Topology and its Applications 98 (1999) 3–17
Quasifibrations and Bott periodicity
M.A. Aguilar 1, Carlos Prieto 2
Instituto de Matemáticas, UNAM, 04510 México, D.F., Mexico Received 10 July 1997; received in revised form 16 July 1998
Abstract We give a proof of the Bott periodicity theorem, along the lines proposed by McDuff, based on the construction of a quasifibration over U with contractible total space and Z × BU as fiber. 1999 Elsevier Science B.V. All rights reserved. Keywords: Bott periodicity; Homotopy groups; Spaces and groups of matrices
AMS classification: Primary 55R45, Secondary 55R65
1. Introduction
The periodicity theorem of Raoul Bott is one of the most important results in algebraic topology. This theorem is used to define K-theory, which is a generalized cohomology theory that has enormous impact in topology, geometry and analysis. Bott’s original proof [5] used Morse theory (see also [11]). The proofs of Toda [21], Cartan and Moore [8] and Dyer and Lashof [10] were based on homological calculations with spectral sequences. Atiyah and Bott [3] (see also [14,22]) obtained the result from a study of bundles over the product space X × S2, in terms of bundles over X. In [2] Atiyah gave a proof using the index of a family of linear elliptic differential operators (cf. also [4]). More recently, other proofs have appeared. Kono and Tokunaga [15] use cohomology and Chern classes; Latour [16] works with the space of Lagrangians; Giffen [12] and Harris [13] use classifying spaces of categories defined via simplicial spaces; and Bryan and Sanders [7] and Tian [20] use moduli spaces of instantons. In this paper, we give a proof which uses only quasifibrations and linear algebra (and some basic differential topology). It is based on a very beautiful idea of McDuff [17], whose program we develop. This proof is both simpler and more elementary than previous proofs.
1 Email: [email protected]. 2 Email: [email protected]. Author partially supported by CONACYT grant 400333-5-25406E.
0166-8641/99/$ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII:S0166-8641(99)00041-3 4 M.A. Aguilar, C. Prieto / Topology and its Applications 98 (1999) 3–17
We shall construct a quasifibration p :E → U over the infinite-dimensional unitary group, such that its total space E is contractible and has Z × BU as fiber. This way, we obtain a long exact sequence
···→πi(Z × BU) → πi(E) → πi (U) (1.1) → πi−1(Z × BU) → πi−1(E) →···, where πi(E) = 0 = πi−1(E),sothat,fori>1, ∼ ∼ πi(U) = πi−1(Z × BU) = πi−1(BU), (1.2) and, for i = 1, ∼ π1(U) = Z. (1.3) ∞ For the time being, one has (locally trivial) fibrations Ek(C ) → BUk with fiber Uk, where the base spaces are the classifying spaces of the unitary groups given by the colimits of Grassmann manifolds, and the total spaces are the corresponding colimits of Stiefel manifolds, such that, by passing again to the colimit, they determine a (locally trivial) fibration EU → BU with EU a contractible space and U as fiber (see [19]). On the other hand, consider the path space PBU = ω : I → BU | ω(0) = x0 of BU, where x0 ∈ BU is the base point. One knows that PBU is contractible and the map q :PBU→ BU, such that q(ω)= ω(1), is a Hurewicz fibration with fiber ΩBU. Clearly, if p : E → B is a quasifibration with fiber F ,andp0 : E0 → B is a Hurewicz fibration with fiber F 0, such that their total spaces are contractible, then there is a weak 0 ∼ homotopy equivalence F → F . Moreover, their homotopy groups satisfy πi−1(F ) = ∼ 0 πi(B) = πi−1(F ), i > 1. Therefore, by [18] and the Whitehead theorem, one obtains homotopy equivalences ΩBU ' UandZ × BU ' ΩU. So we have isomorphisms ∼ ∼ πi−2(U) = πi−2(ΩBU) = πi−1(BU), i > 2. Whence, we obtain the desired theorem, as stated below.
Theorem 1.4 (Bott periodicity). There is a homotopy equivalence Z × BU ' ΩU; hence, for i > 2 there is an isomorphism ∼ πi(U) = πi−2(U), or, equivalently, ∼ πi(BU) = πi−2(BU).
Or, in other terms, again by (1.2) one has that ∼ ∼ ∼ 2 πi(Z × BU) = πi+1(U) = πi+1(ΩBU) = πi(Ω BU); i.e., we obtain an isomorphism ∼ 2 πi(Z × BU) = πi(Ω BU), M.A. Aguilar, C. Prieto / Topology and its Applications 98 (1999) 3–17 5 which is another usual version of Bott periodicity. In particular, from (1.3) and Theo- rem 1.4, we obtain that Z if i is even, πi(BU) = 0ifi is odd.
2. Preliminaries
Let −∞ 6 p 6 q 6 ∞ (not three of them equal) and define q Cp = z : Z → C | zi = 0 for almost all i and if i 6 p or i>q , with the usual topology in the finite-dimensional case and the topology of the union Cq = Cq C∞ = C∞ C1 = C C0 ={ } in the infinite-dimensional case. Therefore, 0 , 0 , 0 , 0 0 , q ∞ etc. All spaces Cp are then subspaces of C−∞. With these definitions, we have that if −∞ 6 ∞ Cq = − 6 6 Cq ⊕ Cr = Cr
Definition 2.1. Given k ∈ Z we define the shift operator by k coordinates ∞ ∞ tk : C−∞ → C−∞ by tk(z)i = zi−k . These shift operators are continuous isomorphisms such that t0 = Iand tk ◦ tl = tl ◦ tk = tk+l .
n+1 → ⊂ C∞ Definition 2.2. Given n,thereisamapjn :BUn BUn+1 sending W to ∞ C ⊕ t1(W) ⊂ C . So, we define BU as
BU = colimBUk. k
In order to compare this definition with a different way of stabilizing, we have to prove a lemma. But before that, we give a definition.
⊂ Cl Cm ⊂ Cm Definition 2.3. Take W k and let m be such that k W.ThenW/ k will denote Cm { } the orthogonal complement of k in W,i.e.,if ek+1,...,em is the canonical basis for Cm { } Cm k , we complete it to an orthonormal basis ek+1,...,em,w1,...,wq of W;thenW/ k { } Cm ⊕ Cm = is spanned by w1,...,wq andwehavethat k (W/ k ) W. 6 M.A. Aguilar, C. Prieto / Topology and its Applications 98 (1999) 3–17
Lemma 2.4. There is a homeomorphism → 0 = ⊂ C∞ | ∞ Ck ⊂ ⇔ = Φ :BU BU W 0 dim W< and 0 W k 0 .
∈ Ck ⊂ = Proof. Take W BUn and let k be the largest integer such that 0 W.DefineΦn(W) Ck ∈ 0 → 0 t−k(W/ 0) BU . Obviously, the map Φn :BUn BU determines in the colimit the desired map Φ. 0 Φ is surjective, since, if W ∈ BU and dim W = n,thenW ∈ BUn and Φn(W) = W, because in this case k = 0. (In fact, the map Ψ : BU 0 → BU such that W 7→ W is the inverse.) It is also injective, since, if V ∈ BUm and W ∈ BUn are such that Φm(V ) = Φn(W), Cp ⊂ Cq ⊂ then, if p and q are the largest integers such that 0 V and 0 W, one has Cp = Cq t−p(V / 0 ) t−q (W/ 0 ). (2.5) Therefore, the dimensions m − p and n − q coincide. Without losing generality, we may 6 − = − > Cq assume that p q, so that, in particular, q p n m 0. Applying tq and adding 0 on the left to both sides of (2.5), yields, on the left, Cq ⊕ Cp = Cq−p ⊕ Cq ⊕ Cp 0 tq−p(V / 0 ) 0 q−p tq−p(V / 0 ) = Cq−p ⊕ Cp ⊕ Cp = Cq−p ⊕ 0 tq−p( 0 V/ 0 ) 0 tq−p(V ), which is the image of V in BUm+q−p = BUn; and, on the right, Cq ⊕ Cq = ; 0 t0(W/ 0 ) W n = n = n ◦···◦ m+1 hence jm(V ) W,wherejm jn−1 jm , and, thus, V and W represent the same element in BU. 2
f p q Definition 2.6. Define BU ={W | C−∞ ⊂ W ⊂ C−∞, −∞ 0 0 f 0 Clearly, the map W 7→ C−∞ ⊕ W determines a homeomorphism BU → BU ; f k f 0 similarly, W 7→ t−k(W) determines a homeomorphism BU → BU , so that one has a canonical homeomorphism Z × BUf 0 → BUf f k f given by the composite (k, W) 7→ tk(W) ∈ BU ,→ BU. By Lemma 2.4, we have proved the following. Theorem 2.7. There is a homeomorphism Z × BU → BUf . ∞ 0 0 We define an operator U in C−∞ as unitary if hUz,Uz i=hz,z i or, equivalently, if ∗ ∞ ∗ UU = I, the identical operator in C−∞,whereU denotes the transposed conjugate operator of U.LetU={U | U is unitary and of finite type}, where we understand by a M.A. Aguilar, C. Prieto / Topology and its Applications 98 (1999) 3–17 7 unitary operator of finite type one for which there exist r ∞ ∞ Remark 2.8. Let S : C−∞ → C be such that 0 e2i if i>0, Sei = e2|i|+1 if i 6 0; n+1 → Cn then the usual inclusions kn :Un Un+1 of the usual unitary groups acting on 0 and Cn+1 7→ ⊕ n+1 0 , respectively, such that U U In , and the induced inclusion in the colimit, → n+1 kn :Un U, up to a shuffling of the intermediate coordinates, induce in and in;that n+1 = n+1 −1 = −1 is, essentially, in (U) Skn (U)S , respectively in(U) Skn(U)S . Therefore, algebraically as well as topologically, the given definition of U coincides with the classical one. Before passing to the proof of the main result of this paper, let us state a criterion to determine when a given map is a quasifibration, which is an easy consequence of theorem [9, 2.15] (see also [1, A.1.19]). S = ⊂ Lemma 2.9. Let B n Bn,whereBn−1 Bn is a closed subspace, with the topology of the union and take a (surjective) map p : E → B. Assume that there are trivializations −1 p (Bn − Bn−1) ≈ (Bn − Bn−1) × F . Furthermore, assume that for every n there is a neighborhood Vn of Bn−1 in Bn and a strong deformation retraction rn : Vn → Bn−1 in Bn −1 −1 such that it has a lifting r˜n : p (Vn) → p (Bn−1), inducing a homotopy equivalence on the fibers. Then p is a quasifibration. 8 M.A. Aguilar, C. Prieto / Topology and its Applications 98 (1999) 3–17 3. Proof of the main theorem Before constructing the desired quasifibration mentioned in the introduction, we shall study, more as a motivation, the finite-dimensional case; afterwards we shall show the stabilization. Recall that an n × n matrix C with complex entries is Hermitian if C = C∗,where C∗ denotes, as before, the transposed conjugate matrix of C.Ifh−, −i denotes the Cn ∈ Cn canonical Hermitian product in 0, then for any v,w 0, C satisfies the identity hCv,wi=hv,Cwi. This implies, in particular, that the eigenvalues of the matrix C are real. The set Hn(C) of all Hermitian n × n matrices has the structure of a real vector space. Let En be the topological subspace of Hn(C) consisting of matrices whose eigenvalues lie in the interval I. The space En is contractible through the homotopy h :En × I → En such that h(C, τ) = (1 − τ)C, which starts with the identity and ends with the constant map with value the matrix 0. Let Mn×n(C) be the complex vector space of complex n× n matrices and let GLn(C) be the subgroup of the invertible ones (general linear group). One has a (differentiable) map exp : Mn×n(C) → GLn(C) given by X∞ i 2 B B B exp(B) ≡ e = = In + B + +···, i! 2! i=0 which fulfills the exponential laws, whenever the matrices taken as exponents commute among themselves. One can easily check the following properties: −1 − eTBT = T eB T 1 for any invertible operator T ,and λ e 1 λ1 D . . e = .. if D = .. . λn e λn a Let M × (C) ⊂ Mn×n(C) be the real subspace of skew-Hermitian matrices, that is, of n n ∗ matrices A such that A∗ =−A.IfA is skew-Hermitian, then (eA)∗ = eA = e−A,sothat A ∗ A −A A 0 (e ) e = e e = e = In. Therefore, the map exp defined above restricts to a C → exp : Mn×n( ) Un. M.A. Aguilar, C. Prieto / Topology and its Applications 98 (1999) 3–17 9 C → a C 7→ One has an isomorphism Hn( ) Mn×n( ),givenbyC 2πiC. We define a map pn :En → Un,bypn(C) = exp(2πiC); then the diagram a C exp Mn×n( ) Un =∼ C Hn( ) pn En commutes. Proposition 3.1. The map pn is surjective. Proof. Take U ∈ Un; we diagonalize this matrix taking another matrix T ∈ Un and the product T −1UT. Since the eigenvalues of a unitary matrix have norm 1, we have that e2πiλ1 0 2πiλ2 −1 e T UT = . , .. 0e2πiλn where λi ∈ I, i = 1, 2,...,n.Take λ1 0 λ2 D = . .. 0 λn −1 −1 ∗ and consider the matrix TDT .SinceT ∈ Un,thenT = T and, therefore, (T DT −1)∗ = (T DT ∗)∗ = TD∗T ∗ = TDT−1, −1 −1 that is, TDT is Hermitian; thus, TDT ∈ En. Whence, we have that − −1 −1 − p (T DT 1) = e2πi(T DT ) = eT(2πiD)T = T e2πiDT 1 n e2πiλ1 0 2πiλ2 e −1 = T . T = U. 2 (3.2) .. 0e2πiλn Let us now analyze the fibers of pn. For this, given a matrix C ∈ En, consider the subspaces ker(C − I) and ker(pn(C) − I). If v ∈ ker(C − I),thenCv = v and one has that 2 2πiC (2πi) 2 pn(C)v = (e )v = I + 2πiC + C +··· v 2! (2πi)2 = Iv + 2πiCv + C2v +··· 2! 10 M.A. Aguilar, C. Prieto / Topology and its Applications 98 (1999) 3–17 (2πi)2 = v + 2πiv + v +··· 2! (2πi)2 = 1 + 2πi + +··· v = e2πiv = v. 2! Therefore, ker(C − I) ⊂ ker(pn(C) − I). Hence, if U ∈ Un,wemaydefineamap −1 → − g : pn (U) G(ker(U I)),theGrassmann space of all finite-dimensional vector − ∈ −1 − = E subspaces of ker(U I), which maps C pn (U) to the subspace ker(C I) 1(v). −1 → − Lemma 3.3. The map g : pn (U) G(ker(U I)) is bijective. Proof. g is surjective; indeed, let V ⊂ ker(U − I) be a subspace. We wish to construct a ∈ −1 ⊂ − = matrix CV pn (U) En such that ker(CV I) V . For this, we construct a matrix T diagonalizing U adequately. Let {v1,...,vr ,vr+1,...,vr+s } be an orthonormal basis of E1(U) such that {v1,...,vr } n ⊥ is a basis of V .SinceU is unitary, the orthogonal complement of E1(U) in C , E1(U) , ⊥ is an invariant subspace under U; indeed, if w ∈ E1(U) and v ∈ E1(U),thenhUw,vi= ∗ −1 ⊥ ⊥ hw,U vi=hw,U vi=hw,vi=0. That is, U(E1(U) ) ⊂ E1(U) and, therefore, we ⊥ may find an orthonormal basis {vr+s+1,...,vn} of E1(U) built up by eigenvectors of U, with eigenvalues different from 1. i i Let T ∈ Un be such that T e = vi , i = 1,...,n,wheree denotes the vectors of the canonical basis of Cn.ThenT −1UT = Db is a diagonal matrix, 10 . .. 1 Db = , 2πiλr+s+1 e . .. 0e2πiλn with r + s ones in the diagonal and all other eigenvalues different from 1. Take now 10 . .. 1 0 . D = .. , 0 λr+s+1 . .. 0 λn with r ones in the diagonal and zeros in the places r + 1 through r + s. −1 2πiD b We shall see that CV = TDT is the desired matrix. Clearly e = D,sothat −1 2πiCV T(2πiD)T 2πiD −1 b −1 pn(CV ) = e = e = T e T = T DT = U. M.A. Aguilar, C. Prieto / Topology and its Applications 98 (1999) 3–17 11 On the other hand, it is immediate to verify that E1(CV ) = T(E1(D)).ButE1(D) ={z ∈ n C | zj = 0forr 2πiλ which is a diagonal matrix with an entry e for each entry λ of D.LetnowC1,C2 ∈ En 2πiC 2πiC be such that e 1 = e 2 and that E1(C1) = E1(C2),thenC1 = C2. To see this, let λ1,...,λn be the eigenvalues of C1 and µ1,...,µn those of C2.SinceE1(C1) = E1(C2), we may assume that λk = µk = 1for16 k 6 r = dim E1(C1) and, then, λk 6= 1 6= µk if r We may summarize all in the following theorem. Theorem 3.4. Let En be the space of Hermitian n × n matrices whose eigenvalues lie in 2πiC the unit interval and let pn :En → Un be such that pn(C) = e .ThenEn is contractible, pn is surjective and the fiber over each matrix U ∈ Un is homeomorphic to the Grassmann space of G(E1(U)). Let us now study two ways of stabilizing this result. The usual one is to take the n → n → canonical embeddings ρn+1 :En En+1 and τn+1 :Un Un+1,givenby C 0 ρn(C) = ∈ En+1, 0 0 and by U 0 τn(U) = ∈ Un+1, 0 I or as it was described above. It is immediate to check that the diagram n ρn+1 En En+1 pn pn+1 Un n Un+1 τn+1 12 M.A. Aguilar, C. Prieto / Topology and its Applications 98 (1999) 3–17 0 0 commutes. This way, one obtains a map p : colimn En → colimn Un such that p ◦ ρn = 0 p |En = pn. 0 Let us now analyze the fibers of p . It is clear that if U ∈ Un,thenE1(τ(U)) = E1(U) ⊕C,andthatE1(ρ(C)) = E1(C) ⊕0; therefore, we have the following commutative diagram: ρn −1 n+1 −1 pn (U) pn+1(τ(U)) =∼ =∼ G(E (U)) G(E (U) ⊕ C) 1 δ 1 = ⊕ = Cn → Cn+1 where δ(V ) V 0. For instance, if we take` U I, we have`δ :G( 0) G( 0 ).This 0 C∞ = way, the fibers of p are homeomorphic to r>0 Gr ( 0 ) r>0 BUr . Now we shall construct a new map p :E→ U whose fiber will be Z × BU, which is, 0 ∞ in a certain way, a completion of p . We define an operator C in C−∞ as Hermitian if hCz,z0i=hz,Cz0i or, equivalently, if C = C∗.LetE={C | C is Hermitian, of finite type and with eigenvalues in I}, where we understand for a Hermitian operator of finite type one for which there exist r Lemma 3.5. For each U ∈ U, there is a homeomorphism f ϕU :G∞(ker(U − I)) ≈ BU. Analogously to Lemma 3.3, we have the following result. Proposition 3.6. Let U ∈ U.Thenp−1(U) ≈ BUf = Z × BU. Proof. It is enough to show that there is a homeomorphism −1 gU : p (U) → G∞ ker(U − I) . ∈ = − = e− s ⊂ Cs ∈ −1 Observe first that if C E, then VC ker(C I) ker(C Ir ) r .TakeC p (U), r then define gU (C) = C−∞ ⊕ VC ∈ G∞(ker(U − I)). M.A. Aguilar, C. Prieto / Topology and its Applications 98 (1999) 3–17 13 We shall now show that gU is surjective. Take W ∈ G∞(ker(U − I)); hence, W = r0 e e C−∞ ⊕ W , with dim W<∞. 0 r Without loss of generality, we may assume r = r.SinceW ⊂ ker(U − I) = C−∞ ⊕ e − s ⊕ C∞ e ⊂ e − s = ker(U Ir ) s , then, taking s sufficiently large, one has that W ker(U Ir ) E e ⊂ Cs 1(U) r . e As in Lemma 3.3, let {v1,...,vm} be an orthonormal basis of W , {vm+1,...,vm+n} an e e orthonormal basis of the orthogonal complement of W in E1(U) and {vm+n+1,...,vs−r } E e Cs an orthonormal basis of the orthogonal complement of 1(U) in r (which is invariant under Ue), this last built up by eigenvectors with eigenvalues different from 1. If we define T ∈ U such that ei if i 6 r, i = 6 Te vi−r if rs we have that T −1UT = Db is diagonal of the form e2πiλm+n+1 r+m+n ⊕ .. ⊕ ∞ I−∞ . Is . e2πiλs−r If we take now λm+n+1 = r ⊕ r+m ⊕ r+m+n ⊕ . ⊕ ∞ D 0−∞ Ir 0r+m .. 0s−r , λs−r −1 2πiC 2πiD −1 b −1 we define CW = TDT , which is such that p(CW ) = e W = T e T = T DT = r U. Moreover, gU (CW ) = C−∞ ⊕ker(CW −I). The same argument as in Lemma 3.3 shows e that ker(CW − I) = W, whence gU (CW ) = W. The map gU is injective, since if C1 and C2 are such that exp(2πiC1) = exp(2πiC2) = U and E1(C1) = ker(C1 − I) = ker(C2 − I) = E1(C2); then we can prove in the same way as in the corresponding part of Lemma 3.3 that C1 = C2. 2 In order to show that p :E→ U is a quasifibration applying Lemma 2.9, we need two | facts. First, we shall see that p Un−Un−1 is trivial, that is, we will define a homeomorphism −1 f h : p (Un − Un−1) → (Un − Un−1) × BU, ◦ = such that proj1 h p. For this, assume that n is even; the odd case is analogous. Let C beamatrixin −1 −n/2 e p (Un − Un−1); therefore, U = p(C) is of the form I−∞ ⊕ U,where−n/2 is maximal, e −n/2+1 ⊕ e0 i.e., U is not of the form I−n/2 U . −1 Take the homeomorphism gU : p (U) → G∞(ker(U − I)), given in the proof of − = C−n/2 ⊕ e − ∞ Proposition 3.6. Since ker(U I) −∞ ker(U I−n/2), it depends continuously on U, as does the homeomorphism gU . Analogously, one can choose the homeomorphism f ϕU :G∞(ker(U − I)) → BU of Lemma 3.5, depending continuously on U. 14 M.A. Aguilar, C. Prieto / Topology and its Applications 98 (1999) 3–17 Consequently, the map h defined by h(C) = (p(C), ϕ(C)),whereϕ(C) = ϕU (gU (C)), is a homeomorphism, since, fiberwise, it is one. Second, we have to show that there is a neighborhood Vn of Un−1 in Un and a strong deformation retraction of Vn onto Un−1, which lifts to a strong deformation retraction of −1 −1 −1 p (Vn) onto p (Un−1) in p (Un). To see this, since Un−1 is a submanifold of Un,we shall construct a tubular neighborhood Vn of the first in the second as follows. Recall Hn(C), the space of Hermitian n × n matrices, and define f :GLn(C) → Hn(C) by f(A)= A∗A. One can easily verify that f is smooth and has I as a regular value; −1 therefore, Un = f (I) is a smooth manifold and if W ∈ Un, then the tangent space of Un at W,TW (Un), is the kernel of the differential of f at W,thatis, ∗ ∗ TW (Un) = A ∈ Mn×n(C) | A W =−W A . ∗ Now recall that there is a Hermitian product in Mn×n(C),givenbyhA,Bi=trace(AB ); thus, taking the real part of this product, we get an inner product Mn×n(C) × Mn×n(C) → R. The restriction of this inner product to each tangent space TW (Un) ⊂ Mn×n(C) defines a Riemannian metric on Un.Leti :Un−1 ,→ Un be the inclusion, such that i(U) = U ⊕ I; then the differential di :TU (Un−1) → Ti(U)(Un) is an inclusion mapping a matrix R to R ⊕ 0, that is R 0 di(R) = . 00 One can easily check that the orthogonal complement of TU (Un−1) in Ti(U)(Un) is given by ( ) b1 ⊥ 0 b . n−1 T (U − ) = ∈ M × (C) b = . ∈ C and t ∈ R , U n 1 −b∗U it n n . bn−1 S ⊥ which is a real (2n−1)-dimensional vector space. We denote byN = T (U − ) , U∈Un−1 U n 1 the normal bundle of Un−1 in Un. Any vector space basis of TI(Un) provides a parallelization of Un which defines a connection on it. This connection does not depend on the chosen basis and determines a spray on Un. By [6], there exists ε>0, such that Nε ={v ∈ N |kvk <ε} is an open neighborhood of the 0-section, and the exponential map associated to the spray, Exp: Nε → Un, is an embedding onto a neighborhood of Un−1 in Un. Now, since the geodesics of this spray are the integral curves of the left-invariant vector fields, then −1 ⊥ Exp(A) = LU exp((dLU ) (A)),whereA ∈ TU (Un−1) , LU :Un → Un is given by LU (W) = UW, and exp is the usual exponential map defined above. Evaluating the ∗ differential of LU , we obtain Exp(A) = U exp(U A). Therefore, we have the following description of a tubular neighborhood Vn = Exp(Nε) of Un−1 in Un as ( ) ∗ 0 b n−1 U exp U U ∈ U − ,(b,t)∈ C × R and k(b, t)k <ε . −b∗U it n 1 M.A. Aguilar, C. Prieto / Topology and its Applications 98 (1999) 3–17 15 ∗ 0 b In order to compute U exp(U ∗ ), first note that −b U it 0 b 0 U ∗b U ∗ = . −b∗U it −b∗U it Set 0 b A(b, t) = . −b∗ it Assume b 6= 0. To diagonalize this matrix, one takes an orthonormal basis of eigenvectors and uses it to form a matrix. The n × n matrix A(b, t) has n − 2 eigenvalues equal to 0 and two eigenvalues λ , λ , such that 1 2p t + (−1)ν 4|b|2 + t2 λν = i, 2 so that the matrix v ··· v − µ bµb W(b,t)= 1 n 2 1 2 , 0 ··· 0 µ1λ1 µ2λ2 n−1 ⊥ where {v1,...,vn−2}⊂C is an orthonormal basis of the space b ={v | v ⊥ b}⊂ n−1 2 2 −1/2 C and µν = (|b| +|λν | ) , is a unitary n × n matrix which satisfies 00 . .. ∗ D(b,t) = W(b,t) A(b,t)W(b,t) = 0 . λ1 0 λ2 Since we can write D(b,t) = W(b,t)∗UU∗A(b, t)UU ∗W(b,t),andA(U ∗b,t) = U ∗A(b, t)U,thenA(U ∗b,t) = U ∗W(b,t)D(b,t)(U∗W(b,t))∗. Therefore, the points in the tubular neighborhood are of the form U exp(A(U ∗b,t)) = UU∗W(b,t)exp(D(b, t)) W(b,t)∗U = exp(A(b, t))U. Hence, every element in Vn coming from the fiber over U in Nε is right translation by U of an element coming from the fiber over I. It is thus enough to study the situation over the identity matrix. Since we may linearly deform the neighborhood Nε to the zero section, simply by 7→ − 6 6 τ → v (1 τ)v,0 τ 1, we obtain a strong deformation retraction rn : Vn Vn,such that τ = − − rn exp(A(b, t))U exp A((1 τ)b,(1 τ)t) U. = 1 = = = Observe that for τ 1, rn (exp(A(b, t))U) exp(0)U IU U,sothatitisa retraction of Vn onto Un−1. ˜τ −1 → −1 In what follows, we define the lifting rn : p (Vn) p (Vn). Since fiberwise, −1 0 0 p (Vn) consists of spaces homeomorphic to the Grassmannians G∞(E1(U )), U ∈ Vn, ˜τ = we will show how rn acts on these spaces. It is clearly enough to study the case τ 0. 0 0 Take U = exp(A(b, t))U ∈ Vn and let GU,b,t = G∞(E1(U ));wealsohavetoshow ˜1 ˜1| → = E that the restriction of the lifting rn , rn : GU,b,t GU,0,0 G∞( (U)) is a homotopy equivalence. 16 M.A. Aguilar, C. Prieto / Topology and its Applications 98 (1999) 3–17 ∗ ∗ Since E1(exp(A(b, t))U) = UE1(exp(A(U b,t)) and for b 6= 0, t 6= 0, E1(exp(A(U b, (n−4)/2 ∞ = C ⊕ C λ1 6= 6= λ2 t))) −∞ n/2, because e 1 e , we have that the Grassmannians GU,b,t and GI,b,t differ only by left multiplication by U. It is thus enough to study the case U = I, namely the map ˜ C(n−4)/2 ⊕ C∞ → C∞ r : G∞( −∞ n/2) G∞( −∞). ⊂ C(n−4)/2 ⊕ C∞ ˜ = ⊂ C∞ If V −∞ n/2 is a subspace, then we define r(V) V −∞,i.e.,themap C(n−4)/2 ⊕ C∞ → C∞ induced by the inclusion −∞ n/2 , −∞. The result now follows from the next proposition. Cr ⊕ C∞ → C∞ 6 ∈ Z Proposition 3.7. The inclusion −∞ s , −∞, r s , induces a homotopy equivalence between the Grassmannians Cr ⊕ C∞ → C∞ α : G∞( −∞ s ) G∞( −∞). ∞ r Proof. Take V ∈ G∞(C−∞) and decompose it as V = V1 ⊕ V2,whereV1 ⊂ C−∞ and ⊂ C∞ C∞ → Cr ⊕ C∞ = ⊕ V2 r , and define β : G∞( −∞) G∞( −∞ s ) such that β(V) V1 ts−r V2, where ts−r is the shift by s − r coordinates (see Definition 2.1). Then αβ(V ) = V1 ⊕ ⊂ C∞ = ⊕ = ⊕ ⊂ Cr ⊕ C∞ ts−r V2 −∞ and βα(W) W1 ts−r W2,ifW W1 W2 −∞ s .The proposition now follows immediately from the next lemma. 2 Cr ⊕ C∞ → Cr ⊕ C∞ 6 ∈ Z Lemma 3.8. The map γ : G∞( −∞ s ) G∞( −∞ s ), r s , given by = ⊕ > = ⊕ ⊂ Cr ⊂ C∞ γ(V) V1 tk(V2), k 0,whereV V1 V2, V1 −∞ and V2 s , is homotopic to the identity. 1 = 1 + 1 C∞ → C∞ 6 6 Proof. The homotopy hτ sin( 2 πτ)I cos( 2 πτ)t1 : s s ,0 τ 1, starts with k = 1 ◦···◦ 1 t1 and ends with the identity through monomorphisms, and hτ hτ hτ (k times) is k = k = ˆ = ⊕ k 2 such that h0 tk and h1 I. Then hτ (V ) V1 hτ (V2) is a homotopy as desired. ˜1 −1 → −1 We have thus shown that rn : p (Vn) p (Un−1) is fiberwise a homotopy equiva- ˜s −1 → −1 lence. It should be remarked that for s<1, the deformation rn : p (Vn) p (Vn) is fiberwise a homeomorphism, since after identifying the fibers with the associated Grass- mannians, it is the identity. 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