η, n)) h (g3, γ, k)) is equal to

· · · (43) (g1, ξ, m) h ((g2, η, n) h (g3, γ, k)) h a(g1,ξ),(g2,η),(g3,γ ), that is, the following diagram commutes:

((g1, ξ) ·h (g2, η)) ·h (g3, γ ) a , (g1, ξ) ·h ((g2, η) ·h (g3, γ ))

((g1, ξ + dm) ·h (g2, η + dn)) ·h (g3, γ + dk) a , (g1, ξ + dm) ·h ((g2, η + dn) ·h (g3, γ + dk)).

By straightforward computations, we obtain that (42) is equal to

(g1 · g2 · g3,

ξ+F1(g1)(η)+F1(g1·g2)(γ ), m+F1(g1)(n)+F1(g1·g2)(k)+F2(g1, g2)(γ +dk)), and (43) is equal to

(g1 · g2 · g3,

ξ+F1(g1)(η)+F1(g1·g2)(γ ), m+F1(g1)(n)+F1(g1)·F1(g2)(k)+F2(g1, g2)(γ )).

Hence (42) is equal to (43) by (31). This implies that a(g1,ξ),(g2,η),(g3,γ ) as deﬁned by (39) is a natural isomorphism. SEMIDIRECT PRODUCTS OF REPRESENTATIONS UP TO HOMOTOPY 229

By (31) and the fact that F1(1G)=Id, the unit given by (40) is indeed a morphism from (1G, 0) to (g, ξ) ·h inv(g, ξ). To see that it is natural, we need to prove that (g, ξ, m) ·h inv(g, ξ, m) ·h i(g,ξ) = i(g,ξ+dm), that is, that the diagram

(1G, 0) i(g,ξ+dm) i(g,ξ)

u (g,ξ,m)·hinv(g,ξ,m) ' (g, ξ + dm) ·h inv(g, ξ + dm)(o g, ξ) ·h inv(g, ξ) commutes. This follows from

−1 −1 −1 −1 F2(g, g )(dm) = F1(g)· F1(g )(m)− F1(g·g )(m) = F1(g)· F1(g )(m)−m, which is a special case of (31). Since F(1G) = Id, we have

(1G, 0) ·h (g, ξ) = (g, ξ) and (g, ξ) ·h (1G, 0) = (g, ξ). Hence the left unit and the right unit can also be taken as the identity isomorphism. The counit e(g,ξ) : inv(g, ξ) ·h (g, ξ) → (1G, 0) can be taken as the identity morphism since

−1 −1 inv(g, ξ) ·h (g, ξ) = (g , −F1(g )(ξ)) ·h (g, ξ) = (1G, 0). Finally, we need to show the pentagon identity for the associator, the triangle identity for the left and right unit laws, and the ﬁrst and second zig-zag identities. We only give the proof of the pentagon identity; we leave the similar proofs of the others to the reader. In fact, the pentagon identity is equivalent to · a(g1,ξ),(g2,η),(g3,γ )·h(g4,θ) v a(g1,ξ)·h(g2,η),(g3,γ ),(g4,θ) = · · · · ((g1, ξ) ha(g2,η),(g3,γ ),(g4,θ)) va(g1,ξ),(g2,η)·h(g3,γ ),(g4,θ) v(a(g1,ξ),(g2,η),(g3,γ ) h(g4, θ)). By straightforward computations, the left hand side is equal to

(g1 · g2 · g3 · g4, ξ + F1(g1)(η) + F1(g1 · g2)(γ ) + F1(g1 · g2 · g3)(θ),

F2(g1 · g2, g3)(θ) + F2(g1, g2)(γ + F1(g3)(θ))), and the right hand side is equal to

(g1 · g2 · g3 · g4, ξ + F1(g1)(η) + F1(g1 · g2)(γ ) + F1(g1 · g2 · g3)(θ),

F2(g1, g2)(γ ) + F2(g1, g2 · g3)(θ) + F1(g1) ◦ F2(g2, g3)(θ)).

By (32), they are equal. 230 YUNHE SHENG AND CHENCHANGZHU

4. Integrating the Lie 2-algebra of string type ޒ → g ⊕ g∗

As an application of Theorem 3.7, we consider the integration of the Lie 2-algebra of string type ޒ → g ⊕ g∗ given by (19). Now we restrict to the case that g is ﬁnite-dimensional. Obviously, given a quadruple (K1, K2, 8, 2) that represents a special Lie 2-group (see Theorem 3.3), we obtain by differentiation a quadruple (k1, k2, φ, θ), which represents a 2-term skeletal L∞-algebra.

Deﬁnition 4.1. A special Lie 2-group that is represented by (K1, K2, 8, 2) is an integration of a 2-term skeletal L∞-algebra that is represented by (k1, k2, φ, θ) if the differentiation of (K1, K2, 8, 2) is (k1, k2, φ, θ). d If the differential d in a 2-term complex V1 −→ V0 is 0, a representation up 0 to homotopy of Lie algebra g on V1 −→ V0 consists of two strict representations µ1 and µ0, and a linear map ν : g ∧ g → Hom(V0, V1) satisfying equation (6). This equation implies that ν is a Lie algebra 2-cocycle representing an element 2 in H (g, Hom(V0, V1)), with the representation [µ( · ), ·] of g on Hom(V0, V1) deﬁned by

[µ( · ), ·](X)(A) , [µ(X), A]

= µ1(X) ◦ A − A ◦ µ0(X) for all X ∈ g, A ∈ Hom(V0, V1). V3 Lemma 4.2. Deﬁne ν˜ : (g ⊕ V0) → V1 by

ν(˜ X1 + ξ1, X2 + ξ2, X3 + ξ3) = ν(X1, X2)(ξ3) + c.p. Then ν is a 2-cocycle if and only if ν˜ is a 3-cocycle where the representation µ˜ of g ⊕ V0 on V1 is given by µ(˜ X + ξ)(m) = µ(X)(m).

Proof. By direct computations, for any Xi +ξi ∈ g⊕V0 with i = 1, 2, 3, 4, we have

dν(˜ X1 + ξ1, X2 + ξ2, X3 + ξ3, X4 + ξ4) = dν(X1, X2, X3)(ξ4) + c.p.

The Lie algebra homomorphism µ from g to End(V0) ⊕ End(V1) integrates to a Lie group homomorphism F1 from the simply connected Lie group G of g to GL(V0) ⊕ GL(V1), with d µ(X) = F1(exp t X) for all X ∈ g. dt t=0

Consequently, Hom(V0, V1) is a G-module with G action −1 g · A = F1(g) ◦ A ◦ F1(g) for all g ∈ G, A ∈ Hom(V0, V1).

The Lie algebra 2-cocycle ν : g ∧ g → Hom(V0, V1) can integrate to a smooth Lie group 2-cocycle F2 : G × G → Hom(V0, V1), satisfying −1 (44) F1(g1) ◦ (F2)(g2, g3) ◦ F1(g1) − (F2)(g1 · g2, g3) + (F2)(g1, g2 · g3) − (F2)(g1, g2) = 0, SEMIDIRECT PRODUCTS OF REPRESENTATIONS UP TO HOMOTOPY 231 and F2(1G, 1G) = 0. Let us explain how. The classical theory of cohomology of discrete groups says that the equivalence classes of extensions of G by a G module M are in bijection with the elements 2 2 of H (G, M). In our case, the same theory tells us that Hsm(G, Hom(V0, V0)) classiﬁes the equivalence classes of splitting extensions of G by the G-module Hom(V0, V1), which is a splitting short exact sequence of Lie groups in which Hom(V0, V1) is endowed with an abelian group structure ˆ (45) Hom(V0, V1) → G → G.

In a general extension, Gˆ is a principal bundle over G; thus it usually does not permit a smooth lift G −→σ Gˆ . It permits such a lift if and only if the sequence splits. However in our case, since the abelian group Hom(V0, V1) is a vector space, 1 we have H (X, Hom(V0, V1)) = 0 for any manifold X. The proof makes use of a partition of unity and is similar to the proof showing that H 1(X, ޒ) = 0 for the sheaf cohomology. Hence all Hom(V0, V1) principal bundles are trivial. Therefore (45) always splits. On the other hand it is well known that when G is simply connected, there is a one-to-one correspondence between extensions of G [Brahic 2010, Theorem 4.15] and extensions of its Lie algebra g, which in turn 2 are classiﬁed by the Lie algebra cohomology H (g, Hom(V0, V1)). Hence in our 2 2 case the differentiation map Hsm(G, Hom(V0, V1)) → H (g, Hom(V0, V1)) is an isomorphism. Hence ν always integrates to a smooth Lie group 2-cocycle unique up to exact 2-cocycles. Then F2(1G, 1G) = 0 can be arranged too, because we can ˆ always modify the section σ : G → G to satisfy σ (1G) = 1Gˆ and the modiﬁcation of sections results in an exact term. Then combined with (44), it is not hard to see that

(46) F2(1G, g) = F2(g, 1G) = 0 for all g ∈ G.

Thus F2 is a normalized 2-cocycle. Proposition 4.3. For any 2-term representation up to homotopy (µ, ν) of a Lie 0 algebra g on the complex V1 −→ V0, there is an associated representation up to 0 homotopy (F1, F2) of the Lie group G on the complex V1 −→ V0, where F1 is the integration of µ and F2 : G × G → End(V0, V1) is deﬁned by

(47) F2(g1, g2) = F2(g1, g2) ◦ F1(g1 · g2).

Proof. Obviously, (31) is satisﬁed. To see (32) is also satisﬁed, combine (47) with (44). By the fact that F1 is a homomorphism, we obtain

−1 −1 −1 F1(g1)◦ F2(g2, g3)◦ F1(g2 ·g3) ◦ F1(g1) − F2(g1 ·g2, g3)◦ F1(g1 ·g2 ·g3) −1 −1 + F2(g1, g2 ·g3)◦ F1(g1 ·g2 ·g3) − F2(g1, g2)◦ F1(g1 ·g2) = 0. 232 YUNHE SHENG AND CHENCHANGZHU

Composing this with F1(g1 · g2 · g3) on the right hand side, we obtain (32). By Proposition 4.3 and Theorem 3.7, we have: Theorem 4.4. Let G be the simply connected Lie group integrating g. Then the Lie 2-algebra of string type ޒ −→0 g ⊕ g∗ given by (19) integrates to the Lie 2-group G × g∗ × ޒ

(48) s t G × g∗, in which the source and target are given by

(49) s(g, ξ, m) = t(g, ξ, m) = (g, ξ), the vertical multiplication ·v is given by

(h, η, n) ·v (g, ξ, m) = (g, ξ, m + n), where h = g, η = ξ, the horizontal multiplication ·h of objects is given by (g , ξ) · (g , η) = (g · g , ξ + Ad∗ η), 1 h 2 1 2 g1 the horizontal multiplication ·h of morphisms is given by (g , ξ, m) · (g , η, n) = (g · g , ξ + Ad∗ η, m + n), 1 h 2 1 2 g1 the inverse map inv is given by = −1 − ∗ = −1 − ∗ − inv(g, ξ) (g , Adg−1 ξ), inv(g, ξ, m) (g , Adg−1 ξ, m), the identity object is (1G, 0), and the associator : · · → · · a(g1,ξ),(g2,η),(g3,γ ) (g1, ξ) h (g2, η) h (g3, γ ) (g1, ξ) h (g2, η) h (g3, γ ) is given by

a = (g · g · g , ξ + Ad∗ η + Ad∗ γ, F (g , g )(γ )). (g1,ξ),(g2,η),(g3,γ ) 1 2 3 g1 g1·g2 2 1 2 All the other structures are identity isomorphisms.

Proof. Since F1 is a usual associative action, we may modify the unit (40) given in Theorem 3.7 to be the identity natural transformation. It turns out that (48) is a ∗ ∗ special Lie 2-group and is represented by (G n g , ޒ, Id, Fe2), where G n g is the semidirect product with the coadjoint action of G on g∗, Id is the constant map ∗ G n g → Aut(ޒ) that maps everything to Id ∈ Aut(ޒ), and Fe2 is given by

(50) Fe2((g1, ξ1), (g2, ξ2), (g3, ξ3))= F2(g1, g2)(ξ3)= F2(g1, g2)◦F1(g1·g2)(ξ3). SEMIDIRECT PRODUCTS OF REPRESENTATIONS UP TO HOMOTOPY 233

0 ∗ Since F2 is normalized, so is Fe2. The Lie 2-algebra of string type ޒ −→ g ⊕ g is skeletal and is represented by (g ⊕ g∗, ޒ, 0, ν)˜ , where ν˜ is given by (20). Thus to show that our Lie 2-group is one of its integrations, we only need to show that the differential of the Lie group 3-cocycle Fe2 is the Lie algebra 3-cocycle ν˜. By direct computations [Brylinski 1993, Lemma 7.3.9], we have X 3 (σ )F (etσ (1) Xσ (1) , t ξ ), (etσ (2) Xσ (2) , t ξ ), (etσ (3) Xσ (3) , t ξ ) ∂ e2 σ (1) σ (1) σ (2) σ (2) σ (3) σ (3) σ∈S3 ∂t1 ∂t2 ∂t3 ti =0 3 ∂ t1 X1 t2 X2 t1 X1 t2 X2 = F 2(e , e ) ◦ F1(e · e )(t3ξ3) + c.p. ∂ ∂ ∂ t1 t2 t3 ti =0 2 ∂ t1 X1 t2 X2 t1 X1 t2 X2 = F 2(e , e ) ◦ F1(e · e )(ξ3) + c.p. ∂ ∂ t1 t2 ti =0 ∂ ∂ t1 X1 t2 X2 t1 X1 = F 2(e , e ) ◦ F1(e )(ξ3) ∂t ∂t 1 t1=0 2 t2=0 t1 X1 ∂ t1 X1 t2 X2 + F 2(e , 1G ) ◦ F1(e · e ) + c.p. ∂ t2 t2=0 ∂ ∂ ∂ t1 X1 t2 X2 ∂ t2 X2 t1 X1 = F2(e , e )(ξ3) + F2(1G , e ) ◦ F1(e )(ξ3) ∂ ∂ ∂ ∂ t1 t2 ti =0 t2 t2=0 t1 t1=0 + c.p. (by (46))

= ν(X1, X2)(ξ3) + c.p.

=ν( ˜ X1 + ξ1, X2 + ξ2, X3 + ξ3) (by (20)), which completes the proof.

Corollary 4.5. If Lie algebra g is semisimple, the Lie group 3-cocycle Fe2 is not 3 ∗ exact, that is, [Fe2] 6= 0 in Hsm(G n g , ޒ).

Proof. By Theorem 4.4, the differentiation of the Lie group 3-cocycle Fe2 is the Lie algebra 3-cocycle ν˜. We only need to show that when g is semisimple, the Lie algebra 3-cocycle ν˜ is not exact. This fact is proved in Proposition 2.13. ∗ p,q = Remark 4.6. Since Gng is a ﬁbration over G, the spectral sequence with E2 p q ∗ 3 ∗ ∗ Hsm(G, Hsm(g , ޒ)) calculates the group cohomology Hsm(G n g , ޒ). Since g q ∗ Vq ∗ is an abelian group, we have Hsm(g , ޒ) = g . Thus when G is compact, p q ∗ Vq ∗ G Vq ∗ G Hsm(G, Hsm(g , ޒ)) = ( g ) if p = 0 and 0 otherwise, where ( g ) denotes the set of invariant elements of Vq g∗ under the coadjoint action of G. Thus when 3 ∗ V3 ∗ G G is compact, Hsm(G n g , ޒ) = ( g ) 6= 0 because the Cartan 3-form is an element of (V3 g∗)G.

Our 2-cocycle F2 is unique only up to exact terms. Hence by Theorem 3.3, we need this lemma to verify that our construction is unique up to isomorphism: 234 YUNHE SHENG AND CHENCHANGZHU

Lemma 4.7. If F2 = dα is exact, then Fe2 = dβ is also exact with

β((g1, ξ1), (g2, ξ2)) = α(g1)F1(g1)(ξ2).

Proof. It is a direct calculation. Since F2 = dα, we have −1 F2(g1, g2) = F1(g1)α(g2)F1(g1) − α(g1g2) + α(g1).

From the deﬁnition of F2, we know that −1 F2(g1, g2) = F1(g1)α(g2)F1(g1) F1(g1g2) − α(g1g2)F1(g1g2) + α(g1)F1(g1g2). By (50), we have

Fe2((g1, ξ1), (g2, ξ2), (g3, ξ3)) −1 = F1(g1)α(g2)F1(g1) F1(g1g2)(ξ3)−α(g1g2)F1(g1g2)(ξ3)+α(g1)F1(g1g2)(ξ3)

= dβ((g1, ξ1), (g2, ξ2), (g3, ξ3)), since F1 is a group homomorphism. Remark 4.8. Our Lie 2-group as a stacky group has the underlying differential ∗ stack G × g × Bޒ. Thus it is 0, 1, 2-connected (that is, it has π0 = π1 = π2 = 0) since π2(Bޒ) = π1(ޒ) = 0 and π1(Bޒ) = π0(ޒ) = 0. Thus it is the unique 0, 1, 2- connected stacky Lie group integrating the Lie 2-algebra of string type ޒ −→0 g⊕g∗ in the sense of [Zhu 2007].

Acknowledgements

We give our warmest thanks to Zhang-Ju Liu, Jiang-Hua Lu, Giorgio Trentinagia and Marco Zambon for useful comments and discussion. Y. Sheng thanks the Courant Research Centre “Higher Order Structures” at Gottingen¨ University, where this work was during his visit there.

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Received October 14, 2009. Revised November 25, 2009. 236 YUNHE SHENG AND CHENCHANGZHU

YUNHE SHENG DEPARTMENT OF MATHEMATICS JILIN UNIVERSITY CHANGCHUN,JILIN 130012 CHINA [email protected]

CHENCHANG ZHU COURANT RESEARCH CENTRE “HIGHER ORDER STRUCTURES” MATHEMATISCHES INSTITUT UNIVERSITYOF GÖTTINGEN BUNSENSTRASSE 3-5 D-37073 GÖTTINGEN GERMANY [email protected] PACIFIC JOURNAL OF MATHEMATICS Vol. 249, No. 1, 2011

HOMOLOGY SEQUENCE AND EXCISION THEOREM FOR EULER CLASS GROUP

YONG YANG

Let R be a Noetherian commutative ring with dim R = d and let l be an ideal of R. For an integer n such that 2n ≥ d + 3, we deﬁne a relative Euler class group En(R, l; R). Using this group, in analogy to homology sequence of the K0-group, we construct an exact sequence

E( p ) E(ρ) En(R, l; R) −−−→2 En(R; R) −−→ En(R/l; R/l), called the homology sequence of the Euler class group. The excision theorem in K-theory has a corresponding theorem for the Euler class group. An application is that for polynomial and Laurent polynomial rings, we get short split exact sequences

E( p ) E(ρ) 0 → En(R[t],(t); R[t]) −−−→2 En(R[t]; R[t]) −−→ En(R; R) → 0 and E( p ) 0 → En(R[t, t−1],(t − 1); R[t, t−1]) −−−→2 En(R[t, t−1]; R[t, t−1]) E(ρ) −−→ En(R; R) → 0.

1. Introduction

Let R be a Noetherian commutative ring of dimension d. The notion of the Euler class group of R was introduced by Nori around 1990, with the aim of develop- ing an obstruction theory for algebraic vector bundles over smooth affine varieties [Mandal 1992]. Later, the Nori’s definition was extended by S. M. Bhatwadekar and Raja Sridharan [2000]. Given a Noetherian commutative ring R with dimension d ≥ 2, they defined an obstruction group Ed (R; R) also called the Euler class group (ECG). For ޑ ⊆ R and any projective R-module P of rank d with orientation χ : R =∼ Vd P, they defined an obstruction class e(P; χ) ∈ Ed (R; R) and proved that P =∼ Q ⊕ R if and only if e(P; χ) = 0. After that, much work

MSC2000: primary 13C10; secondary 13B25. Keywords: Euler class group, K-theory, excision theorem. Partially supported by NCET (NCET06-09-23), and NUDT(JC08-02-03) to Professor Lianggui Feng.

237 238 YONG YANG on ECGs and weak ECGs was done. K. D. Das [2003; 2006] defined the ECG Ed (R[t]; R[t]) for a Noetherian commutative ring R with dim R = d, and proved that for a general such ring, the ECG Ed (R; R) of R is a direct summand of the ECG Ed (R[t]; R[t]), whereas if R is a smooth affine domain over some perfect field k, then Ed (R[t]; R[t]) =∼ Ed (R; R). The ECG Ed (R[t, t−1]; R[t, t−1]) of a Laurent polynomial ring R[t, t−1] was defined in [Keshari 2007]. On the other hand, in K-theory we have a homology theory for the category of projective R-modules. The K0-group K0(R) is closely related to the ECG of R. For example, Murthy’s Chern class of the projective R-module P, which is one of the sources of Euler class theory, is defined by an element in the K0- group. For any Noetherian commutative ring R of dimension d, there is a sub- d group F K0(R) of K0(R) [Mandal 1998]. If R is regular and contains the field of d ⊗ ∼ rational numbers ޑ, we have a Riemann–Roch theorem saying that E0 (R) ޑ = d ⊗ ∼ d ⊗ d d F K0(R) ޑ =CH (R) ޑ, in which E0 (R) is the weak ECG of R and CH (R) is the Chow group of codimension d of Spec(R) [Das and Mandal 2006]. Now let l ⊆ R be an ideal of R. K-theory gives for K0-groups the homology sequence K0(R, l) → K0(R) → K0(R/l). If the ring homomorphism ρ : R → R/l is split, then this homology sequence reduces to the short split exact sequence

0 → K0(R, l) / K0(R) / K0(R/l) → 0 , which is said to be the excision sequence for K0-groups. Inspired by the correspondence between K-theory and ECGs, in this paper we establish ECG counterparts to the homology sequence and excision theorem of K0-groups. These counterparts are Theorem 4.2 and Theorem 4.3, respectively. Let R be a Noetherian commutative ring with dimension d, and let l be an ideal of R with dim R/l = d − m. For any integer n such that 2n ≥ d + 3, we deﬁne in Section 3 a group homomorphism E(ρ) : En(R; R) → En(R/l; R/l), called the restriction map of the ECG. In analogy to the relative K0-group K0(R, l) (denoted n by K0(l) in [Rosenberg 1994]), we deﬁne in Section 4 the relative ECG E (R, l; R) n = and the relative weak ECG E0 (R, l). In particular, when l R the relative ECG n ; n E (R, l R) and the relative weak ECG E0 (R, l) are the same as the generalized n ; n ECG E (R R) and the weak ECG E0 (R), respectively. Using these groups, we construct an exact sequence

E(p2) E(ρ) En(R, l; R) / En(R; R) / En(R/l; R/l), the homology sequence of the ECG. If the ring homomorphism ρ : R → R/l has a splitting β satisfying a dimensional condition (see Theorem 4.3 and Remark 4.4), then the homology sequence above reduces to the short split exact sequence

E(p2) E(ρ) 0 → En(R, l; R) / En(R; R) / En(R/l; R/l) → 0, HOMOLOGY SEQUENCE AND EXCISION THEOREM FOR EULER CLASS GROUP 239 called the excision sequence of the ECG. Under these conditions, we have an isomorphism En(R; R) =∼ En(R, l; R) ⊕ En(R/l; R/l). In Section 5, we use the results in Section 4 to get the excision sequences of the ECG for the polynomial extension and the Laurent polynomial extension:

E(p2) E(ρ) 0 → En(R[t],(t); R[t]) / En(R[t]; R[t]) / En(R; R) → 0 and

E(p2) 0 → En(R[t, t−1],(t − 1); R[t, t−1]) / En(R[t, t−1]; R[t, t−1])

E(ρ) / En(R; R) → 0.

Both of these are split exact; see Corollaries 5.1 and 5.3.

2. Some preliminary results

Here, recall the deﬁnition of the generalized ECG and collect some related results.

Definition 2.1. Let R be a Noetherian commutative ring, n be an integer such that 2n ≥ d + 3. In [Bhatwadekar and Sridharan 2002], the generalized Euler class group En(R; R) is defined as follows: Let J ⊂ R be an ideal of height n, such that J/J 2 is generated by n elements. Two surjections α and β from (R/J)n to J/J 2 are said to be related if and only if there exists an elementary matrix δ ∈ Ᏹln(R/J) such that αδ = β. This defines an equivalence relation on the set of surjections from (R/J)n to J/J 2.

n • Let G be the free Abelian group on the set of pairs (J; ωJ ), where J ⊆ R is an ideal of height n, having the property that Spec(R/J) is connected and 2 n 2 J/J is generated by n elements, and ωJ : (R/J) J/J is an equivalence class of surjections. • Now assume that J ⊆ R be an ideal of height n and J/J 2 is generated by n elements. By [Bhatwadekar and Sridharan 2002, Lemma 4.1], J has a Tr unique decomposition J = i=1 Ji where ideals Ji are pairwise comaximal n 2 and Spec(R/Ji ) is connected. Let ωJ : (R/J) → J/J be a surjection. Then : n → 2 ωJ gives rise in a natural way to surjections ωJi (R/Ji ) Ji /Ji . By ; Pr ; n ; (J ωJ ) we mean the element i=1(Ji ωJi ) in G , and (J ωJ ) is called a local orientation. n n • Let H be the subgroup of G generated by set of pairs (J; ωJ ), where J is n 2 an ideal of height n generated by n elements and ωJ : (R/J) J/J has n the property that ωJ can be lifted to a surjection 2 : R J. The generalized Euler class group En(R; R) is deﬁned by En(R; R) = Gn/H n. 240 YONG YANG

n n 2 n Let {ei } be the standard basis of R and α : R J/J be a surjection from R 2 2 to J/J that sends e¯i to a¯i for 1 ≤ i ≤ n, where ai ∈ J and {¯ai } generate J/J . In rest of this paper, we always use (a1,..., an) to denote α. The generalized weak ECG was deﬁned in [Mandal and Yang 2010]:

• n 2 Let L0 denote the set of all ideals J of height n such that Spec(J/J ) is : n 2 n connected and there is a surjection α (R/J) J/J . Let G0 be the free n group generated on the set L0. • For any ideal J ⊆ R of height n such that J/J 2 is generated by n elements, Tr there is a unique decomposition J = i=1 Ji , where the ideals Ji are pairwise comaximal and Spec(R/Ji ) is connected. By (J) we mean the element Pr n i=1(Ji ) in G0. • n n Let H0 be the subgroup of G0 generated by (J), where J could be generated by n elements. Then the generalized weak Euler class group is deﬁned by n = n n E0 G0/H0 . Theorem 2.2 [Bhatwadekar and Sridharan 2002, Theorem 4.2]. Suppose R is a d- dimensional Noetherian commutative ring, and let n be an integer with 2n ≥ d +3. Let J ⊆ R be an ideal of height n such that J/J 2 is generated by n elements, and let n 2 ωJ : (R/J) J/J be an equivalence class of surjections. Suppose that (J; ωJ ) n is zero in the ECG E (R; R). Then, J is generated by n elements and ωJ can be n lifted to a surjection 2 : R J. The following lemma is easy to prove, so we omit the proof. n Lemma 2.3. Let (I ; ωI ) and (J; ωJ ) be two elements of E (R; R). Let surjections

n (a1,...,an) 2 n (b1,...,bn) 2 ωI : R / / I/I and ωJ : R / / J/J be representatives of the equivalence classes of ωI and ωJ , respectively. Suppose I and J are comaximal ideals of R. Then by the Chinese remainder theorem, we can ﬁnd a unique surjection

n (c1,...,cn) 2 ωI ∩J : R / / I ∩ J/(I ∩ J) ,

2 2 where the ci are elements of I ∩ J such that ci = ai (mod I ) and ci = bi (mod J ). n Then (I ∩ J; ωI ∩J ) ∈ E (R; R) is independent of choice representative in the n equivalence classes ωI and ωJ , and (I ; ωI )+(J; ωJ ) = (I ∩ J; ωI ∩J ) ∈ E (R; R). The next lemma is an adapted version of [Mandal and Yang 2010, Lemma 4.3]. We give a proof for this new form. Lemma 2.4 (transversal lemma). Suppose R is a Noetherian commutative ring n with dim R = d. Assume I is an ideal of R with height I = n and ω : R I/J is HOMOLOGY SEQUENCE AND EXCISION THEOREM FOR EULER CLASS GROUP 241

2 a surjection in which J is an ideal of R contained in I . Let l1,..., lr be ﬁnitely n many ideals of R. Then we can ﬁnd a surjective lift v : R I ∩ K such that

K + J = R, height K ≥ n, height((K +li )/li ) ≥ n for any li with 1 ≤ i ≤ r.

Proof. We use standard generalized dimension theory. First, there is a lift v0 : n n R → I of ω. Then I = (v0(R ), a) for some a ∈ J Let ᏼn−1 ⊆ Spec(R) be the set of all prime ideals p with height p ≤ n − 1 and a ∈/ p. For any li such that 1 ≤ i ≤ r, let ᏽi,n−1 ⊆ Spec(R) be the set of all prime ideals p such that p ⊇ li and a ∈/ p, and height(p/li ) ≤ n − 1. Write Sr ᏼ = i=1 ᏽi,n−1 ∪ ᏼn−1. Let d0 : ᏼn−1 → ގ be the restriction of the usual dimension function and let di : ᏽi,n−1 → ގ be the dimension function induced by that on Spec(Ra/lia) for 1 ≤ i ≤ r. Then d0 and di for 1 ≤ i ≤ r induce a generalized dimension function d : ᏼ → ގ; see [Mandal 1997] or [Plumstead 1983]. n∗ n Now (v0, a) ∈ R ⊕ R is a basic element on ᏼ. Since rank(R ) = n > d(p) for n∗ all p ∈ ᏼ, there is a φ ∈ R such that v = v0 + aφ is basic on ᏼ. Clearly, v is a lift of ω and I = (v(Rn), a). Since v is a lift of ω, we can write v(Rn) = I ∩ K, such that K + J = R. It is routine to check that height(K ) ≥ n and height((K + li )/li ) ≥ n for 1 ≤ i ≤ r. Lemma 2.5 (avoid lemma). Let R be a Noetherian commutative ring such that dim R = d, and let l ⊆ R be an ideal of R. Assume that I is an ideal of R and n 2 n φ : R I/I is a surjective map. If there is a surjective map ψ : R (I + l)/l such that φ˜ = ψ ⊗(R/(I +l)) = ψ ⊗(R/I¯), in which the bar denotes the reduction ˜ ˜ n 2 ∼ ¯ ¯2 modulo l, and φ is the surjective map φ : R (I + l)/(I + l) = I /I induced ˆ n 2 by φ, then we can ﬁnd a surjective lift φ : R I/(I l) of φ. n n Proof. Let φ1 : R → I and ψ1 : R → I be lifts of φ and ψ, respectively. Since ˜ n 2 φ = ψ ⊗ R/(I + l), we have φ1 − ψ1 ∈ Hom(R , I + l). Then we can ﬁnd n 2 n α ∈ Hom(R , I ) and β ∈ Hom(R , l) such that φ1 −ψ1 = α +β. This can be seen from the commutative diagram Rn u (α,β) u u φ1−ψ1 u zu I 2 ⊕ l / / I 2 + l,

n 2 in which (α, β) ∈ Hom(R , I ⊕ l) is a lift of φ1 − ψ1. n Now we construct a map φ2 = φ1 − α ∈ Hom(R , I ). Of course, φ2 is still a lift of φ. Let the bar denote the reduction modulo l. Then since φ2 = φ1 −α = ψ1 +β, n and β ∈ Hom(R , l), we have φ2 = ψ1 = ψ. Recall that ψ is surjective, so it n 2 2 is clear that φ2(R ) + I ∩ l = I . Now consider the ideal φ2(R ) + I l. Since n n 2 φ2(R ) + I ∩l = φ2(R ) + I = I , it follows that any prime ideal p of R contains 242 YONG YANG

2 2 n 2 I if and only if it contains φ2(R ) + I l. Note that since φ2(R ) + I = I , we 2 2 have (φ2(R ) + I l)p = I p for any prime ideal p of R containing I . So we get n 2 φ2(R ) + I l = I . ˆ n 2 ˆ Now let φ : R I/(I l) be the map induced by φ2. It’s obvious that φ is a surjective lift of φ.

3. Restriction and extension map

In this section, we construct two group homomorphisms, the restriction map and extension map for the ECG. Let φ : R → A be a ring homomorphism and I ⊆ R be an ideal of R. In the rest of the paper, φ(I ) without special decorations will always denote the ideal φ(I )A, which is the ideal of A generated by φ(I ). Definition 3.1 (restriction map). Let R be a Noetherian commutative ring with dim R = d, and l ⊆ R be an ideal of R with dim R/l = d−m. Let the bar denote the reduction modulo l, and let ρ : R → R/l denote the natural ring homomorphism. For an integer n such that 2n ≥ d + 3, let En(R; R) and En(R/l; R/l) denote the generalized ECG of R and R/l, respectively, as defined in [Bhatwadekar and Sridharan 2002]. Then we can define a group homomorphism E(ρ) : En(R; R) → En(R; R), called the restriction map of ECG, as follows: For any element x ∈ En(R; R), from the properties of the group En(R; R), we n know that x can be written as a pair of (I ; ωI ) ∈ E (R; R), where I is an ideal of n 2 R with height I ≥ n, and ωI is an equivalence class of surjections ωI : R I/I . 0 n Moreover by Lemma 2.4, we can find (I ; ωI ) = (I ; ωI 0 ) ∈ E (R; R) such that 0 0 n height I + l ≥ n in R. Then we define E(ρ)(I ; ωI ) = (I + l; ωI 0+l ) ∈ E (R; R), in which ωI 0+l is the equivalence class of induced surjection defined as

ω 0 0 ¯ 0 0 n I I γ I +l ∼ (I +l) ω 0 : R / / / / = , I +l I 0 2 I 0 2 +l (I 0 +l)2

0 0 2 0 0 2 where γ¯ is the natural map from I /I to (I + l)/(I + l), and ωI 0 is any representative of the equivalence class ωI 0 . 0 (1) Since the map Ᏹln(R) → Ᏹln(R/I ) is surjective, we know that the element E(ρ)(I ; ωI ) is independent of choice of the representative of ωI 0 . n n (2) If (I ; ωI ) = 0 ∈ E (R; R), then E(ρ)(I ; ωI ) = 0 ∈ E (R; R). n Proof of (2). Since (I ; ωI )=0∈ E (R; R), there exists by Theorem 2.2 a surjective n 0 0 lift of ωI 0 , denoted by vI 0 . Then it is easy to check that v¯I 0 : R I (I + l)/l n is a surjective lift of ωI 0+l . So we have E(ρ)(I ; ωI ) = 0 ∈ E (R; R). 0 (3) E(ρ)(I ; ωI ) is independent of choice of the element (I ; ωI 0 ). HOMOLOGY SEQUENCE AND EXCISION THEOREM FOR EULER CLASS GROUP 243

00 n 00 Proof of (3). If there is another element (I ; ωI 00 ) ∈ E (R; R) such that (I ; ωI 00 ) 00 = (I ; ωI ), and height I + l ≥ n in R, then by Lemma 2.4 we can ﬁnd (K ; ωK ) in En(R; R) such that

K + I = K + I 0 = K + I 00 = R, height K + l ≥ n, n (K ; ωK ) + (I ; ωI ) = 0 ∈ E (R; R).

0 0 00 By Lemma 2.3, (K ; ωK ) + (I , ωI 0 ) = (K ∩ I ; ωK ∩I 0 ) = (K ; ωK ) + (I ; ωI 00 ) = 00 n (K ∩ I ; ωK ∩I 00 ) = 0 ∈ E (R; R). Then, from the properties of the ECG and the result (2) above, it can be easily checked that

0 0 0 E(ρ)(K ∩ I ; ωK ∩I 0 ) = (K ∩ I + l; ωK ∩I 0+l ) = (K + l; ωK +l ) + (I + l; ωI 0+l ) 0 00 00 = E(ρ)(K ; ωK )+ E(ρ)(I ; ωI 0 ) = E(ρ)(K ∩ I ; ωK ∩I 00 ) = (K ∩ I + l; ωK ∩I 00+l ) 00 00 = E(ρ)(K ; ωK ) + E(ρ)(I ; ωI 00 ) = (K + l; ωK +l ) + (I + l; ωI 00+l ),

0 00 which is equal to zero. Therefore, E(ρ)(I ; ωI 0 ) = E(ρ)(I ; ωI 00 ). This shows that 0 E(ρ)(I, ωI ) is independent of the choice of the element (I ; ωI 0 ).

n (4) If (I ; ωI ) = (J; ωJ ) ∈ E (R; R), then

n E(ρ)(I ; ωI ) = E(ρ)(J; ωJ ) ∈ E (R/l; R/l).

(5) For any elements x, y ∈ En(R; R), we have

E(ρ)(x) + E(ρ)(y) = E(ρ)(x + y).

n Proof of (5). Let x = (I ; ωI ), y = (J; ωJ ) be two elements of E (R; R). By the method we used above, we may further assume that I + J = R. Now deﬁne maps

n (i1,...,in) 2 ωI : R / I/I , where ir ∈ I for 1 ≤ r ≤ n,

j j n ( 1,..., n) 2 ωJ : R / / J/J , where jr ∈ J for 1 ≤ r ≤ n,

Since I and J are comaximal, by Lemma 2.3, we can ﬁnd a surjection

n (k1,...,kn) I ∩ J ω ∩ : R / / I J (I ∩ J)2

2 2 such that kr ∈ I ∩ J and kr = ir (mod I ) and kr = jr (mod J ) for 1 ≤ r ≤ n, n that is, x + y = (I ∩ J; ωI ∩J ) ∈ E (R; R). Hence, if the bar denotes reduction 244 YONG YANG modulo l, we get ¯ ¯ ¯ ¯ ¯ ¯ E(ρ)(x) + E(ρ)(y) = (I ; (i1,..., in)) + (J; ( j1,..., jn)) ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ = (I ; (k1,..., kn)) + (J; (k1,..., kn)) = (I ∩ J; (k1,..., kn)) = E(ρ)(I ∩ J; ωI ∩J ) = E(ρ)(x + y). (6) If n > d − m, the map E(ρ) vanishes.

n Proof of (6). This comes from the fact that in this case E (R/l; R/l) = 0. By all of the above, the group homomorphism E(ρ) : En(R; R) → En(R; R) is well-deﬁned. Deﬁnition 3.2 (extension map). Let R and A be Noetherian commutative rings with dimension d and s, respectively. Let n be an integer with 2n ≥ d + 3 and 2n ≥ s + 3. If there is a ring homomorphism φ : R → A such that

n 2 (∗) height φ(I ) ≥ n for any local n-orientation ωI : R I/I . then similarly to the above deﬁnition, we can construct a group homomorphism E(φ) : En(R; R) → En(A; A), called the extension map of the ECG, as follows

n Let x = (I ; ωI ) ∈ E (R; R) be any element, and suppose that ωI is the surjective map

n (i1,...,in) 2 R / / I/I in which it ∈ I for 1 ≤ t ≤ n.

n Then we deﬁne E(φ)(x) by (φ(I ); ωφ(I )) ∈ E (A; A), where ωφ(I ) is the surjection

(φ(i1),...,φ(in)) An / / φ(I )/φ(I )2 . By a method similar to the one used in Definition 3.1, it can be checked that E(φ) is indeed a group homomorphism. By forgetting the orientation in Definitions 3.1 and 3.2, we have the following for the weak ECG. Definition 3.3. Let R and l be as in Definition 3.1. Let φ : R → R/l be the natural ring homomorphism. For an integer n with 2n ≥ d + 3, there is a group : n → n homomorphism E0(φ) E0 (R) E0 (R/l), which is called the restriction map of the weak ECG. Similarly, let R, A and n be as in Definition 3.2, and let φ : R → A be a ring homomorphism satisfying the condition (∗). Then there is a group homomorphism : n → n E0(φ) E0 (R) E0 (A), which is called the extension map of the weak ECG. HOMOLOGY SEQUENCE AND EXCISION THEOREM FOR EULER CLASS GROUP 245

4. The relative ECG and homology sequence

In this section, in analogy to related notions in K-theory, we deﬁne the relative and relative weak ECGs. Using these groups, we construct homology sequences for the ECG, which are the counterparts of homology sequence for K0-groups. Also we will give excision theorems for the ECG, which are the counterparts of excision theorem for K0-groups. Deﬁnition 4.1. Let R be a Noetherian commutative ring with dim R = d, and let l be an ideal of R. Then we have the double D(R, l) of R along l as the subring of the Cartesian product R × R, given by D(R, l) = {(x, y) ∈ R × R : x − y ∈ l}.

Note that if p1 denotes the projection onto the first coordinate, then there is a split exact sequence p1 0 → l / D(R, l) / R → 0 in the sense that p1 is split surjective (with splitting map given by the diagonal embedding of R in D(R, l) and with ker p1 identified with l.) Since D(R, l) is finite over the subring R (given by the diagonal embedding), we get D(R, l) that is Noetherian and is integral over the subring R. Moreover, we have dim D(R, l) =dim R = d, and height(ker p1) = 0 (with ker p1 being regarded as an ideal of D(R, l)). Then for any integer n with 2n ≥ d + 3, the relative ECG of R and l is defined by n n n E (R, l; R) = ker(E(p1) : E (D(R, l); D(R, l)) → E (R; R)). and the relative weak ECG of R and l is defined by n n n E0 (R, l) = ker(E0(p1) : E0 (D(R, l)) → E0 (R)). in which E(p1) and E0(p1) are the restriction map of the ECG of Definition 3.1 and the restriction map of the weak ECG of Definition 3.3, respectively. It can be seen easily that when l = R, the relative ECG En(R, R; R) and the n n ; relative weak ECG E0 (R, R) are the same as the generalized ECG E (R R) and n the generalized weak ECG E0 (R), respectively. Theorem 4.2 (homology sequence). Let R be a Noetherian commutative ring with dim R = d, and let l ⊆ R be an ideal of R with dim R/l = d − m. Let p2 denote the projection from D(R, l) to the second coordinate. Then, for any integer n such that 2n ≥ d + 3, we have the exact sequence

E(p2) E(ρ) En(R, l; R) / En(R; R) / En(R/l; R/l), called the homology sequence of the ECG. 246 YONG YANG

Proof. Step I: First, we check that E(ρ)◦ E(p2) = 0. Let ker p1 and ker p2 denote the kernels of projections p1 and p2. Then height(ker p1) = height(ker p2) = 0. On the other hand, we have ring homomorphisms ρ ◦ p1, ρ ◦ p2 : D(R, l) R/l. By the deﬁnition of D(R, l), it can be seen easily that ρ ◦ p1 = ρ ◦ p2. Hence for any element x ∈ En(R, l; R), by the method of used construction of the restriction map and by Lemma 2.4, we can assume that x = (Z; ωZ ), in which Z is an ideal of D(R, l) with properties

height p1(Z) ≥ n, height p2(Z) ≥ n in R,

height ρ ◦ p1(Z) = height ρ ◦ p2(Z) ≥ n in R/l.

Write ωZ as (z1,..., zn), where zi = (xi , yi ) ∈ D(R, l) for 1 ≤ i ≤ n, and let the bar denote the reduction modulo l. Then it can be seen that

E(ρ) ◦ E(p1)(Z; ωZ ) = (p1(Z); (x¯1,..., x¯n)),

E(ρ) ◦ E(p2)(Z; ωZ ) = (p2(Z); (y¯1,..., y¯n)).

Since Z is an ideal of D(R, l) and (xi , yi ) ∈ D(R, l) for 1 ≤ i ≤ n, we have p1(Z) = p2(Z) and x¯i =y ¯i for 1 ≤ i ≤ n. On the other hand, from the deﬁnition of En(R, l; R), we know that

E(ρ) ◦ E(p1)(Z; ωZ ) = E(ρ)(p1(Z); (x1,..., xn)) = (p1(Z); (x¯1,..., x¯n)) = 0.

Thus we get

E(ρ) ◦ E(p2)(Z; ωZ ) = (p2(Z); (y¯1,..., y¯n)) = (p1(Z); (x¯1,..., x¯n)) = 0.

This establishes that E(ρ) ◦ E(p2) = 0, that is, ker E(ρ) ⊇ Im E(p2).

Step II: Next we check that the kernel of E(ρ) is contained in the image of E(p2), that is, ker E(ρ) ⊆ Im E(p2). Let x ∈ En(R; R) such that E(ρ)(x) = 0 ∈ En(R/l; R/l). By the method we used in the construction of restriction map, we can assume that x = (I ; ωI ), in which I is an ideal of R such that properties height I ≥ n, and height(I +l)/l ≥ n in R. n ¯ By the assumption that E(ρ)(x) = 0 ∈ E (R/l; R/l), we have (I ; ωI¯) = n n ¯ ¯2 0 ∈ E (R/l; R/l), in which ωI¯ : R I /I is the map induced by ωI . By [Bhatwadekar and Sridharan 2000, Theorem 4.2], there exists a surjective map n ¯ ¯ n 2 vI¯ : R I such that vI¯ ⊗ R/I = ωI¯. So by Lemma 2.4, ωI : R I/I can be n 2 lifted to a surjective map ωÎ : R I/(I l). Then by Lemma 2.3, we can find a n 2 surjective lifting v : R I ∩K of ωÎ such that K +I l = R and height K ≥n. Since 2 n 2 K +I l = R, v induces a surjective map ωK : R K/K , which defines an element n n (K ; ωK ) ∈ E (R; R). It can be seen easily that (K ; ωK )+(I ; ωI ) = 0 ∈ E (R; R). HOMOLOGY SEQUENCE AND EXCISION THEOREM FOR EULER CLASS GROUP 247

Now write ωK as (x1,..., xn), where xi ∈ K for 1 ≤ i ≤ n. Then we can deﬁne n an element (Z; ωZ ) ∈ E (R, l; R) as follows: • Deﬁne Z ⊆ D(R, l) to be the ideal of D(R, l) that is generated by pairs (r1, r2) ∈ R × R such that r2 ∈ K and r1 − r2 ∈ l.

n (z1,...,zn) 2 • Deﬁne the map ωZ : D(R, l) / Z/Z , in which zi = (xi , xi ) ∈ Z for 1 ≤ i ≤ n.

By the facts that ker p1∩ker p2 = (0, l) ∩ (l, 0) = 0 ⊆ D(R, l) and p1(Z) = R and height p2(Z) =height K ≥ n, we have height I ≥ n. We should check that ωZ is surjective. Let z = (k + l1, k) ∈ Z, where k ∈ K 2 and l1 ∈ l. Since ωK is surjective, there exist ri ∈ R for 1 ≤ i ≤ n and k1 ∈ K , Pn 2 2 2 such that k = i=1 ri xi + k1. Since K + l = R contains l, there exist k2 ∈ K 2 2 2 and l2 ∈ l such that k2 + l2 = l1. By the fact k2 = l1 − l2 ∈ K ∩ l = K l, we get 2 Pm k3t ∈ K , and l3t ∈ l for 1 ≤ t ≤ m, such that k2 = t=1 l3t k3t . Finally, n X z = (k + l1, k) = (ri , ri )(xi , xi ) + (k1, k1) + (l1, 0) i=1 n m X X = (ri , ri )(xi , xi ) + (k1, k1) + (l2, 0) + (l3t , 0)(k3t , k3t ). i=1 t=1

This shows that ωZ is surjective. Since K + l = R, we have p1(Z) = R by the construction of Z. This implies n that E(p1)(Z; ωZ ) = 0 ∈ E (R; R). Putting all of these together, we see (Z; ωZ ) is indeed an element of En(R, l; R). n It can be seen easily that E(p2)(Z; ωZ ) = (K ; ωK ) ∈ E (R; R). Now let y ∈ n E (R, l; R) be such that y + (Z; ωZ ) = 0. Since

E(p2)(y) + E(p2)(Z; ωZ ) = (I ; ωI ) + (K ; ωK ) = 0, we see that E(p2)(y) = (I ; ωI ). This shows that ker E(ρ) ⊆ImE(p2). By steps I and II, the sequence is indeed an exact sequence. Theorem 4.3 (excision theorem). Let R be a Noetherian commutative ring with dim R = d, and let l ⊆ R be an ideal of R with dim R/l = d − m. Let p2 denote the projection from D(R, l) to the second coordinate. If there exists a splitting β of the ring homomorphism ρ : R → R/l such that β satisﬁes condition (∗), then for any integer n with 2n ≥ d + 3, we have the split exact sequence

E(p2) E(ρ) 0 → En(R, l; R) / En(R; R) / En(R/l; R/l) → 0, called the excision sequence of the ECG. In particular, we have an isomorphism En(R; R) =∼ En(R, l; R) ⊕ En(R/l; R/l). 248 YONG YANG

Proof. Step I: First,we check that E(ρ) is a split surjection. Since the ring homomorphism β : R/l → R satisﬁes the condition (∗), by Deﬁnition 3.2 there is a group homomorphism E(β) : En(R/l; R/l) → En(R; R). By the fact that β is a splitting of ρ, it is easy to check that E(β) has the property E(ρ) ◦ E(β) = I dEn(R/l;R/l). This shows that E(ρ) is split surjective.

Step II: We check that E(p2) is injective. Now we have a surjective ring homomorphism ρ ◦ p2 : D(R, l) → R/l and an exact sequence ρ◦p 0 → l × l → D(R, l) −−−→2 R/l → 0 in which l × l ⊂ D(R, l) is the ideal of D(R, l) generated by elements (l1, l2) ∈ n R× R, where l1, l2 ∈l. Then we have the restriction map E(ρ◦ p2) : E (R, l; R) → n E (R/l; R/l) of the ECG. It can be easily checked that E(ρ)◦ E(p2) = E(ρ ◦ p2). n n Let x = (Z; ωZ ) ∈ E (R, l; R) be such that E(p2)(x) = 0 ∈ E (R; R). By the method we used in the construction of restriction map, we can assume that height p1(Z) ≥ n, height p2(Z) ≥ n and height ρ ◦ p2(Z) ≥ n. On the other hand, since E(ρ ◦ p2) = E(ρ) ◦ E(p2) = 0, by the same method we used in the proof of n Theorem 4.2, we can ﬁnd (K ; ωK ) ∈ E (R, l; R) such that

• (K ; ωK ) + (Z; ωZ ) = 0, • K + Z 2(l × l) = D(R, l),

• height K ≥ n, height p1(K ) ≥ n and height p2(K ) ≥ n. n By the assumption that E(p2)(Z; ωZ ) = 0, we get E(p2)(K ; ωK ) = 0 ∈ E (R; R). n 2 Now, write ωK : D(R, l) K/K as (k1,..., kn) where ki = (xi , yi ) ∈ K for 1 ≤ i ≤ n. We have the following. • ; = ; = ; E(p1)(K ωK ) (p1(K ) ωp1(K )) (I1 ωI1 ), where I1 denotes the ideal

p1(K ), and ωI1 denotes the surjection induced by ωK , that is, x x : n ( 1,..., n) 2 ωp1(K ) R / / I1/I1 .

• ; = ; = ; E(p2)(K ωK ) (p2(K ) ωp2(K )) (I2 ωI2 ), where I2 denotes the ideal

p2(K ), and ωI2 denotes the surjection induced by ωK , that is, (y ,...,y ) : n 1 n 2 ωp2(K ) R / / I2/I2

Since K + (l × l) = D(R, l), we see that I1 + l = I2 + l = R. n By the fact that E(p1)(K ; ωK ) = E(p2)(K ; ωK ) = 0 ∈ E (R; R), there exist surjective lifts (x0 ,...,x0 ) (y0 ,...,y0 ) : n 1 n : n 1 n vI1 R / / I1 and vI2 R / / I2 0 ∈ 0 ∈ ≤ ≤ of ωI1 and ωI2 , respectively, in which xi I1, and yi I2 for 1 i n. HOMOLOGY SEQUENCE AND EXCISION THEOREM FOR EULER CLASS GROUP 249

0 − ∈ 2 ≤ ≤ 0 − = Since vI1 is a lift of ωI1 , we have xi xi I1 for 1 i n. Let xi xi ai , ∈ 2 ≤ ≤ = ∈ 2 with ai I1 for 1 i n. Since I1 p1(K ), there exist bi I2 such that 2 00 (ai , bi ) ∈ K for 1 ≤ i ≤ n. Now let yi = yi + bi for 1 ≤ i ≤ n. It follows from 2 0 00 the facts (xi , yi ) ∈ K and (ai , bi ) ∈ K that (xi , yi ) = (xi + ai , yi + bi ) ∈ K and 0 00 2 (xi , yi ) = (xi + ai , yi + bi ) = (xi , yi )(mod K ) for 1 ≤ i ≤ n. So we obtain a surjection (k0 ,...,k0 ) 1 n 1 n 2 ωK : R / / K/K

0 0 00 1 in which ki = (xi , yi ) ∈ K for 1 ≤ i ≤ n. Clearly, (K ; ωK ) = (K ; ωK ). By the same method, we can get a surjection

(k00,...,k00) 2 n 1 n 2 ωK : R / / K/K

00 00 0 2 in which ki = (xi , yi ) ∈ K for 1 ≤ i ≤ n, such that (K ; ωK ) = (K ; ωK ). ; ; n ; Now we construct two elements (Ᏽ1 ωᏵ1 ) and (Ᏽ2 ωᏵ2 ) of E (R, l R). Let the bar denote reduction modulo l.

(I) Define Ᏽ1 to be the ideal of D(R, l) that is generated by pairs (β(a¯), b), where (a, b) ∈ K. (kˆ00,...,kˆ00) : n 1 n 2 ˆ00 = ¯00 0 ∈ (II) Define the map ωᏵ1 D(R, l) / Ᏽ1/Ᏽ1 , where ki (β(xi ), yi ) Ᏽ1 for 1 ≤ i ≤ n. ∗ ¯ (I ) Define Ᏽ2 to be the ideal of D(R, l) that is generated by pairs (a, β(b)), where (a, b) ∈ K. (kˆ0 ,...,kˆ0 ) ∗ : n 1 n 2 ˆ0 = 0 ¯00 ∈ (II ) Define the map ωᏵ2 D(R, l) / Ᏽ2/Ᏽ2 , where ki (xi , β(yi )) Ᏽ2 for 1 ≤ i ≤ n.

We check that height Ᏽ1 ≥ n and height Ᏽ2 ≥ n. By the fact that ker p1∩ker p2 = (0, l) ∩ (l, 0) = 0 ⊂ D(R, l) and ¯ p1(Ᏽ1) = β(I1) = R and height p2(Ᏽ1) = height I2 ≥ n, we get height Ᏽ1 ≥ n. Similarly, we have height Ᏽ2 ≥ n. ; ; n ; We check that (Ᏽ1 ωᏵ1 ) and (Ᏽ2 ωᏵ2 ) equate to zero in E (R, l R). In fact we have a map

(kˆ00,...,kˆ00) : n 1 n vᏵ1 D(R, l) / Ᏽ1,

= ˆ00 ∈ { } ≤ ≤ which is deﬁned by vᏵ1 (ei ) ki Ᏽ1, where ei for 1 i n is the standard basis n ¯ ∈ ∈ of R . It is obviously a lift of ωᏵ1 . Now let (β(a), b) Ᏽ1 for (a, b) K . Since 0 Pn 0 I2 is generated by {yi }, there exist ri ∈ R for 1 ≤ i ≤ n such that i=1 ri yi = b. 250 YONG YANG

00 0 ¯ On the other hand, it follows from the facts (a, b) ∈ K and (xi , yi ) ∈ K that a¯ = b 00 0 and x¯i =y ¯i for 1 ≤ i ≤ n. So we get n n ¯ X 0 X 00 β(a¯) = β(b) = β(r¯i )β(y¯i ) = β(r¯i )β(x¯i ). i=1 i=1 ¯ = Pn ¯ ¯00 0 Thus (β(a), b) i=1(β(ri ), ri )(β(xi ), yi ). This shows that vᏵ1 is a surjective ; = ∈ n ; lift of ωᏵ1 . So (Ᏽ1 ωᏵ1 ) 0 E (R, l R). By the same method, we can prove ; = ∈ n ; that (Ᏽ2 ωᏵ2 ) 0 E (R, l R). ; + ; = ; Next we check that (Ᏽ1 ωᏵ1 ) (Ᏽ2 ωᏵ2 ) (K ωK ). We ﬁrst check that Ᏽ1 +Ᏽ2 = R. Since I1 +l = I2 +l = R, there exists (i1, i2) ∈ K ¯ such that β(i1) = 1. So (1, i2) ∈ Ᏽ1, and 1 − i2 ∈ l. By the same method, we can 0 ∈ 0 ∈ − 0 = − ∈ ﬁnd i1 I1 such that (i1, 1) Ᏽ2. So (0, 1 i2)(i1, 1) (0, 1 i2) Ᏽ2. It follows that (1, i2) + (0, 1 − i2) = (1, 1) ∈ Ᏽ1 + Ᏽ2. Hence Ᏽ1 + Ᏽ2 = R. ¯ Second, we check that Ᏽ1 ∩Ᏽ2 = K . Let (i1, i2) ∈ K . Then (β(i1), i2) ∈ Ᏽ1. Now − ¯ = ∈ + = 0 0 ∈ ¯0 = let i1 β(i1) l1 l. Since I1 l R, there exists (ii , i2) K such that β(i1) 1. ¯0 0 = ∈ ¯ + = ∈ So we have (l1, 0)(β(i1), i2) (l1, 0) Ᏽ1. Since (β(i1), i2) (l1, 0) (i1, i2) Ᏽ1, we get K ⊆ Ᏽ1. Similarly, we can prove K ⊆ Ᏽ2. Thus K ⊆ Ᏽ1 ∩ Ᏽ2. Let (i1, i2) ∈ Ᏽ1. Then there exist r1t , r2t ∈ R, with r1t −r2t ∈l, and (x1t , x2t ) ∈ K for 1 ≤ t ≤ m, where m ∈ ޚ is an integer such that

m m m X X X (i1, i2) = (r1t , r2t )(β(x¯1t ), x2t ) = r1t β(x¯1t ), r2t x2t . t=1 t=1 t=1

So we get i2 ∈ I2. Thus if (i1, i2) ∈ Ᏽ1 ∩ Ᏽ2, we will also have i1 ∈ I1. Now we Pm Pm Pm Pm have ( t=1 r1t x1t , t=1 r2t x2t ) ∈ K and (i1, i2) − ( t=1 r1t x1t , t=1 r2t x2t ) = Pm (l1, 0) ∈ D(R, l). It follows that i1 − t=1 r1t x1t = l1 ∈ l ∩ I1 = l I1. Hence there 0 ∈ 0 ∈ ∈ = Ps 0 0 0 ∈ exist i1λ I1, i2λ I2 and l2λ l such that l1 λ=1 l2λi1λ and (i1λ, i2λ) K for ≤ ≤ ∈ = Ps 0 0 ∈ = 1 λ s, with s ޚ. Thus (l1, 0) λ=1(l2λ, 0)(i1λ, i2λ) K and (i1, i2) Pm Pm ( t=1 r1t x1t , t=1 r2t x2t ) + (l1, 0) ∈ K . This shows that K ⊇ Ᏽ1 ∩ Ᏽ2. So we get Ᏽ1 ∩ Ᏽ2 = K . Third, we check that

⊗ = ⊗ = ωK D(R, l)/Ᏽ1 ωᏵ1 and ωK D(R, l)/Ᏽ2 ωᏵ2 .

; 1 = ; ⊗ = Since (K ωK ) (K ωK ), to prove ωK D(R, l)/Ᏽ2 ωᏵ2 , we only need to 0 00 − 0 ¯00 ∈ 2 show that (xi , yi ) (xi , β(yi )) Ᏽ2. 0 00 − 0 ¯00 = ∈ ∈ 2+ = In fact let (xi , yi ) (xi , β(yi )) (0, l1) D(R, l), where l1 l. Since I2 l R, ∈ 2 ∈ 2 ∈ 2 ¯ = ∈ 2 there exist a I1 and b I2 , such that (a, b) K and (a, β(b)) (a, 1) Ᏽ2. = ∈ 2 ⊗ = So we have (0, l1)(a, 1) (0, l1) Ᏽ2. This shows that ωK D(R, l)/Ᏽ2 ωᏵ2 . ⊗ = By the same way, we can prove ωK D(R, l)/Ᏽ1 ωᏵ1 . HOMOLOGY SEQUENCE AND EXCISION THEOREM FOR EULER CLASS GROUP 251

; + ; = ; = By all of the results above, (Ᏽ1 ωᏵ1 ) (Ᏽ2 ωᏵ2 ) (K ωK ) 0 holds in the n group E (R, l; R). This shows that E(p2) is injective. Putting these results together with those from Step I and Step II, we have a split exact sequence

E(p2) E(ρ) 0 → En(R, l; R) / En(R; R) / En(R/l; R/l) → 0.

n ∼ n n In particular, E (R; R) = E (R, l; R)⊕ E (R/l; R/l). The proof is complete. Remark 4.4. The proof shows that if the ring homomorphism ρ : R → R/l has a splitting β, which may not satisfy condition (∗), the map E(p2) is still injective.

5. Applications for polynomial and Laurent polynomial extensions

In this section, we will use Theorem 4.3 to get excision sequences for Euler class groups of polynomial rings and Laurent polynomial rings.

Excision sequence of polynomial rings. Let R be a Noetherian commutative ring with dim R = d. Let R[t] be the polynomial ring over R. Then for any integer n with 2n ≥ d + 4, we get the following corollary by setting l = (t) in Theorem 4.3. Corollary 5.1. Let R be a Noetherian commutative ring with dim R = d, and let R[t] be the polynomial ring over R. Then for any integer n with 2n ≥ d + 4, we have the short split exact sequence

E(p2) E(ρ) 0 → En(R[t],(t); R[t]) / En(R[t]; R[t]) / En(R; R) → 0 and an isomorphism

En(R[t]; R[t]) =∼ En(R[t],(t); R[t]) ⊕ En(R; R).

Das [2006] constructed a map 9 : Ed (R[t]; R[t]) → Ed (R; R) that is split surjective. In fact, this map is the map E(ρ). Proposition 5.2 [Das 2003]. Let R be a smooth afﬁne domain containing the ﬁeld of rational numbers, with dim R = d. Then there is an isomorphism

8 : Ed (R[t]; R[t]) =∼ Ed (R; R).

Using [Bhatwadekar and Keshari 2003, theorem 4.13], and Corollary 5.1, this result can generalized: Corollary 5.3. Let k be an infinite perfect field, and let R be a d-dimensional regular domain that is essentially of finite type over k. Let n be an integer such that 2n ≥ d + 4. Then En(ρ) : En(R[t]; R[t]) → En(R; R) is an isomorphism. 252 YONG YANG

Proof. By Corollary 5.1, we only need to prove that E(p2)(x) = 0 for any element n n x ∈ E (R[t],(t); R[t]). Suppose (Z; ωZ ) ∈ E (R[t],(t); R[t]), where Z is an n 2 ideal of D(R[t],(t)) and ωZ : D(R[t],(t)) Z/Z is a surjection. Let p1, p2 denote projections from D(R[t],(t)) to respective coordinates. We may assume further that height p1(Z) =height p2(Z) = n in R[t]. Let ; = ; ; = ; E(p1)(Z ωZ ) (p1(Z) ωp1(Z)) and E(p2)(Z ωZ ) (p2(Z) ωp2(Z)), [ ]n 2 where ωp1(Z) and ωp2(Z) are respectively surjections R t p1(Z)/p1(Z) and n 2 n R[t] p2(Z)/p2(Z) induced by ωZ . Since (Z; ωZ ) ∈ E (R[t],(t); R[t]), we ; = ∈ n [ ]; [ ] ⊆ [ ] have (p1(Z) ωp1(Z)) 0 E (R t R t ). For any ideal I R t , let I (0) denote the reduction modulo (t), which is the same as setting t = 0 in I . Now we get ; = ∈ n ; n (p1(Z)(0) ωp1(Z)(0)) 0 E (R R), where ωp1(Z)(0) is the surjection R 2 = p1(Z)(0)/(p1(Z)(0)) that the surjection ωp1(Z) induces by setting t 0. By

[Bhatwadekar and Sridharan 2002, Theorem 4.2], this means that ωp1(Z)(0) can be : n ; lifted to a surjection vp1(Z)(0) R p1(Z)(0). On the other hand, since (Z ωZ ) n [ ] ; [ ] = = is in E (R t ,(t) R t ), we have p1(Z)(0) p2(Z)(0) and ωp1(Z)(0) ωp2(Z)(0). So we get : [ ]n 2 : n ωp2(Z) R t p2(Z)/p2(Z) and vp1(Z)(0) R p2(Z)(0), ⊗ [ ] = ⊗ such that ωp2(Z) R t /(t) vp1(Z)(0) R/p2(Z)(0). We can ﬁnd a surjective lift of ωp2(Z) by [Bhatwadekar and Keshari 2003, Theorem 4.13]. This shows that ; = ; = ∈ n [ ]; [ ] (p2(Z) ωp2(Z)) E(p2)(Z ωZ ) 0 E (R t R t ). Excision sequence of Laurent polynomial rings. Corollary 5.4. Let R be a Noetherian commutative ring with dim R = d, and let n be an integer such that 2n ≥ d + 4. Let R[t, t−1] be the Laurent polynomial ring over R. Then by setting l = (t − 1) in Theorem 4.3, we get the short split exact sequence

E(p2) 0 → En(R[t, t−1],(t − 1); R[t, t−1]) / En(R[t, t−1]; R[t, t−1])

E(ρ) / En(R; R) → 0 and an isomorphism En(R[t, t−1]; R[t, t−1]) =∼ En(R[t, t−1],(t − 1); R[t, t−1]) ⊕ En(R; R). Postscript. In K-theory, we have the following results: When R is a regular Noe- therian commutative ring, the homology sequences for the K0-groups of the polynomial ring R[t] and the Laurent polynomial ring R[t, t−1] reduce to isomorphisms ∼ −1 ∼ K0(R[t]) = K0(R) and K0(R[t, t ]) = K0(R), respectively [Rosenberg 1994]. In HOMOLOGY SEQUENCE AND EXCISION THEOREM FOR EULER CLASS GROUP 253 light of this result and the correspondence between excision sequence for K-theory and the excision sequence for the ECGs, we ask if there exist isomorphisms

En(R[t]; R[t]) =∼ En(R; R) and En(R[t, t−1]; R[t, t−1]) =∼ En(R; R).

For the case of polynomial extension, Corollary 5.3 gives an affirmative answer. For the case of Laurent polynomial extension, we wonder if there is also an isomorphism En(R[t, t−1]; R[t, t−1]) =∼ En(R; R) if the ring R satisfies the conditions of Corollary 5.3. In fact, for the weak ECG, we can prove the following result using Suslin’s cancellation theorem [1977] and [Murthy 1994, Theorem 2.2]. Let R be a smooth affine algebra over some algebraically closed field k, with dim R = d. Let R[t, t−1] be the Laurent polynomial ring over R. Then we have d [ −1] ⊗ ∼ d ⊗ E0 (R t, t ) ޑ = E0 (R) ޑ, which corresponds to what the Riemann–Roch theorem tells us in geometry.

Acknowledgements. I sincerely thank Professors Lianggui Feng and Satyagopal Mandal for many helpful discussions and encouragement. I also thank the referee for carefully going through the manuscript and suggesting improvements.

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Received April 17, 2010. Revised May 27, 2010.

YONG YANG DEPARTMENT OF MATHEMATICS AND SYSTEMS SCIENCE NATIONAL UNIVERSITYOF DEFENSE TECHNOLOGY CHANGSHA,HUNAN 410073 CHINA [email protected] PACIFIC JOURNAL OF MATHEMATICS Vol. 249, No. 1, 2011

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Metabelian SL(n, ރ) representations of knot groups, II: Fixed points 1 HANS U.BODEN and STEFAN FRIEDL Lewis–Zagier correspondence for higher-order forms 11 ANTON DEITMAR Topology of positively curved 8-dimensional manifolds with symmetry 23 ANAND DESSAI Strong Kähler with torsion structures from almost contact manifolds 49 MARISA FERNÁNDEZ,ANNA FINO,LUIS UGARTE and RAQUEL VILLACAMPA Connections between Floer-type invariants and Morse-type invariants of Legendrian 77 knots MICHAEL B.HENRY 2011 A functional calculus for unbounded generalized scalar operators on Banach spaces 135 DRAGOLJUB KECKIˇ C´ and ÐORÐE KRTINIC´ Geometric formality of homogeneous spaces and of biquotients 157 o.29 o 1 No. 249, Vol. D.KOTSCHICK and S. TERZIC´ Positive solutions for a nonlinear third order multipoint boundary value problem 177 YANG LIU,ZHANG WEIGUO,LIU XIPING,SHEN CHUNFANG and CHEN HUA

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