Equivariant Geometric K-Homology with Coefficients

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Equivariant geometric K-homology with coefﬁcients

Michael Walter

Equivariant geometric K-homology with coefﬁcients

Diplomarbeit

vorgelegt von

Michael Walter

geboren in

Lahr

angefertigt am Mathematischen Institut der Georg-August-Universität zu Göttingen

2010

v Equivariant geometric K-homology with coefﬁcients Michael Walter

Abstract

K-homology is the dual of K-theory. Kasparov’s analytic version, where cycles are given by (abstract) elliptic operators over (not necessarily commutative) spaces, has proved to be an extremely powerful tool which, together with its bivariant generalization KK-theory, lies at the heart of many important results at the intersection of algebraic topology, functional analysis and geometry. Independently, Baum and Douglas have proposed a geometric version of K-homology inspired by singular bordism. Cycles for this theory are given by vector bundles over compact Spinc- manifolds with boundary which map to the target, i.e. E

f (M, BM) / (X, Y). There is a natural transformation to analytic K-homology defined by sending such a cycle to the pushforward of the class determined by the twisted Dirac operator. It is well-known to be an isomorphism, although a rigorous proof has appeared only recently. While both theories have obvious generalizations to the equivariant case and coefficients, the question whether these remain isomorphic is far from trivial (and has negative answer in the general case). In their work on equivariant correspondences Emerson and Meyer have isolated a useful sufficient condition for their theory which, while vastly more general, only deals with the absolute case. Our focus is not so much to construct a geometric theory in the most general situation, but to show that in the presence of a group action and coefficients the above picture still gives a generalized homology theory in a very geometrical way, isomorphic to Kasparov’s theory.

In this thesis we will show that equivariant geometric and analytic K-homology for compact Lie group actions and unital coefficient algebras are naturally isomorphic generalized homology theories on a broad category of spaces, following an approach by Baum, Oyono-Oyono and Schick. Chapter 1 gives a modern overview of both analytic K-theory and K-homology in terms of Kas- parov’s equivariant KK-theory. On the way we generalize several well-known results and constructions to the equivariant case and coefficients, in particular Swan’s theorem, Poincaré duality, Thom isomorphisms and functorial Gysin maps. For the latter we work out in some detail the relationship between its geometrical and analytical descriptions. Chapter 2 introduces geometric K-homology groups by defining cycles and equivalence relations. By translating the latter into analytical terms we show that the natural transformation from geometric to analytic cycles is well-defined on the level of homology classes. Using the Mostow embedding theorem we then prove that this map is in fact an isomorphism. vi

Acknowledgements. It is a pleasure to acknowledge Prof. Thomas Schick for inspiring discus- sions and helpful advice, and for sharing his enthusiasm for doing mathematics. Contents

Abstract v Chapter 1. Analytic K-theory and K-homology 1 ¦ 1.1. C -algebras and topological spaces with group actions 1 ¦ 1.2. Hilbert G-C -modules 5 1.3. Equivariant KK-theory 9 1.4. Analytic K-theory 15 1.5. Topological K-theory 18 1.6. Analytic K-homology 25 1.7. Products 25 1.8. Elliptic operators and analytic K-homology 29 1.9. Dirac operators 34 1.10. Spinc-structure and spinor bundles 40 1.11. Fundamental classes and Poincaré duality 51 1.12. The double of a manifold 60 1.13. Equivariant Spinc-structure of the spheres 62 1.14. Bott periodicity and Thom isomorphism 65 1.15. Gysin maps 70 Chapter 2. Geometric K-homology 75 2.1. Cycles and equivalence relations 75 2.2. Geometric K-homology groups 79 2.3. Isomorphism between geometric and analytic K-homology 82

Outlook 87 Bibliography 89 Notation Index 91

Subject Index 95

vii

CHAPTER 1

Analytic K-theory and K-homology

In his landmark article [Kas81], Kasparov introduced a bivariant theory, KK-theory, which uni- fied both K-theory and K-homology. Its main feature is the existence of the Kasparov product, an associative pairing KK(A, B) ¢ KK(B, C) Ñ KK(A, C) which unifies and generalizes most constructions of the preceding theories. Pushforwards and pullbacks, boundary maps and the pairing of K-theory and K-homology, as well also more involved constructions such as Bott periodicity and the Thom isomorphism can all be expressed in terms of Kasparov products with suitable elements. In the subsequent paper [Kas88], KK-theory was extended to the equivariant setting. In this chapter, we start by giving a modern account of analytic K-theory and K-homology which we define in terms of Kasparov’s equivariant KK-bifunctor (Sections 1.1–1.7). We shall also generalize the classical topological picture of K-theory to the equivariant case and coefficients (Section 1.5). Key ingredient to this latter result is a version of Swan’s theorem for equivariant bundles of finitely generated, projective Hilbert modules. The second part of the chapter focuses on connecting analysis with geometry. We recall how symmetric elliptic operators, in particular Dirac operators, define cycles in K-homology and show that it is possible to include into these the action of more general algebras which are similarly well-behaved as C0(M) (Section 1.8). Kasparov’s Dirac element fits naturally into this setting, as do the fundamental classes determined by manifolds equipped with an equivariant Spinc-structure, the K-theory equivalent of an orientation (Sections 1.9–1.11). We will in fact see that both elements are but different sides of the same medal. This will be our way of deducing Poincaré duality. With this machinery in place, we proceed to re-derive Bott periodicity and the Thom isomorphism by a careful analysis of the Spinc-structure on even-dimensional spheres (Sections 1.13 and 1.14). Our proof will give both a topological and an analytic description of these results which will be crucial for the second part of the thesis. In the last section we similarly define “wrong-way” functorial Gysin maps first in an analytical way and then derive concrete geometric descriptions for them (Section 1.15). Although Kasparov’s theory can be developed in a more general context, it is sufficient for our ¦ purposes to only consider separable complex C -algebras, and we can avoid much technical trou- ¦ ble that way. See [Bla98, Chapter 14] for a convenient review of the basic theory of C -algebras and [Bla06] for a more detailed account. We shall also require that all topological spaces be locally compact, second-countable and Hausdorff and that topological groups even be compact.

¦ 1.1. C -algebras and topological spaces with group actions

¦ ¦ In this section we will recall the fundamental notion of a G-C -algebra, that is, a C -algebra equipped with a continuous group action by automorphisms. We shall always identify a group element with the automorphism by which it acts. ¦ ¦ 1.1.1 DEFINITION (G-CONTINUITY, G-C -ALGEBRA). Let A be a C -algebra equipped with a (left) action of a (compact topological) group G by automorphisms. An element a P A is called G- continuous if the orbit map g ÞÑ g(a) is continuous.

1 2 1. ANALYTIC K-THEORY AND K-HOMOLOGY

¦ The category of G-C -algebras (G-algebra in [Kas88], covariant system in [Bla98]) is then defined ¦ as follows. Its objects are (separable) C -algebras with an action of G by automorphisms such that every element is G-continuous. In other words, the associated group homomorphism G Ñ Aut(A) is strongly continuous. Morphisms are G-equivariant ¦-homomorphisms. ¦ The category of (Z/2-)graded G-C -algebras is defined similarly except that we require both the ¦ group actions and morphisms to preserve the grading. Unless noted otherwise, all G-C -algebras are assumed to be ungraded (or, equivalently, trivially graded). We will always denote the degree of homogeneous elements a by Ba. ¦ We will always consider the complex numbers C as a trivial G-C -algebra. ¦ Recall that the tensor product A bp B of graded G-C -algebras A and B can be formed in different ways by completing the skew-commutative algebraic tensor product A bp B. We will always use the minimal/spatial tensor product from [Kas81, Section 2]. There exist canonical braiding iso- B B morphisms A bp B B bp A sending a tensor a b b to (¡1) a bb b a such that the class of graded ¦ G-C -algebras forms a symmetric monoidal category. We write A b B if both algebras are trivially ¦ graded. A detailed treatment of tensor products of C -algebras can be found in [WO93, Appen- dix T]; for the graded case see [Bla98, Section 14]. ¦ Most of the time, one of the factors in our tensor products will be a nuclear C -algebra so that all ¦ C -tensor products are in fact equal. We will make repeated use of the following facts: ¦ 1.1.2 PROPOSITION ([Bla06, Theorem 15.8.2]). The following classes of C -algebras are nuclear: Com- ¦ ¦ ¦ mutative C -algebras, finite-dimensional C -algebras, C -algebras of compact operators on a separable Hilbert space. ¦ Moreover, the class of nuclear C -algebras is closed under direct limits, extensions, ideals and quotients.

1.1.3 PROPOSITION ([WO93, Theorem T.6.26]). Let 1 2 0 / A / A / A / 0 ¦ ¦ ¦ 2 be a short exact sequence of C -algebras and B another C -algebra. If at least one of the C -algebras A or B is nuclear then the tensored sequence 1 2 0 / A b B / A b B / A b B / 0 remains exact. ¦ 1.1.4 EXAMPLE ([WO93, Corollary T.6.17, Theorem T.6.20]). In particular, the C -algebra C0(X) of complex-valued functions vanishing at inﬁnity on a locally compact Hausdorff space X is nuclear. Let us also recall that there exist canonical isomorphisms bp Ñ b ÞÑ ÞÑ C0(X) A C0(X, A), f a (x f (x)a), b Ñ ¢ b ÞÑ ÞÑ C0(X) C0(Y) C0(X Y), f g ((x, y) f (x)g(y)) ¦ for all graded C -algebras A and spaces Y. Here and in the following, the space C0(X, A) of A-valued functions vanishing at inﬁnity is always equipped with the supremum norm.

We will now formalize our notion of topological spaces and pairs with a group action.

1.1.5 DEFINITION (G-SPACE, G-PAIR).A G-space is a (locally compact, second-countable, Haus- dorff) space X with a (left) action of a group G such that the associated map G ¢ X Ñ X is continuous. Morphisms will always be proper G-maps, i.e. proper G-equivariant continuous maps. If Y X is a closed G-invariant subspace then (X, Y) is called a G-pair. A morphism of G-pairs 1 1 1 1 (X, Y) Ñ (X , Y ) is a proper G-map X Ñ X sending Y to Y . We deﬁne the Cartesian product 1 1 1 1 1 of G-pairs (X, Y) and (X , Y ) to be the pair (X ¢ X , X ¢ Y Y Y ¢ X ). Similarly, we deﬁne their 1 1 disjoint union to be the pair (X > X , Y > Y ). ¦ 1.1. C -ALGEBRAS AND TOPOLOGICAL SPACES WITH GROUP ACTIONS 3

1.1.6 LEMMA. The group action of a non-compact G-space X extends naturally to its one-point compact- iﬁcation X+ = X Y t8u (with the trivial action on 8). ¢ + Ñ + 8 +z PROOF. It sufﬁces to verify continuity of G X X in the point (1G, ). Let X K with K X compact be a neighborhood of 8 P X+. Then GK, the orbit of K, is a compact set in X and ¢ +z 8 +z G (X GK) is a neighborhood of (1G, ) that maps into X K.

In the following, we will often consider bifunctors deﬁned for G-pairs together with graded G- ¦ C -algebras, i.e. functors whose domain is the product of the category of G-pairs with the cat- ¦ egory of graded G-C -algebras (or its opposite). We will write objects in this category in the form (X, Y; A) and morphisms as (ϕ; Φ) where ϕ is a morphism of G-pairs and Φ a morphism of ¦ ¦ graded G-C -algebras. We will usually omit the G-C -algebras if A = C and the subspace Y if it is empty.

1.1.7 PROPOSITION. The assignment # Ñ z z bp (X, Y; A) C0(X Y, A) C0(X Y) A C : 0 ( ) ÞÑ ( ÞÑ ¥ ¥ ) ϕ; Φ f Φ f ϕ XzY deﬁnes a bifunctor which is contravariant in the G-pair and covariant in the algebra variable. z Here, we extend functions by zero so that the second composition makes sense, and G acts on C0(X Y, A) ¡ by the formula g( f )(x) := g( f (g 1(x))).

PROOF. This is obvious once we have established that the functor is well-defined on objects and morphisms. ¦ z ¦ Let (X, Y) be a G-pair and A a graded G-C -algebra. It is well-known that C0(X Y, A) is a C - z z bp ¦ algebra and that C0(X Y, A) C0(X Y) A in the sense of graded C -algebras (Example 1.1.4). It is separable because the space X is assumed to be second-countable. Since the G-action defined by the above formula corresponds precisely to the diagonal action on the tensor product it suffices z ¦ to show that C0(X Y) is in fact a G-C -algebra, i.e. that every element is G-continuous. Clearly H P + we may assume that Y = . Then the orbit map of any function f C0(X) C(X ) can be written as the composition

¡ g 1 f ¥¡ G / C(X+, X+) / C(X+)

+ + + which in fact maps into C0(X). Here, the function spaces C(X , X ) and C(X ) are equipped with the compact-open topology so that both maps are continuous. But the compact-open topology on the latter space agrees with the topology induced by the supremum norm. This shows that f is G-continuous.

It remains to verify that C0 is well-deﬁned on morphisms. We only need to show that the preim- 1z 1 | age of a compact set K X Y under ϕ XzY is compact (we do not have to worry about Φ because ¡ ¡ it is norm-decreasing). But this preimage is precisely ϕ 1(K) X (XzY) = ϕ 1(K), hence compact in X (since ϕ is proper), and thus compact in the open subset XzY containing it.

1.1.8 REMARK. The well-known Gelfand-Naimark theorem asserts that this functor implements a ¦ duality between the category of spaces and the category of commutative C -algebras (see e.g. [Bla06, Theorem II.2.2.4]).

1.1.9 REMARK. The functor C0 maps disjoint unions to direct sums and Cartesian products to tensor products. Indeed, > 1 z > 1 z > 1z 1 z ` 1z 1 C0((X X ) (Y Y )) = C0((X Y) (X Y )) = C0(X Y) C0(X Y ), ¢ 1 z ¢ 1 Y ¢ 1 z ¢ 1z 1 z b 1z 1 C0((X X ) (X Y Y X )) = C0((X Y) (X Y )) = C0(X Y) C0(X Y ). 4 1. ANALYTIC K-THEORY AND K-HOMOLOGY

¦ Let us write ϕ := C0(ϕ; idA) and Φ¦ := C0(id(X,Y); Φ) for the pullback of pairs and the pushforward of algebras, respectively. There are two natural morphisms associated to every G-pair (X, Y), the inclusion incl: Y X of the closed subspace and the relativization map rel: X Ñ (X, Y). ¦ 1.1.10 COROLLARY. We have natural short exact sequences of G-C -algebras ¦ ¦ z rel incl (1.1.11) 0 / C0(X Y, A) / C0(X, A) / C0(Y, A) / 0 which are semisplit, i.e. there exists a completely positive, norm-decreasing, grading-preserving C-linear ¦ section for the projection incl .

PROOF. It is is easy to see that the above sequence is exact for A = C, and because C0(Y) is nuclear it is also semisplit [Bla98, Theorem 15.8.3]. Moreover, by nuclearity the sequence remains exact after tensoring with A (Proposition 1.1.3) and it is clear that the resulting sequence is still semisplit (tensor the split with the identity map of A). 1 1 For naturality, observe that every morphism (ϕ; Φ) : (X, Y; A) Ñ (X , Y ; B) gives rise to a commutative diagram (Y, H; A) / (X, H; A) / (X, Y; A)

(ϕ ;Φ) (ϕ;Φ) (ϕ;Φ) Y 1 1 1 1 (Y , H; B) / (X , H; B) / (X , Y ; B) in the product category. The claim thus follows from applying the functor C0.

These basic results will enable our later passage from topology to analysis.

¦ Our last two examples are ﬁnite-dimensional graded G-C -algebras. They will play a crucial role in what follows.

1.1.12 EXAMPLE (ENDOMORPHISMSOFTHEEXTERIORALGEBRA). Let W be a ﬁnite-dimensional ¦ unitary representation1 of the group G with isometric antilinear involution x ÞÑ x such that the group action commutes with the involution. We denote by ¡W the same representation ¦ ¦ with involution x ÞÑ ¡x . The inner product and G-action extend to the exterior algebra W which then also becomes a ﬁnite-dimensional unitary representation of G, graded into the span ¦ of even and odd monomials. In particular, the space L( W) of linear endomorphisms of the exterior algebra, graded into grading-preserving and grading-reversing operators, is a graded G- ¦ C -algebra — this is easy to see directly, but also follows from Example 1.2.3 in the next section.

1.1.13 EXAMPLE (CLIFFORD ALGEBRA). Let W as in the previous example. The complex Clifford 2 algebra CW is defined as quotient of the tensor algebra of W with respect to the relation x = ¦ xx , xy. It inherits the G-action and Z/2-grading from the tensor algebra. We obtain an antilinear ¤ ¤ ¤ ¦ ¦ ¤ ¤ ¤ ¦ P involution on CW by setting (x1 xk) := xk x1 for x1,..., xk W. ¦ The Clifford algebra CW satisfies the following universal property: Every G-equivariant -homomorphism Φ from W into an arbitrary complex unital ¦-algebra A with a linear G-action that 2 x ¦ y ¦ Ñ satisfies Φ(x) = x , x extends uniquely to a G-equivariant -algebra homomorphism CW A. In particular, every G-equivariant isometric ¦-endomorphism of W extends to a G-equivariant ¦ -algebra automorphism of CW.

We can deﬁne a canonical inner product on CW by taking any orthonormal basis (ei) of W and ( ¤ ¤ ¤ ) ¤ ¤ requiring that the monomials ei1 eik for i1 ... ik and 0 k dim W form an orthonormal basis. Then we have a canonical isometric isomorphism of unitary G-representations ©¦ Ñ C ^ ^ ÞÑ ¤ ¤ ¤ (1.1.14) W W, ei1 ... eik ei1 eik , where (ei) is any orthonormal basis of W.

1We will often identify a representation with its corresponding representation space. ¦ 1.2. HILBERT G-C -MODULES 5

¦ However, we will usually consider CW to be equipped with the C -norm constructed as follows: Denote by λx the operator of left multiplication by x P W on the exterior algebra. By the universal property, the G-equivariant ¦-homomorphism ©¦ ` ¡ Ñ ` ÞÑ ¦ (1.1.15) W ( W) L( W), x y λx+y + λx¦¡y¦ Ñ ¦ ¦ extends to an equivariant homomorphism CW`(¡W) L( W) of -algebras which for reasons ¦ of dimension has to be an isomorphism. We can use it to equip CW with a C -norm via the ãÑ ` ¡ canonical embedding CW CW`(¡W) extending the inclusion W W ( W). Thus CW is ¦ canonically a graded G-C -algebra. + 1.1.16 EXAMPLE (STANDARD CLIFFORD ALGEBRA). Consider W = Cp q equipped with the canonical inner product, trivial G-action and involution given by complex conjugation on the first p summands and negated complex conjugation on the last q summands. We write Cp,q for the Clifford algebra of W. This is the complex Clifford algebra with respect to the quadratic form ÞÑ 2 2 ¡ 2 ¡ ¡ 2 (zk) z1 + ... + zp zp+1 ... zp+q. We will refer to the algebras Cp,q as the standard Clifford algebras. All these algebras are canonically isomorphic for fixed sum p + q = n. Indeed, an isomor- Ñ phism Cp,q+1 Cp+1,q is induced by the universal property from mapping the coordinate basis vector ep+1 to iep+1 and fixing all other coordinate basis vectors. We also have isomorphisms bp 1 1 Ñ 1 1 Cp,q Cp ,q Cp+p ,q+q extending the assignment $ b ÞÑ ¤ ' ei 1 ei (i p) ' & b ÞÑ 1 ¡ ei 1 ep +i (i p) ' b ÞÑ ¡ q ¤ 1 ' 1 ei ( 1) ep+i (i p ) %' b ÞÑ ¡ 1 1 ei ep+q+i (i p ) p p+0 In the following we will write Cp := Cp,0 = CCp and C¡p := C¡Cp where C = C as above. ¡ i¡1 We shall always identify C0,n C¡n using the isomorphism induced by sending ei to ( 1) ei.

1.1.17 REMARK. Our notation and conventions are as in Kasparov’s article [Kas81, 2.11–2.17]. It is also used in the articles [Kas88, CS84, KS91] and [Bla98], the standard textbook on KK-theory, as well as the lecture notes [HR04, Kas09]. However, the opposite convention is equally popular. It has been used in Kasparov’s original article on analytical K-homology [Kas75] as well as in the popular monographs [HR00, LM89] and many articles, in particular [BHS07, BHS08, BOOSW10].

¦ 1.2. Hilbert G-C -modules

¦ In this section we will deﬁne the equivariant version of a Hilbert C -module and the corresponding bimodules. These objects are basic to the formulation of Kasparov’s equivariant KK-theory in [Kas88] which will be presented in the next section, and they provide the correct framework ¦ for the generalization of Morita equivalence from [Rie74] to G-C -algebras. ¦ ¦ 1.2.1 DEFINITION (HILBERT G-C -MODULE). Let A be a G-C -algebra. A Hilbert G-A-module is a Hilbert A-module E with a left action of a compact group G by C-linear bounded operators such ¢ that the action G Ñ B(E) is strongly continuous and the following conditions are satisﬁed: – g(va) = g(v)g(a) and – g(xv, wy) = xg(v), g(w)y for all g P G, v, w P E, a P A. A morphism of Hilbert G-A-modules is a G-equivariant operator with adjoint. There is an obvious notion of the direct sum of Hilbert modules and we get an additive category. 6 1. ANALYTIC K-THEORY AND K-HOMOLOGY

Let us denote by B(V, W) the set of bounded C-linear operators between two normed vector spaces V and W, and by 1 := 1B(V) = idV the identity operator of V. If E and F are two Hilbert G-A-modules then we write BA(E, F) for the set of those operators T which have an adjoint ¦ T with respect to the A-valued inner products. Any such operator is automatically A-linear and bounded, and there is a natural G-action on BA(E, F) given by the formula g(T)(x) := ¡ g(T(g 1(x))). Note that an operator is G-equivariant precisely if it is G-invariant under this action. The subspace KA(E, F) of compact operators is defined to be the closed ideal generated by the “rank one” operators Tv,w := xv, ¡yw where v P E and w P F. ¦ The space BA(E) := BA(E, E) is always a C -algebra (with the operator norm), but the set of G- continuous elements is usually a proper subset of BA(E). However, every G-equivariant operator in BA(E) is of course G-continuous (its orbit map is constant). In particular, the closed subset of ¦ G-equivariant operators in BA(E) is a G-C -algebra. We also remark that the ideal of compact ¦ operators KA(E) := KA(E, E) is always a G-C -algebra. ¦ The category of (Z/2-)graded Hilbert G-A-modules for a graded G-C -algebra A is defined in the obvious way: We take as objects graded Hilbert A-modules E with grading-preserving G-action and we also require morphisms to be grading-preserving. Every operator in BA(E) can of course be uniquely decomposed into a grading-preserving and a grading-reversing part. This way we ¦ can consider BA(E) and KA(E) as graded C -algebras. ¦ n 1.2.2 EXAMPLE. Every graded G-C -algebra A and, more generally, every° finite direct sum A is x y n ¦ a graded Hilbert G-A-module over A with inner product (ai), (bi) := i=1 ai bi and diagonal G-action. It is well-known that every A-linear map An Ñ Am can be written as left multiplication by a matrix with entries in the left multiplier algebra Mleft(A) of A. It follows that every such map is automatically continuous and has an adjoint. The compact operators on An are precisely those which are represented by a matrix with entries in A. In particular, if A is trivially graded and unital then we have the following chain of isomorphisms ¦ of G-C -algebras: n m n m n m b n m b LA(A , A ) = BA(A , A ) = KA(A , A ) Mm¢n(A) Mm¢n(C) A L(C , C ) A.

Here, we have used the notation LA(V, W) for the set of A-linear maps from V to W, and we write L(V, W) := LC(V, W).

1.2.3 EXAMPLE (REPRESENTATIONS). A graded Hilbert G-C-module W is simply a (strongly continuous) unitary representation of G on a graded Hilbert space. Consequently, for ﬁnite-dimen- ¦ sional W all the algebras B(W), K(W) and L(W) are identical and hence graded G-C -algebras.

1.2.4 DEFINITION (EXTERIORTENSORPRODUCT). Let E be a graded Hilbert G-A-module and F a graded Hilbert G-B-module. The skew-commutative (exterior) tensor product E bp F is deﬁned by completing the algebraic skew-commutative tensor product with respect to the A bp B-valued inner product 1 1 1 1 xx b y, x b y y := xx, x y b xy, y y. It is a Hilbert G-A bp B-module with diagonal G-action, right A bp B-action given by B B (x b y)(a b b) := (¡1) y a(xa b yb) and grading B(x b y) = Bx + By. Moreover, we have a canonical embedding bp Ñ bp bp ÞÑ b ÞÑ ¡ BSBx b BA(E) BB(F) BA bp B(E F), T S (x y ( 1) T(x) S(y)) bp Ñ bp which restricts to an isomorphism KA(E) KB(F) KA bp B(E F). We will usually write b instead of bp for tensor products of trivially graded Hilbert modules. ¦ 1.2. HILBERT G-C -MODULES 7

¦ 1.2.5 EXAMPLE (FREE HILBERT G-A-MODULE). Let A be a graded G-C -algebra and W a finite- dimensional unitary representation of G on a graded Hilbert space. Then, combining the preceding examples, we find that W bp A is a graded Hilbert G-A-module which we will call a finite- dimensional free Hilbert G-A-module (we will see in Proposition 1.4.7 that this is indeed the correct notion of a free Hilbert module). Note that we recover An if W is the trivial representation on Cn. 1 Now suppose that A and W are trivially graded and that A is unital. If W is another trivially- graded finite-dimensional unitary representation of G then, generalizing Example 1.2.2, we find that b 1 b b 1 b 1 b LA(W A, W A) = BA(W A, W A) L(W, W ) A. ¦ 1.2.6 EXAMPLE. Let A and B be graded G-C -algebras. We can either form their exterior tensor product, interpreting the individual factors as graded Hilbert modules, or we can first form the ¦ tensor product in the sense of C -algebras and then interpret the result as a graded Hilbert module. One easily verifies that both procedures in fact yield the same result since one completes with respect to the same norm. In particular, the notation A bp B is not ambiguous.

1.2.7 DEFINITION (INTERIORTENSORPRODUCT). Again, let E be a graded Hilbert G-A-module and F a graded Hilbert G-B-module. Every G-equivariant morphism Φ : A Ñ BB(F) of graded ¦ C -algebras endows F with the structure of a left A-module so that we can form the algebraic tensor product over A. Its completion with respect to the B-valued pre-inner product 1 1 1 1 xx b y, x b y y := xy, Φ(xx, x y)y y (we may have to quotient out vectors of norm zero ﬁrst) is a graded Hilbert G-B-module with bp the diagonal G-action and B-action on the second factor, called the (interior) tensor product E ΦF along Φ. It is graded in the same way as the exterior tensor product. Note that for every other 1 Hilbert G-A-module E we have a canonical map 1 Ñ bp 1 bp ÞÑ b BA(E, E ) BB(E ΦF, E ΦF), T T 1. In general there is no analogue for the second factor. ¦ The interior tensor product can in particular be used to modify the underlying G-C -algebra of a Hilbert module:

1.2.8 DEFINITION (PUSHFORWARD). The pushforward of a graded Hilbert G-A-module E along Ñ ¦ b a morphism Φ : A B of graded G-C -algebras is deﬁned to be Φ¦(E) := E Φ B. Together with the canonical map from the previous deﬁnition we get a functor Φ¦ from graded Hilbert G-A-modules to graded Hilbert G-B-modules which is also functorial in Φ (up to isomorphism). The following lemma is useful for identifying simple pushforwards. ¦ 1.2.9 LEMMA ([Bla98, Example 13.5.2, (a)]). Let Φ : A Ñ B be a morphism of graded G-C -algebras. Then the canonical map a b b ÞÑ Φ(a)b induces an isomorphism Φ¦(A) Φ(A)B of Hilbert G-B- modules. In particular, if Φ is essential, i.e. the range of Φ contains an approximate unit for B, then Φ¦(A) B.

The following is the equivariant version of a Hilbert bimodule. ¦ ¦ 1.2.10 DEFINITION (HILBERT G-C -BIMODULE). Let A and B be graded G-C -algebras. A graded Hilbert G-A-B-bimodule is a graded Hilbert G-B-module E together with a G-equivariant mor- ¦ phism Φ : A Ñ BB(E) of graded C -algebras. We will always consider E as a left A-module via Φ. Morphisms are morphisms of Hilbert G-B-modules which intertwine the left A-action. The operations of direct sum, exterior tensor product and pushforward can be extended in the obvious way to graded Hilbert bimodules. We will always take the interior tensor product of a graded Hilbert G-A-B-bimodule E and a graded Hilbert G-B-C-bimodule F with respect to the left B- p module action of F and write E b BF for the result. 8 1. ANALYTIC K-THEORY AND K-HOMOLOGY

¦ We also deﬁne the pullback Φ (E) of a Hilbert G-B-C-bimodule E along a morphism Φ : A Ñ B ¦ Ñ of graded G-C -algebras by precomposing the morphism B BC(E) with Φ. In other words, the left action is given by the formula ax := Φ(a)x. ¦ 1.2.11 EXAMPLE. Every morphism Φ : A Ñ B of graded G-C -algebras determines a graded Hilbert G-A-B-bimodule BΦ in the following way. As a Hilbert G-B-module, BΦ is just B considered as a Hilbert module over itself, and the left action by A is deﬁned as above by the formula ab := Φ(a)b. In particular, note that A acts by compact operators in KB(B) B (Example 1.2.2).

In fact, it is easy to see that the interior tensor product with the bimodule BΦ implements the pullback and pushforward of Hilbert bimodules along Φ. The following lemma is somewhat of an analogon of Lemma 1.2.9 for Hilbert bimodules: ¦ 1.2.12 LEMMA. Let Φ : A Ñ B be an essential morphism of graded G-C -algebras (that is, the range of ¦ Φ contains an approximate identity for B). Then Φ¦(A) Φ (B) as Hilbert G-A-B-bimodules.

PROOF. The isomorphism a b b ÞÑ Φ(a)b of graded Hilbert G-B-modules from Lemma 1.2.9 clearly intertwines the left A-action. ¦ Recall that in Example 1.1.13 we have deﬁned both an inner product norm and a C -norm on Clifford algebras CW. Their interplay can be understood as follows:

1.2.13 EXAMPLE. Let CW be a complex Clifford algebra. Then the Hilbert space CW is a Hilbert G-CW-C-bimodule with respect to each of the following actions of CW: 1. left multiplication, or 2. right multiplication by the transpose (this is the linear involution (¡)t defined by reversing the order of products). Ñ PROOF. The nontrivial part of the assertion is that the respective morphisms CW B(CW ) are ¦-preserving. In order to see this, we will identify the Hilbert space C with the exterior algebra ¦ W W (Equation (1.1.14)). P For the first action, let x W, choose a unit vector e1 = sx in the span of x and extend it to an ( ) = ¤ ¤ ¤ ^ ^ orthonormal basis ei of W. Then for every v ei1 eik ei1 ... eik with i1 ... ik we # have ¦ ¤ ¤ ¤ ( + ¦ ) = sei2 eik λx λx v, if i1 1 xv = ¦ ¤ ¤ ¤ ( + ¦ ) ¡ se1ei1 eik λx λx v, if i1 1 This shows that the left multiplication action corresponds to the canonical left action of CM on the exterior algebra as defined in Equation (1.1.15). ¤ ¤ ¤ ^ ^ ¡ ¡ For the second action, consider v = ei ei ei ... ei with i1 ... ik. Then # 1 k 1 k ¤ ¤ ¤ ¡ k ¡ ¦ sei ei ¡ ( 1) (λx λx¦ )v, if ik = 1 vx = 1 k 1 k k ¦ (¡ ) ¤ ¤ ¤ (¡ ) ( ¡ ¦ ) ¡ 1 se1ei1 eik 1 λx λx v, if ik 1 Consequently, the second action of some vector x P W corresponds to the action of x considered ¡ as an element of W C¡W as defined in Equation (1.1.15) times the grading operator. It is evident from these descriptions that both actions are ¦-preserving.

1.2.14 DEFINITION (G-IMPRIMITIVITYBIMODULE,MORITA EQUIVALENCE). A graded Hilbert G- A-B-bimodule E is called a G-imprimitivity bimodule between A and B if Φ is an isomorphism onto KB(E) and if E is full, i.e. if the range of the B-valued inner product is dense in B. In this case, we say that A and B are Morita equivalent (cf. [Rie74]). It can be seen as follows that Morita equivalence deﬁnes an equivalence relation: Evidently, A is a G-imprimitivity bimodule between A and A. If E is a G-imprimitivity bimodule between A and B then its dual imprimitivity 1.3. EQUIVARIANT KK-THEORY 9

¦ bimodule E := KB(E, B) is a G-imprimitivity bimodule between B and A (with left multiplication on the result and right action of A KB(E) by precomposition; see e.g. [Bla98, Exercise b ¦ ¦ b 13.7.1]). In fact, E B E A and E A E B as G-imprimitivity bimodules. Finally, it can be shown that the interior tensor product of a G-imprimitivity bimodule between A and B and a G-imprimitivity bimodule between B and C is a G-imprimitivity bimodule between A and C.

1.2.15 EXAMPLE (IRREDUCIBLE CLIFFORD MODULES). Recall the deﬁnition of the standard Clif- ¦ ford algebras from Example 1.1.16. We can consider the exterior algebra Cp as a Hilbert Cp,p-C-bimodule using the left action deﬁned by the composition ©¦ ÝÑ bp ÝÑ bp bp ÝÑ ÝÑ p (1.2.16) Cp,p Cp,0 C0,p Cp C¡p = CCp C¡Cp CCp`(¡Cp) L( C ) All isomorphisms are induced by the canonical ones from Examples 1.1.13 and 1.1.16. Because their composition is again an isomorphism and every Hilbert C-module is full, we conclude that ¦ p Wp := C is an imprimitivity bimodule between Cp,p and C. Note that in this case the dual imprimitivity bimodule is just the dual space with its canonical right Cp,p-action by left multiplication in the argument. We will later need the following fact. p 1.2.17 LEMMA. We have Wp+q Wp b Wq as Hilbert Cp+q,p+q-C-bimodules. Here, we use the canon- p ical isomorphism Cp,p b Cq,q Cp+q,p+q from Example 1.1.16 to consider the exterior tensor product as a left Cp+q,p+q-module.

PROOF. Clearly, it sufﬁces to show that both representations of Cp+q,p+q are isomorphic. It is well-known from the representation theory of complex Clifford algebras that all irreducible 2p graded representations of Cp,p have dimension 2 and that they are classiﬁed by the action of ¤ ¤ ¤ the orientation element ωp := e1 e2p (cf. [LM89, Section I.5]). This element commutes with even and anticommutes with odd monomials and squares to 1. Hence it is determined by its action on the even part which is by either of the scalars ¨1.

2(p+q) Both representations of Cp+q,p+q we want to compare have dimension 2 , hence both are irreducible. It thus sufﬁces to show that the orientation element acts by the same scalar on both representations. Indeed, by chasing the isomorphisms in Equation (1.2.16) we ﬁnd that ωp+q acts by the identity on 1 P Wp+q, hence on any even vector in Wp+q. On the other hand, the canonical p isomorphism Cp+q,p+q Cp,p b Cq,q sends ωp+q to (ωp b 1)(1 b ωq), hence the volume element p also acts by the identity on even vectors in Wp b Wq. We conclude that both representations are isomorphic and the claim is proved.

1.3. Equivariant KK-theory

In the following we give a brief overview of Kasparov’s equivariant KK-theory. Our treatment is emphatically not self-contained. Instead, we will try to make precise the most important deﬁ- nitions and theorems which are then used in the course of this thesis. Authoritative sources for ¦ this section are [Kas88, Bla98]. Recall that we assume that all C -algebras be separable, hence in particular σ-unital, which simpliﬁes the presentation of what follows. We start by giving an abstract characterization of the equivariant KK-bifunctor. ¦ 1.3.1 THEOREM ([Kas88]). There exists a bifunctor KK¦G from pairs of graded G-C -algebras to Z-graded Abelian groups such that the following properties hold:

A.KK ¦G is contravariant in the ﬁrst variable and covariant in the second. B.KK G is homotopy-invariant in each variable. ¦ We write Φ and Φ¦ for the pullback and pushforward along a morphism Φ. 10 1. ANALYTIC K-THEORY AND K-HOMOLOGY

C. There exists an associative bilinear product bp G bp b G bp Ñ G bp bp D : KKp (A1, B1 D) KKq (D A2, B2) KKp+q(A1 A2, B1 B2)

called the Kasparov product which is contravariant in A1 and A2, covariant in B1 and B2 and ¦ ¦( ) bp = bp ( ) Ñ functorial in D in the sense that Φ x D2 y x D1 Φ x for every morphism Φ : D1 D2.

The most important special cases are the composition Kasparov product where A2 = B1 = C and the exterior Kasparov product where D = C. D. The exterior Kasparov product is gradedly commutative, i.e. B B x bp y = (¡1) x yy bp x. for homogeneous elements x and y. Here and in the following we identify KK-groups via pullback and pushforward along the braiding isomorphisms of the respective skew-symmetric tensor products. This ¦ is never a problem because the category of G-C -algebras forms a symmetric monoidal category with respect to the tensor product. ¦ Ñ P G E. Every morphism of graded G-C -algebras Φ : A B defines an element [Φ] KK0 (A, B). This assignment is functorial with respect to the composition Kasparov product, and the tensor product Φ bp Ψ of morphisms is sent to the exterior Kasparov product [Φ] bp [Ψ]. G ¦ We can regard KK0 as a category whose objects are graded G-C -algebras, where morphisms from A to B G are given by the elements of KK0 (A, B) and where composition is implemented by the Kasparov product. Isomorphisms in this category are called KK-equivalences. Property E then asserts that we have a natural ¦ transformation from the ordinary category of G-C -algebras to KKG. F. Pullback and pushforward along a morphism Φ : A Ñ B are implemented in terms of left and right Kasparov multiplication by the KK-element [Φ] (over B and A, respectively). G G In particular, KK¦ (A, A) is a graded ring with unit 1A := [idA], and all the groups KK¦ (A, B) are graded KK¦G(A, A)-KK¦G(B, B)-bimodules and graded two-sided KK¦G(C, C)-modules. We will often refer to the exterior Kasparov product with 1A as tensoring with A. G. We have the identity p p p p x b Dy = (x b 1D ) b bp bp (1D b y) 2 D1 D D2 1 P G bp bp P G bp bp for all x KK¦ (A1, B1 D1 D) and y KK¦ (D D2 A2, B2). It follows using properties C and D that the Kasparov product is gradedly distributive in the sense that 1 1 B 1B 1 1 bp bp bp 1 ¡ x y bp bp bp (x Dy) (x D y ) = ( 1) (x x ) D bp D1 (y y ). In particular, pullback and pushforward of an external Kasparov product x bp y along a tensor product of morphisms Φ bp Ψ is the external tensor product of the pullbacks or pushforwards of the individual factors, and tensoring is distributive in the sense that bp bp bp bp bp bp bp bp bp bp 1E (x Dy) = (x Dy) 1E = (1E x) E bp D(1E y) = (x 1E) D bp E(y 1E) (up to canonical identification of the KK-groups via the braiding isomorphisms). Moreover, we see that every Kasparov product can be decomposed into first tensoring and then performing a composition Kasparov product (use B1 = A2 = C). H.KK G is σ-additive in the first and finitely additive in the second variable. Together with the above we see that the Kasparov product is compatible with direct sum decomposition in both variables. I.KK G is stable in each variable, i.e. there are natural isomorphisms G G p G p G p p KK¦ (A, B) KK¦ (A b K, B) KK¦ (A, B b K) KK¦ (A b K, B b K) induced by tensoring elements with a fixed rank-one projection in the algebra K of compact operators on a separable Hilbert space. These isomorphisms are compatible with the Kasparov product. 1.3. EQUIVARIANT KK-THEORY 11

P G J. Every Hilbert G-A-B-bimodule E with compact A-action determines an element [E] KK0 (A, B). In fact, the KK-element [Φ] assigned to a morphism Φ : A Ñ B according to Property E is induced by the bimodule BΦ from Example 1.2.11, so that pullback and pushforward of [E] are determined by pullback and pushforward of E. Moreover, exterior and interior tensor product correspond to the respective Kasparov products, i.e. p p p p [E b F] = [E] b [F] and [E b BF] = [E] b B[F] in the situation of Deﬁnition 1.2.10, and every ¦ G G-C -algebra A considered as a bimodule over itself determines the unit element of KK0 (A, A). In particular, every G-imprimitivity bimodule determines a KK-equivalence (this follows from the discussion after Deﬁnition 1.2.14). We write R¦(G) := KK¦G(C, C).

K.R 0(G) is isomorphic to the complex representation ring of the compact group G. The isomorphism is implemented as follows: Take a G-representation W, make it unitary by choosing an appropriate inner P product, note that we now have a Hilbert G-C-C-bimodule and send it to the element [W] R0(G). Furthermore, R1(G) is always zero. L. Every short exact sequence

j q 0 / J / A / Q / 0 ¦ of graded G-C -algebras which is semisplit (i.e. there exists a completely positive, norm-decreasing, grading-preserving C-linear section for the projection morphism q) determines a natural boundary BP G ¦ element KK¡1(Q, J). Moreover, for every G-C -algebra D we get long exact sequences

j¦ q¦ B ... / G / G / G / G / ... KK¦ (D, J) KK¦ (D, A) KK¦ (D, Q) KK¦¡1(D, J)

¦ ¦ q j B ... / G / G / G / G / ... KK¦ (Q, D) KK¦ (A, D) KK¦ (J, D) KK¦¡1(Q, D)

The boundary maps in these sequences, which are also denoted by B, are implemented by right and left Kasparov multiplication by the KK-element B, respectively. ¦ If we tensor such a sequence from the left or right with a nuclear G-C -algebra D then the boundary element of the resulting sequence is equal to the left or right exterior Kasparov product of B with the unit element 1D. It follows from naturality of the boundary element and associativity of Kasparov multiplication that the long exact sequence is natural in both the short exact sequence and the variable D. ¦ M. For every G ¢ H-space X with free H-action and graded H-C -algebras A, B there exists a natural induction homomorphism H Ñ G H H KK¦ (A, B) KK¦ (C0(X, A) , C0(X, B) ) compatible with Kasparov product and unit elements (for A = B). Here, (¡)H denotes the subalgebra ¦ H of H-invariant elements. Note that if A is a ﬁnite-dimensional H-C -algebra then C0(X, A) can be ¦ considered as the G-C -algebra of sections of the G-vector bundle X ¢H A over X/H. In particular, the image of a KK-equivalence under induction is again a KK-equivalence. G G N. We have natural isomorphisms KK¦ KK¦+2 (Clifford periodicity) which commute with the Kasparov product and induction.

We can thus safely think of KK¦G as a Z/2-graded theory since the analoga of the above properties still hold. In this picture, the long exact sequences from Property L can in fact be considered to be natural cyclic six-term exact sequences. 12 1. ANALYTIC K-THEORY AND K-HOMOLOGY

Let us now sketch the construction of KK-theory in terms of cycles and equivalence relations and indicate how the various operations and properties are implemented.

1.3.2 DEFINITION (CYCLE).A cycle for the G-equivariant KK-theory of (A, B) is a pair (E, T) containing the following data – E is a countably generated Hilbert G-A-B-bimodule, and – T is a grading-reversing operator in BB(E) such that 2 ¦ [T, a], a(T ¡ 1), a(T ¡ T ), g(T) ¡ T P KB(E)(@a P A). Here and in the following, [¡, ¡] is the graded commutator. A cycle is degenerate if all the above expressions vanish. We remark that the operator T is automatically G-continuous by [Tho99, footnote on page 228]. Two cycles (E, T) and (F, S) are equivalent if there is an isometric isomorphism of Hilbert G-A-B- bimodules U : E Ñ F intertwining the operators T and S. The direct sum, exterior tensor product, pullback and pushforward of a cycle are deﬁned in the obvious way using their analoga for Hilbert bimodules and bounded operators thereon.

1.3.3 DEFINITION (HOMOTOPY).A homotopy for (A, B) is a cycle for (A, C([0, 1], B)), and we say Ñ ÞÑ that the pushforwards along the evaluation maps evalk : C([0, 1], B) B, f f (k), k = 0, 1 are homotopic. We can now deﬁne the equivariant KK-groups.

G 1.3.4 DEFINITION (EQUIVARIANT KK-GROUPS). The equivariant KK-group KK0 (A, B) is deﬁned as the set of equivalence classes of G-equivariant cycles for (A, B) modulo homotopy. It is an Abelian group with respect to the operation of direct sum. The identity element is represented by the zero cycle (0, 0) and in fact by every degenerate cycle as there is an obvious homotopy to the zero cycle. The inverse of a cycle (E, T) is represented by its opposite. This is the cycle (E op, ¡T) given by reversing the grading of the Hilbert module, negating the operator and precomposing the action of A with the grading operator. The higher KK-groups are then deﬁned by G G bp KKn (A, B) := KK0 (A Cn,0, B) G G bp KK¡n(A, B) := KK0 (A C0,n, B) P G for n N¥0, and this way we can consider KK¦ (A, B) as a Z-graded Abelian group. 1 1.3.5 REMARK. A cycle (E, T ) is a compact perturbation of another cycle (E, T) for (A, B) if 1 a(T ¡ T) P KB(E)(@a P A). 1 Such cycles are homotopic via the linear path from T to T (in the obvious sense) and hence G determine the same element in KK0 (A, B). The next lemma gives another useful means of identifying cycles. 1 1.3.6 LEMMA ([Bla98, Proposition 17.2.7]). Two cycles (E, T), (E, T ) for (A, B) with 1 ¦ a[T, T ]a ¥ 0 (mod KB(E)) P G for all a A determine the same element in KK0 (A, B). 1.3.7 REMARK. By averaging over the compact group G we can in fact assume that the operator T in an element [E, Φ, T] is G-equivariant. 1.3. EQUIVARIANT KK-THEORY 13

1.3.8 REMARK. We can always assume that the left action of A on E is essential. The general proof of this fact is quite involved (see [Bla98, Proposition 18.3.6]). However, if the submodule AE E is complemented (e.g. if B = C) then we have [E, T] = [E, PTP + (1 ¡ P)T(1 ¡ P)] = [PE, PTP] ` [(1 ¡ P)E, (1 ¡ P)T(1 ¡ P)] = [PE, PTP]. Here, P is the projection E Ñ AE, the ﬁrst identity is by compact perturbation, and the last cycle in the third expression is degenerate. The cycle (PE, PTP) is called the compression of [E, T] to the range of the projection P. We will now sketch how the individual properties of the KK-theory are realized in terms of these deﬁnitions, but concentrate on the aspects which we will use in this thesis and refer to the literature for details and proofs.

1.3.9 SKETCHOF PROPERTIES A AND B. Pullback and pushforward are implemented in terms of the same operations on Hilbert bimodules (Deﬁnition 1.2.10). Homotopy invariance follows from the homotopy equivalence relation we have quotiented out in the deﬁnition of the KK-groups. Ñ Indeed, if Φ : A C([0, 1], B) is a homotopy of morphisms Φ0 and Φ1 then the pushforward of an arbitrary cycle along Φ is a homotopy of the pushforwards of that cycle along Φ0 and Φ1 (similarly for pullbacks, see [Bla98, Proposition 17.9.1]).

1.3.10 SKETCHOF PROPERTY C. One first defines the composition Kasparov product bp G b G Ñ G D : KK0 (A, D) KK0 (D, B) KK0 (A, B). On the level of bimodules, the product is simply given by the interior tensor product over D, p but because there is no canonical embedding BB(F) Ñ BB(E b DF) (cf. Definition 1.2.7) the correct definition of the product operator is highly non-trivial. It is usually defined implicitly by a number of properties and it is a difficult theorem to show its well-definedness ([Kas88, Theorem 2.11]). In fact, unless one of the operators is zero the product cannot be explicitly computed in most situations. ¦ One now defines the exterior tensor product of a cycle (E, T) and a G-C -algebra D as (E bp D, T bp 1), considering D as a Hilbert G-bimodule over itself, and establishes that this assignment induces a G Ñ G bp bp homomorphism σD : KK0 (A, B) KK0 (A D, A D). This map is then used to define a more general Kasparov product by the formula # G bp b G bp Ñ G bp bp KK0 (A1, B1 D) KK0 (D A2, B2) KK0 (A1 A2, B1 B2) b ÞÑ bp x y σA (x) bp bp σB (y) 2 B1 D A2 1 where we have omitted the braiding isomorphisms swapping tensor product factors (cf. [Kas88, Theorem 2.12]). We will soon see how to extend this to the higher KK-groups. For the proof of graded commutativity of the exterior Kasparov product (Property D) we refer to [Kas81, Theorem 5.6].

1.3.11 SKETCHOF PROPERTIES E, F, G AND J. Let E be a graded Hilbert G-A-B-bimodule with ¦ compact A-action and Φ : A Ñ B a morphism of graded G-C -algebras. We set P G [E] := [E, 0], [Φ] := [BΦ] = [BΦ, 0] KK0 (A, B). The compactness assumption ensures that these assignments define valid cycles. Up to the operator, it is clear from the preceding sketch that exterior and interior tensor products are compatible with the respective Kasparov products and that pullbacks and pushforwards are implemented in terms of left and right composition Kasparov products. As we have not made precise the correct definition of the product operator we shall only say that the first assertion is true almost by definition (the zero operator is a 0-connection in the sense of [Bla98, Definition 18.4.1]) and refer to [Bla98, Examples 18.4.2, (a) and (b)] for a proof of the second fact. One similarly verifies that the 14 1. ANALYTIC K-THEORY AND K-HOMOLOGY exterior tensor product with 1D agrees with the tensoring homomorphism σD so that Property G holds almost by definition. The proofs of Properties H, I, K and L are not relevant for what follows and we refer to [Kas88, Theorem 2.9], [Bla98, Corollary 17.8.8], [Kas88, Remark after Definition 2.15] and [Bla98, Section 19.5 and Exercise 20.10.2] together with [BS89, Theorème 7.2], respectively.

1.3.12 SKETCHOF PROPERTY M. The induction homomorphism is deﬁned as follows H Ñ G H H ÞÑ H ¥ ¡ KK¦ (A, B) KK¦ (C0(X, A) , C0(X, B) ), [E, T] [C0(X, E) , T ] H for cycles with H-equivariant operator T (cf. Remark 1.3.7). Here, C0(X, E) denotes the subspace of H-invariant elements of the Hilbert bimodule. By analyzing [Kas88, Theorems 3.4 and 3.5] we see that this assignment is well-deﬁned and compatible with Kasparov product and unit elements (in fact, Kasparov deals with a more general situation, but in our case of compact groups one can simply choose c = 1 and S = T ¥ ¡ in the proof of [Kas88, Theorem 3.4]).

1.3.13 SKETCHOF PROPERTY N. One first proves that the canonical isomorphisms bp 1 1 Ñ 1 1 Cp,q Cp ,q Cp+p ,q+q , (1.3.14) Ñ Cp,q+1 Cp+1,q from Example 1.1.16 are compatible with each other in the sense that any combination of them yields identical pushforward and pullback homomorphisms in KK-theory ([Kas81, Theorem bp 5.3]). In particular these isomorphisms allow us to identify Cn+2,0 Cn,0 C1,1 and C0,n+2 bp P C0,n C1,1 for all n N¥0. G Consequently, it suffices to fix a single KK-equivalence in KK0 (C1,1, C) since it follows from the above considerations and graded commutativity that exterior Kasparov multiplication by any G G such element implements natural isomorphisms KK¦ KK¦+2 compatible with the Kasparov product. These are also compatible with induction since the latter commutes with exterior Kas- parov multiplication. Moreover, any such choice of KK-equivalence allows us to extend the Kasparov product to the higher KK-groups. To see this, note that our previous definition of the Kasparov product together with the canonical isomorphisms in (1.3.14) determines a map bp G bp bp b G bp bp 1 1 ÝÑD G bp bp bp 1 1 bp KK0 (A1 Cp,q, B1 D) KK0 (D A2 Cp ,q , B2) KK0 (A1 A2 Cp,q Cp ,q , B1 B2) ÝÑ G bp bp 1 1 bp KK0 (A1 A2 Cp+p ,q+q , B1 B2).

By repeatedly applying the inverse of the KK-equivalence we can reduce Cp+p1,q+q1 to either of the forms Cp+p1¡q¡q1,0 or C0,q+q1¡p¡p1 . Observe that this specializes to a product on all KK- groups. The KK-equivalence we shall ﬁx is the one induced by the imprimitivity bimodule coming from the canonical representation W1 of C1,1 on the exterior algebra of C (Example 1.2.15). Note that p it follows from the isomorphism of Hilbert bimodules Wp+q Wp b Wq (Lemma 1.2.17) that G G the induced higher periodicity isomorphisms KK¦ KK¦+2p can also be written as Kasparov P G multiplication by the element [Wp] KK0 (Cp,p, C). 1.3.15 REMARK. An equivalent picture of the higher KK-groups is given by the canonical isomorphisms G G bp ÝÑ G bp KKp (A, B) = KK0 (A Cp,0, B) KK0 (A, B C0,p) G G bp ÝÑ G bp KK¡p(A, B) = KK0 (A C0,p, B) KK0 (A, B Cp,0) 1.4. ANALYTIC K-THEORY 15 which are implemented by ﬁrst tensoring with the right-hand side Clifford algebra and then applying Clifford periodicity. Their inverses are given by an analogous recipe. They are of course compatible with the operations of KK-theory, and Clifford periodicity in this picture can be im- ¦ plemented by the dual imprimitivity bimodule Wp .

P G We will now connect KK-theory to the index theory of Fredholm operators. Let [E, T] KK0 (C, C). By averaging, we may assume that T is G-equivariant (Remark 1.3.7). We may also assume that the left action of C is unital (Remark 1.3.8). Note that in this case T is invertible modulo compacts, hence Fredholm. In fact it is also unitary modulo compacts, hence its index is zero. However this ¡ does not necessarily apply to its restriction T : E + Ñ E which of course is still Fredholm. G Ñ 1.3.16 LEMMA. The isomorphism KK0 (C, C) R0(G) is implemented by sending an element [E, T] with G-equivariant T and unital left action to the (graded) G-index of T, i.e. ¡ ¡ [E, T] ÞÑ indexG T := [ker(T : E + Ñ E )] ¡ [coker(T : E + Ñ E )]. ¡ In particular, if G = 1 then this is just the ordinary Fredholm index of T : E + Ñ E (under the identiﬁ- ÞÑ cation R0(1) Z, [W] dim W). PROOF. It is clear that this assignment is inverse to the isomorphism from Property K.

1.3.17 COROLLARY. Denote by incl the inclusion C A of the subalgebra spanned by the unit of a graded ¦ G-C -algebra (which is always ﬁxed by G and hence this map is G-equivariant). Then the composition

¦ G inclÝÑ G Ñ KK0 (A, C) KK0 (C, C) R0(G) sends an element [E, T] with G-equivariant T and unital left action to the G-index of T.

1.4. Analytic K-theory

¦ 1.4.1 DEFINITION (ANALYTIC K-THEORYFOR C -ALGEBRAS). Equivariant analytic K-theory of ¦ graded G-C -algebras is the functor K¦G := KK¦G(C, ¡), i.e.

G G K¦ (A) := KK¦ (C, A).

It follows from the properties of KK-theory presented in the last section that K¦G is a stable ho- ¦ mology theory on the category of graded G-C -algebras in the sense of [Bla98, Section 21.1] (with respect to semisplit extensions). We can extend (or rather specialize) this deﬁnition to topological G-pairs and coefﬁcients.

1.4.2 DEFINITION (ANALYTIC K-THEORYFORTOPOLOGICALPAIRS). Equivariant analytic K-theory ¦ ¦ G ¥ of G-pairs with coefficients in a graded G-C -algebra is the bivariant functor KG := K¡¦ C0, i.e. ¦ G z G z bp KG(X, Y; A) := K¡¦(C0(X Y, A)) = KK¡¦(C, C0(X Y) A). It is contravariant in (X, Y) and covariant in A. ¦ ¦ We write ϕ := (ϕ )¦ for the pullback of pairs and Φ¦ := (Φ¦)¦ for the pushforward of coef- ¦ ficients. In fact, we will also use the notation ϕ for arbitrary pushforwards along the pullback ¦ morphism ϕ in KK-theory. ¦ 1.4.3 THEOREM. Equivariant K-theory with coefficients in a graded G-C -algebra is Z-graded generalized cohomology theory on the category of G-pairs which can also be considered as a Z/2-graded theory.

PROOF. Combine Theorem 1.3.1 with the natural short exact sequences from Corollary 1.1.10. 16 1. ANALYTIC K-THEORY AND K-HOMOLOGY

Every inclusion incl: U X of an open G-invariant subset induces a morphism incl! : C0(U) ¦ C0(X) of G-C -algebras which is given by extension by zero. Note that for any G-pair (X, Y) and U := XzY this is precisely the second map in the canonical short exact sequence (1.1.11): z 0 / C0(X Y) = C0(U) / C0(X) / C0(Y) / 0. More generally, for every inclusion incl: (U, V) (X, Y) of G-pairs where UzV is a G-invariant z z z open subset of X Y we write incl! for the corresponding morphism C0(U V) C0(X Y).

Let us also denote by incl! := (incl!)¦ the corresponding pushforward in KK-theory, in particular in K-theory. The shriek symbol signiﬁes that the map appears to go “in the wrong way”. We will later see that this map can be interpreted as a Gysin map (Lemma 1.15.2).

1.4.4 EXAMPLE (DISJOINTUNION). Let (X, Y) be the disjoint union of G-pairs (X1, Y1) and (X2, Y2). Denote by incli the respective inclusion maps. Then we have a decomposition z z ` z C0(X Y) C0(X1 Y1) C0(X2 Y2) ¦ where the inclusions are given by (incli)! (extension by zero) and the projections by incli (restriction to the respective component). By additivity of the KK-functor (Property H) and contravari- ance we get a similar decomposition ¦ ¦ ` ¦ KG(X, Y; A) KG(X1, Y1; A) KG(X2, Y2; A) where the inclusions and projections are induced by the above.

1.4.5 REMARK (ONE-POINT COMPACTIFICATION). It follows from the axioms that we can express the K-theory of a non-compact G-space X in terms of its one-point compactiﬁcation X+. Indeed, consider the short exact sequence of G-C+-algebras associated to the pair (X+, 8): ¦ + incl 8 0 / C0(X, A) / C(X , A) / C( , A) / 0

Here, the second arrow is extension of a function in C0(X) by zero and the third is induced by the inclusion incl: 8 P X+ of the point at inﬁnity. This sequence is right split by the pullback along the collapse map collapse: X+ Ñ 8 so that from the long exact sequence of K-theory we ﬁnd that ¦ + ¦ ` ¦ 8 ¦ ` ¦ KG(X ; A) KG(X; A) KG( ; A) KG(X; A) KG(A) and in particular that ¦ ¦ ¦ + Ñ ¦ 8 KG(X; A) ker(incl : KG(X ; A) KG( ; A)).

We will now discuss a convenient representation of the analytic K-theory of a (trivially graded) ¦ unital G-C -algebra A in terms of ﬁnitely generated, projective (right) A-modules, following [Phi87, Section 2.2]. Recall that, algebraically, every ﬁnitely generated A-module P is but a direct summand in some An so that there exists an A-linear projection p with range P. Let us equip An with its canonical topology as a Hilbert A-module. It turns out that the induced topology on P does not depend on its embedding as a direct summand. Furthermore, every A-linear map (which corresponds to left multiplication by a matrix in Mn¢n(A)) is continuous (see Example 1.2.2). Thus, in particular the projection p is continuous, hence has closed range, and P is automatically a closed direct summand.

1.4.6 DEFINITION (FINITELY GENERATED, PROJECTIVE G-A-MODULE).A ﬁnitely generated, projective G-A-module is a ﬁnitely generated, projective A-module P (equipped with its canonical topology as described above) with a left action of the group G by C-linear bounded operators ¢ such that the action G Ñ B(P) is strongly continuous and g(va) = g(v)g(a)(@g P G, v P P, a P A). 1 A morphism of two such modules P, P is a G-equivariant A-linear map. It is automatically continuous by the above argument. We get an additive category using the direct sum (with diagonal G-action). 1.4. ANALYTIC K-THEORY 17

The following proposition due to Julg is the equivariant version of our above discussion.

¦ 1.4.7 PROPOSITION.[ Bla98, Proposition 11.2.3] Let A be a trivially graded unital G-C -algebra. Then every ﬁnitely generated, projective G-A-module is a closed direct summand in a (trivially graded) free Hilbert G-A-module W b A (in the sense of Example 1.2.5).

1.4.8 DEFINITION (PUSHFORWARD). The pushforward of a finitely generated, projective G-A-module ¦ P along a unital morphism Φ : A Ñ B of G-C -algebras is the algebraic interior tensor product b Φ¦(P) := P Φ B with the diagonal G-action and right B-action on the second factor. b Since Φ is assumed to be unital, A Φ B B, so the preceding proposition shows that Φ¦(P) is automatically a finitely generated, projective G-B-module. As in Definition 1.2.8 we get a functor Φ¦ by sending morphisms T to T b 1.

1.4.9 DEFINITION (EXTERIORTENSORPRODUCT). If P and Q are ﬁnitely generated, projective G-modules over A and B, respectively, then their algebraic exterior tensor product P b Q with the diagonal G-action is a direct summand in some (V b W) b (A b B) (V b W) b (A b B). Its completion, the exterior tensor product P b Q, is therefore a ﬁnitely generated, projective G-A b B- module. It does not depend on the choice of embedding as a direct summand.

1.4.10 REMARK. Using Proposition 1.4.7 we can equip every ﬁnitely generated, projective G-A- module P with a Hilbert G-A-module structure. Note that we can also assume that the direct sum decomposition is orthogonal (use 1 ¡ p for the complement if p is the projection onto P W b A).

Moreover, it is clear that, using this Hilbert module structure, any pushforward Φ¦(P) in the sense just deﬁned is already complete with respect to its pushforward (pre-)Hilbert module structure (it is in fact non-degenerate), and thus agrees with the ordinary pushforward of Hilbert modules as deﬁned earlier. Similarly, our construction of the exterior tensor product shows that it agrees with the exterior tensor product in the sense of Hilbert modules. Hence there is no confusion in our notation.

Moreover, it is clear that every morphism Φ : P Ñ Q has an adjoint (again from the “matrix” representation) and if it is an isomorphism then it can be made isometric by replacing it with ¦ ¡ ¦ Φ(Φ Φ) 1/2 (which is still G-equivariant because it is contained in the C -algebra generated ¦ by Φ and Φ ). This shows that the Hilbert module structure on P is canonical up to isometric isomorphism.

Let us formalize the preceding and connect it to equivariant KK-theory.

1.4.11 DEFINITION (COEFFICIENT CATEGORY, ALGEBRAIC K-FUNCTOR). The coefﬁcient category is ¦ the category of trivially graded unital G-C -algebras (coefﬁcient algebras) with unital morphisms.

For any such algebra A, the set FinGenProjG(A) of isomorphism classes of finitely generated, projective G-A-modules is an Abelian semigroup with respect to the direct sum. Together with the pushforward we get a functor FinGenProjG from the coefficient category to Abelian semigroups. By the argument in the preceding remark we get an isomorphic functor if we instead take the isometric isomorphism classes of finitely generated, projective Hilbert G-A-modules.

The G-equivariant algebraic K-functor KG is then deﬁned as the composition of the Grothendieck functor with FinGenProjG.

1.4.12 THEOREM ([Jul81, Theorem 2.3], [Kas88, Theorem 3.11]). The algebraic K-functor KG is natu- G rally isomorphic to the analytic K-theory functor K0 on the coefﬁcient category, via the natural homomorphism sending an equivalence class [P] to the cycle [P, 0] (after equipping P with its essentially canonical Hilbert G-C(X, A)-module structure). 18 1. ANALYTIC K-THEORY AND K-HOMOLOGY

1.5. Topological K-theory

Atiyah and Hirzebruch’s topological K-theory of a compact space X is defined as the Grothen- dieck group of isomorphism classes of complex vector bundles over X (see [AH59]). It is well- 0 known that this group is isomorphic to the analytic K-theory group K (X) = KK0(C, C0(X)). Indeed, a famous theorem of Swan from [Swa62] asserts that the section functor implements an equivalence between the category of complex vector bundles over X and the category of finitely generated, projective C(X)-modules, and the Grothendieck group of the latter category is in turn isomorphic to K0(X) by Theorem 1.4.12. In the literature one finds the following generalizations of this result: If one uses G-equivariant vector bundles then the topological description of equivariant K-theory from [Seg68] is recovered, and if one instead uses locally trivial bundles of finitely generated, projective A-modules as in [MF79, Sch05] then the resulting Grothendieck group is isomorphic to K-theory with coefficients in A. The purpose of this section is to generalize Swan’s theorem to the case of equivariant K- 0 theory with coefficients. We will then get a topological description of the analytical K-functor KG on G-spaces with opposite coefficients which will be one of the building blocks for our definition of geometric K-homology cycles in Chapter 2. ¦ In this section, A will always be a coefficient algebra, i.e. a trivially graded unital G-C -algebra. The following is our generalization of the concept of a vector bundle.

1.5.1 DEFINITION (G-A-VECTORBUNDLE).A G-A-vector bundle (or finitely generated, projective G- A-module bundle) over a G-space X is a G-space E (the total space) together with a surjective G-map πE : E Ñ X (the projection) such that the following conditions are satisfied: ¡1 t u – every fiber Ex := πE ( x ) carries the structure of a finitely generated, projective A-module, – the group acts fiberwise by C-linear bounded operators, Ñ ¢ P – we have local trivializations ψU : E U U which are fiberwise A-linear isomorphisms (where Pi is a finitely generated, projective A-module), and – g(va) = g(v)g(a) for all g P G, v P E, a P A.

A morphism of G-A-vector bundles E, F over X is a G-map φ : E Ñ F over the identity, i.e. πF ¥ φ = πE, which is fiberwise A-linear. The direct sum of G-A-vector bundles is defined in the obvious way, and we get an additive category. Note that we recover the notions of a G-vector bundle and of a finitely generated, projective A- module bundle if we take A = C or G = 1. The following lemma shows that G-A-vector bundles and their morphisms are also vector bundles and morphisms in the sense of [Lan01]: EMMA Ñ ¢ P Ñ ¢ Q 1.5.2 L . Let E be a G-A-vector bundle over X. If ψU : E U U and ψV : E V V are X Ñ two local trivializations then the associated transition function U V BA(P, Q) is norm continuous. Ñ Ñ ¢ R Moreover, if φ : E F is a morphism of G-A-vector bundles, ψW : F W W a local trivialization Ñ of F and φ(U) W then the induced map U BA(P, R) is also norm continuous.

PROOF. It suffices to verify the following assertion: For every continuous map f : U ¢ P Ñ Q where P and Q are finitely generated, projective A-modules and which is A-linear on each fiber P Ñ ÞÑ ¡ (i.e. for each fixed x U), the induced map F : U BA(P, Q), x f (x, ) is norm continuous. Indeed, using projectivity we may assume that P = An. Then we have the representation ¸n x ¡y F(x) = f (x, ei) ei, i=1 x¡ ¡y from which it is immediate that F is continuous (here, ei = (δi,j1A) and , is the canonical A-valued inner product of An). 1.5. TOPOLOGICAL K-THEORY 19

1.5.3 DEFINITION (HILBERT G-A-VECTORBUNDLE).A Hilbert G-A-vector bundle is a G-A-vector bundle with the following additional requirements: – each fiber is a finitely generated, projective Hilbert A-module, – for any two sections s, t the induced map x ÞÑ xs(x), t(x)y is continuous, and – g(xv, wy) = xg(v), g(w)y for all g, v, w. A morphism of Hilbert G-A-vector bundles is defined just as above, direct sums are equipped with the obvious inner product rendering the summands orthogonal, and again we get an additive category. Observe that in the case of Hilbert G-vector bundles where A = C we recover the notion of a Hermitian G-vector bundle with isometric G-action. The following lemma shows that the additional structure makes no difference on the level of isomorphism classes (the forgetful functor implements a category equivalence):

1.5.4 LEMMA. Every G-A-vector bundle can be equipped with the structure of a Hilbert G-A-vector bundle. Moreover, if two Hilbert G-A-vector bundles are isomorphic then they are also isometrically isomorphic (i.e. there exists an isomorphism which is ﬁberwise isometric).

PROOF. Let E be any G-A-vector bundle. Using convexity of the space of A-valued inner products and a partition of unity we can construct A-valued inner products [¡, ¡] on the ﬁbers of E satisfying the ﬁrst two requirements. After averaging over the compact group G via the formula » ¡ xv, wy := g 1([g(v), g(w)])dg G all three requirements hold. The second assertion is proved in [Sch05, Lemma 3.12].

1.5.5 REMARK (SMOOTHSTRUCTURE). If M is a smooth manifold then one can also introduce the notion of a smooth structure on a (Hilbert) G-A-vector bundle over M. It consists of an atlas of local X Ñ trivializations such that the corresponding transition functions U V BA(P, Q) are smooth functions with values in the Banach space BA(P, Q). In [Sch05, Theorem 3.13] it is proved by an approximation argument that we can arrange for every (isometric) isomorphism to be smooth, so up to (isometric) isomorphism there is at most one smooth structure on every (Hilbert) G-A- vector bundle. In this thesis we will only need smooth structures in the classical case A = C where one recovers the notion of a smooth (Hermitian) G-vector bundle (with isometric G-action).

1.5.6 DEFINITION (SECTION). We denote by Γ0(E) the Banach space of continuous sections vanishing at infinity, endowed with the supremum norm. It is an (algebraic) G-C(X, A)-module with ¡ respect to the left G-action defined by the formula g(s)(x) := g(s(g 1x)). Every morphism φ : E Ñ F of Hilbert G-A-vector bundles over a G-space X induces a G-equi- Ñ variant C(X, A)-linear map Γ0(φ) : Γ0(E) Γ0(F) by postcomposition, and this assignment is functorial. If the base space is compact, every continuous section vanishes at infinity and we write just Γ.

1.5.7 LEMMA. The space of sections of a Hilbert G-A-vector bundle over a compact space is a ﬁnitely generated, projective G-C(X, A)-module.

PROOF. Let E be a Hilbert G-A-vector bundle. As in the proof of [Sch05, Proposition 3.17] we see that Γ(E) is a ﬁnitely generated, projective C(X, A)-module. One then proves as in the case of G-vector bundles that every element acts by bounded C-linear operators (see e.g. [Phi87, 2.3.1]). This is essentially due to the operator norm of the action of a group element varying continuously 20 1. ANALYTIC K-THEORY AND K-HOMOLOGY with the ﬁber and the base space being compact. For strong continuity, write the orbit map of a section s P Γ(E) C(X, E) as a composition ¥¡ ¢ (s ) idC(E,E) ¥ G / C(X, X) ¢ C(E, E) / C(X, E) ¢ C(E, E) / C(X, E) which in fact takes values in Γ(E). If we equip all function spaces with the compact-open topology, this is a composition of continuous maps, and hence continuous (cf. proof of Proposition 1.1.7). But the compact-open topology on C(X, E) is induced by the supremum norm and thus restricts to the correct topology on Γ(E). This shows that the action is strongly continuous.

Thus we can think of Γ as a functor from Hilbert G-A-vector bundles over a ﬁxed compact space X to ﬁnitely generated, projective G-C(X, A)-modules.

1.5.8 EXAMPLE (FREETRIVIALBUNDLE). For every G-space X and every ﬁnite-dimensional unitary G-representation W we get a Hilbert G-A-vector bundle X ¢ (W b A), endowed with the diagonal G-action. We will refer to it as a free trivial Hilbert G-A-vector bundle (cf. Example 1.2.5). Its module of sections is isomorphic to C(X, W b A) equipped with the G-action given by the ¡ formula g( f )(x) = g( f (g 1(x))).

1.5.9 LEMMA. Let E be a Hilbert G-A-vector bundle E over a compact G-space X and x P X. Then the evaluation map evalx : s ÞÑ s(x) induces an A-linear isomorphism of Banach A-modules

Γ(E)/Γx(E) Ex where Γx(E) := ts P Γ(E) : s(x) = 0u is the associated vanishing submodule. Moreover, Γx(E) = Γ(E)Ix where Ix C(X) is the ideal of functions vanishing at x.

PROOF. It is clear that all three modules are Banach A-modules. Since the evaluation map is a bounded A-linear operator with kernel Γx(E), it induces an A-linear isomorphism Γ(E)/Γx(E) ran(evalx). Using local triviality and the existence of bump functions one can show that evalx indeed surjects onto the ﬁber Ex. For the last claim we may use local triviality and a partition of unity to reduce to the case of trivial E. In fact, using Proposition 1.4.7 and additivity we may assume that E = X b A. Now note that, ¦ by nuclearity of C, the short exact sequence of C -algebras

evalx 0 / Ix / C(X) / C / 0 remains exact after tensoring with A (Proposition 1.1.3). Moreover, it does not matter whether we ¦ ¦ tensor the individual C -algebras in the sense of C -algebras or in the sense of Hilbert modules (Example 1.2.6). Consequently, Γx(E) Ix b A, and we conclude from

Γx(E) Γ(E)Ix (C(X) b A)Ix = Ix b A that Γ(E)Ix is indeed dense in Γx(E).

1.5.10 LEMMA. The section functor is fully faithful on the category of Hilbert G-A-vector bundles over a ﬁxed compact G-space X.

PROOF. For any two Hilbert G-A-vector bundles E and F over X, we have to consider the map Hom(E, F) Ñ Hom(Γ(E), Γ(F)), φ ÞÑ (s ÞÑ φ ¥ s) given by the section functor on the set of morphisms from E to F. This map is injective because using the preceding lemma we can find a section attaining any fixed vector v P Ex. For surjectivity, let Φ : Γ(E) Ñ Γ(F) be a morphism of G-C(X, A)-modules. The preceding lemma also shows that Φ sends the vanishing submodule Γx(E) to the corresponding vanishing submodule Γx(F) so that we get morphisms φx on each fiber. If we can show that for any trivializing open 1.5. TOPOLOGICAL K-THEORY 21

¢ P ¢ P1 set U X with E U U and F U U the induced map Ñ 1 ÞÑ ψU : U BA(P, P ), x φx is continuous then the totality of the maps φx defines a continuous map φ : E Ñ F which is easily seen to be G-equivariant and fiberwise A-linear, hence a morphism, and a preimage of Φ (use [Lan01, Proposition III.1.3]). Let K U be a compact neighborhood of an arbitrary point x P U. The preceding lemma shows in fact that Φ does not enlarge the support of sections, hence we can consider Φ as a G-C(K, A)- 1 module morphism C(K, P) Ñ C(K, P ). It is clear that the above procedure produces the same 1 maps φx. Since P and P are finitely generated projective A-modules we may also assume that 1 P = An and P = Am. Then the image of Φ under the isomorphism n m n m BC(K,A)(C(K, A ), C(K, A )) BC(K,A)(C(K, A) , C(K, A) ) C(K, A) b Mn¢m(C)

C(K) b A b Mn¢m(C)

C(K) b Mn¢m(A) b n m C(K) BA(A , A ) n m C(K, BA(A , A )) is precisely the restriction of ψU to K. This shows that ψU is continuous in x. Since x was an arbitrary point of U, ψU is continuous everywhere.

1.5.11 GENERALIZED SWAN THEOREM. The section functor Γ implements an additive equivalence of the category of Hilbert G-A-vector bundles over a compact G-space X and the category of ﬁnitely generated, projective G-C(X, A)-modules.

PROOF. We already know that Γ is a fully faithful additive functor mapping into the target category (Lemmas 1.5.7 and 1.5.10). Thus in order to show that Γ implements a category equivalence we only need to prove that it is essentially surjective. We will now do this, following the lines of the proof of [Sch05, Proposition 3.17]. Let P be a finitely generated, projective G-C(X, A)-module. By Proposition 1.4.7 there exists a finite-dimensional G-representation W such that P ` Q W b C(X, A) C(X, W b A) Γ(X ¢ (W b A)) for some complement module Q. Define bundles E := t(x, v) P X ¢ (W b A) : p(x) = v for some p P Pu X ¢ W b A, F := t(x, v) P X ¢ (W b A) : q(x) = v for some q P Qu X ¢ W b A where the projection is just onto the first component. Since Ex ` Fx = W b A, the fibers are canonically finitely generated, projective Hilbert A-modules, and as in the proof of [Sch05, Proposition 3.17] one shows that both bundles are Hilbert A-vector bundles and E ` F = X ¢ W b A. Both bundles are in fact Hilbert G-A-vector bundles. Indeed, they are invariant subsets under the diagonal G-action and thus inherit all desired properties from the free trivial Hilbert G-A-vector bundle X ¢ W b A (cf. Example 1.5.8). Moreover, note that Γ(E) t f P C(X, W b A) : @x P X : Dp P P : f (x) = p(x)u P, Γ(F) t f P C(X, W b A) : @x P X : Dq P Q : f (x) = q(x)u Q. But P ` Q Γ(E ` F), and since we already know that Γ is additive we conclude that the above inclusions have to be equalities, whence Γ(E) P.

The analogous statement holds if we consider the category of all G-A-vector bundles and/or the category of ﬁnitely generated, projective Hilbert G-C(X, A)-modules — this follows from our previous discussion in Remark 1.4.10 and Lemma 1.5.4. 22 1. ANALYTIC K-THEORY AND K-HOMOLOGY

1.5.12 COROLLARY. Every Hilbert G-A-vector bundle over a compact G-space is isometrically isomorphic to an orthogonal direct summand of a free trivial Hilbert G-A-vector bundle (in the sense of Example 1.5.8). The analogous statement holds for ordinary G-A-vector bundles.

We can now give a topological description of equivariant K-theory with coefﬁcients in terms of G-A-vector bundles, generalizing the descriptions in [Seg68, Phi87, Sch05].

1.5.13 DEFINITION (PULLBACK, PUSHFORWARD). The pullback of a G-A-vector bundle F over Y along a G-map ϕ : X Ñ Y is deﬁned by

¦ ϕ (F) := t(x, w) P X ¢ F : ϕ(x) = πF(w)u.

As in the case of vector bundles one shows that this is a G-A-vector bundle over X in a natural ¦ way. Note that if incl: X Y is the inclusion of a subspace then incl (E) is isomorphic to the restriction E X of the bundle to X. We can also construct the pushforward of a G-A-vector bundle E along a unital morphism Φ : A Ñ ¢ B. To do so, choose an open trivializing cover (Ui) for E such that E Ui Pi for some ﬁnitely Ui generated, projective A-modules Pi and denote by ψi,j the corresponding transition functions. ¢ Then glue together the trivial bundles Ui Φ¦(Pi) with the transition functions deﬁned by

X ÝÑψi,j ÝÑΦ¦ Ui Uj BA(Pi, Pj) BB(Φ¦(Pi), Φ¦(Pj)).

This works as in the case of ordinary vector bundles, and the resulting bundle is canonically a G-B-vector bundle.

Pushforward and pullback commute and respect direct sums and isomorphism. We can also perform the same constructions for Hilbert G-A-vector bundles.

1.5.14 REMARK. The pushforward construction can also be described more abstractly using the language of [Lan01]: It is the operation induced by the continuous functor Φ¦ sending ﬁnitely generated, projective (Hilbert) A-modules to their pushforward and a morphism T to the morphism T b 1.

1.5.15 DEFINITION (TOPOLOGICAL K-FUNCTOR). The set VectG(X; A) of isometric isomorphism classes of Hilbert G-A-vector bundles over a G-space X is an Abelian semigroup with respect to direct sums. We get a bifunctor VectG, contravariant in the space and covariant in the coefﬁcient algebra, if we send morphisms (ϕ; Φ) to the homomorphism induced by pullback along ϕ and pushforward along Φ. By Lemma 1.5.4, we get an isomorphic functor if we instead take the set of isomorphism classes of ordinary G-A-vector bundles.

The G-equivariant topological K-bifunctor KG is then deﬁned as the composition of the Grothen- dieck functor with VectG.

1.5.16 THEOREM. The section functor induces a natural isomorphism between the topological K-bifunctor 0 KG and the analytic K-theory bifunctor KG on the category of compact G-spaces with coefﬁcient algebras.

PROOF. The Generalized Swan Theorem 1.5.11 shows that the section functor induces an isomorphism VectG(X; A) FinGenProjG(C(X, A)) of Abelian semigroups. In particular, the corre- 0 sponding Grothendieck groups are isomorphic so that KG(X; A) KG(X; A) by Theorem 1.4.12. We will show naturality on the level of Abelian semigroups. That is, if (ϕ; Φ) : (X, A) Ñ (Y, B) ¦ is a morphism and if F is a Hilbert G-B-vector over Y then we want to show that Γ(Φ¦φ F) ¦ (ϕ b Φ)¦Γ(F). Let us ﬁrst consider the restriction of F to a trivializing open subset V X where 1.5. TOPOLOGICAL K-THEORY 23

¢ P = ¡1( ) F V V . Set U : ϕ V . Then ¦ ( ¦ ) ( P b ) Γ0 Φ φ F V C0 U, Φ A b b C0(U) (P Φ A) 1.2.9 b ¦ b b (C0(V) ϕ C0(U)) (P Φ A) b b ¦ b (C0(V) P) ϕ bΦ (C0(U) A) ¦ = ( b )¦ ( ) ϕ Φ Γ0 F V ¦ (ϕ b Φ)¦Γ(F). ¦ We can use this construction and a partition of unity to deﬁne a homomorphism Γ(Φ¦φ F) Ñ ¦ (ϕ b Φ)¦Γ(F). This homomorphism does not depend on the choice of partition of unity. More- over, it is an isomorphism if F is a free trivial bundle since then we can take U = X. The general case now follows from Corollary 1.5.12 and additivity.

We will always use this natural isomorphism to identify isomorphism classes of G-A-vector bundles E over compact G-spaces X with their images in analytic K-theory. To that end we deﬁne P 0 [E] := [Γ(E)] = [Γ(E), 0] KG(X; A) 0 ¡ Note that every element in KG(X; A) can be represented by a difference bundle [E] [F], but of course this representation is not unique. Let us also remark that if incl: X Y denotes the inclusion of a union of connected components of a compact G-space Y then incl![E] is the K-theory element determined by the G-A-vector bundle given by extending E by zero to all of Y.

1.5.17 REMARK (VECTORBUNDLESOVERNON-COMPACT SPACES). If X is not a compact G-space then this definition does not make sense, as in that case [Γ0(E), 0] is not a valid KK-cycle since the identity operator is not compact. This reflects the fact that our analytical version of K-theory for topological spaces is compactly supported. The philosophically correct choice would be to use G equivariant representable K-theory RKK0 (X; C, A). This is the natural analytical theory in which every G-A-vector bundle determines a class (cf. [Seg70, Kas88]). Emerson and Meyer have in- vestigated the correct definition of representable K-theory for very general groupoid-equivariant situations as well as when this theory is isomorphic to the Grothendieck group of isomorphism classes of G-vector bundles ([EM09d, Sections 5–6]). In this thesis we have chosen not to use representable K-theory so as not to introduce additional technical machinery (although the reader acquainted with the theory will see that its use is implicit in several constructions). There is however another way to get a topological description of the K-theory of non-compact spaces. Indeed, from our discussion in Remark 1.4.5 we know that 0 ¦ 0 + Ñ 0 8 KG(X; A) ker(incl : KG(X ; A) KG( ; A)). Since pulling back along the inclusion map 8 X+ corresponds to restricting bundles over X+ to their fiber at 8 (Definition 1.5.13), the K-theory 0 t ¡ P 0 + u KG(X; A) [E] [F] KG(X ; A) : [E 8] = [F 8] is isomorphic to the subgroup of formal differences of bundles whose fiber at infinity determines 0 the same class in KG(A). In the case of A = C and G = 1 this means precisely that the fibers at infinity have the same dimension so that we can think of the elements of K0(X) as those difference bundles with formal dimension zero at infinity. One can similarly describe the K-theory of an arbitrary compact G-pair (X, Y) as difference bundles over X which agree on Y, but we will not need that in the following. However, we need the following structure: 24 1. ANALYTIC K-THEORY AND K-HOMOLOGY

1.5.18 DEFINITION (GRADEDVECTORBUNDLES).A graded G-A-vector bundle E is equipped with a decomposition E = E+ ` E¡ into two subbundles. In the case of Hilbert G-A-vector bundles we also require that both subbundles be orthogonal. Then Γ0(E) inherits a grading and we can deﬁne ` op ¡ P 0 (1.5.19) [E] := [Γ(E)] = [Γ(E+) Γ(E¡) ] = [E+] [E¡] KG(X; A) if the base space X is compact. We will only need this in the case of trivial coefﬁcients.

Finally, let us generalize the tensor product to G-A-vector bundles.

1.5.20 DEFINITION (TENSORPRODUCT). Let E be a G-A-vector bundle and F a G-B-vector bundle over the same base space. Their tensor product is the G-A b B-vector bundle E b F defined in the obvious way by tensoring fibers in local trivializations and transition functions, similarly to how we constructed the pushforward (Definition 1.5.13). One shows by the usual arguments that the tensor product is associative, distributive with respect to direct sums and respects isomorphisms. Again, it is clear that we can perform the same constructions in the “category” of Hilbert G-A-vector bundles.

1.5.21 DEFINITION (CUPPRODUCT). The tensor product induces a product on the K-theory of a compact space X in the following way: Y 0 b 0 Ñ 0 b b Ñ b : KG(X; A) KG(X; B) KG(X; A B), [E] [F] [E F]. We will call it the cup product of K-theory in order to distinguish it from the exterior Kasparov product and to stress its formal similarity with the cup product in cohomology. 0 Note that the cup product induces on KG(X) the structure of a commutative ring with unit ¢ 0 0 [X C] and KG(X; A) becomes a two-sided KG(X)-module. We postpone a more careful study of the product structure to Section 1.7 where we will have introduced K-homology, and conclude by deriving an analytic expression for the cup product (again, note the formal similarity to cohomology).

1.5.22 PROPOSITION. The cup product can be implemented by ﬁrst taking the exterior Kasparov product and then pulling back along the diagonal map ∆ : X Ñ X ¢ X, i.e. Y ¦ bp @ P 0 P 0 x y = ∆ (x y)( x KG(X; A), y KG(X; B)). Here, bp denotes the exterior Kasparov product.

PROOF. It sufﬁces to verify the formula for classes represented by bundles. Using distributivity and Corollary 1.5.12, we can reduce to the case of free trivial bundles, say E = X ¢ P and F = X ¢ Q. Then the assertion follows from the isomorphism Γ(E b F) = Γ(X ¢ P b Q) = C(X, P b Q) ¦ (∆ )¦(C(X ¢ X)) b P b Q ¦ b b (∆ idAbB)¦(C(X, P) C(X, Q)) ¦ b b = (∆ idAbB)¦(Γ(E) Γ(F)).

It is clear from the last representation in (1.5.19) that this proposition also holds for graded vector bundles if we grade the tensor product bundle in the obvious way. 1.7. PRODUCTS 25

1.6. Analytic K-homology

In this section we will introduce analytic K-homology and derive some basic results which are in some sense dual to those of Section 1.4. ¦ 1.6.1 DEFINITION (ANALYTIC K-HOMOLOGYFOR C -ALGEBRAS). Equivariant analytic K-homolo- ¦ ¦ G ¡ gy of graded G-C -algebras is the functor KG := KK¡¦( , C), i.e. ¦ G KG(A) := KK¡¦(A, C). ¦ It follows from the properties of KK-theory presented in the last section that K is a stable coho- ¦ G mology theory on the category of G-C -algebras in the sense of [Bla98, Section 21.1] (with respect to semisplit extensions). The correct deﬁnition for topological G-pairs and coefﬁcients however is the following:

1.6.2 DEFINITION (ANALYTIC K-HOMOLOGYFORTOPOLOGICALPAIRS). Equivariant analytic K- ¦ homology of G-pairs with coefficients in a graded G-C -algebra is the bivariant functor K¦G := ¡¦ ¡ ¡ KKG (C0( ; C); ), i.e. G G z K¦ (X, Y; A) := KK¦ (C0(X Y); A). ¦ ¦ It is covariant in both variables, and we simply write ϕ¦ := (ϕ ) and Φ¦ for the pushforward of pairs and coefficients. We will also use the notation ϕ¦ for arbitrary pullbacks along the pullback ¦ morphism ϕ in KK-theory. ¦ 1.6.3 THEOREM. Equivariant K-homology with coefficients in a graded G-C -algebra is Z-graded generalized homology theory on the category of G-pairs which can also be considered as a Z/2-graded theory. This theorem is proved similarly to its dual, Theorem 1.4.3, by collecting the relevant properties of KK-theory and the C0-functor.

Recall that for certain inclusion maps incl we have deﬁned incl! to denote both the morphism given by extension of functions by zero and the induced pushforward in KK-theory. Let us also ! ¦ deﬁne incl := (incl!) to denote the corresponding pullback in KK-theory, in particular in K- homology.

1.6.4 EXAMPLE (DISJOINTUNION). Let (X, Y) be the disjoint union of G-pairs (X1, Y1) and (X2, Y2). Again denote by incli the respective inclusions. Recall from Example 1.4.4 that we have a decomposition z z ` z C0(X Y) C0(X1 Y1) C0(X2 Y2) ¦ where the inclusions are given by (incli)! and the projections by incli . By additivity of the KK- functor (Property H) and covariance we get a similar decomposition G G ` G K¦ (X, Y; A) K¦ (X1, Y1; A) K¦ (X2, Y2; A) ! where, dually to Example 1.4.4, the projections and inclusions are given by incli and (incli)¦.

1.7. Products

In Section 1.5 we have defined a cup product Y: K0(X; A) b K0(X; B) Ñ K0(X; A bp B) for the 0-th equivariant K-theory group over a compact G-space X with coefficients in a trivially graded, uni- ¦ tal G-C -algebra. We shall use the analytical description from Proposition 1.5.22 to generalize the cup product to all K-theory groups of not necessarily compact G-pairs and arbitrary coefficients. ¦ 1.7.1 DEFINITION (CUPPRODUCT). Let (X, Y), (X, Z) be G-pairs and A, B be graded G-C -algebras. We define the cup product by # ¦ ¦ ¦ K (X, Y; A) b K (X, Z; B) Ñ K (X, Y Y Z; A bp B) Y G G G : ¦ x Y y := ∆ (x bp y). 26 1. ANALYTIC K-THEORY AND K-HOMOLOGY

Here, ∆ : (X, Y Y Z) Ñ (X, Y) ¢ (X, Z) is the diagonal map (cf. Remark 1.1.9). In the following we will denote all diagonal maps by ∆. This is of course analogous to the usual deﬁnition of the cup product in terms of the cross product (whose role in K-theory is assumed by the exterior Kasparov product).

1.7.2 LEMMA. The cup product is associative and graded commutative. If XzZ is compact, then cup z ¢ P 0 z product with [(X Z) C] KG(X, Z), the class determined by the trivial vector bundle over X Z, is the ¦ identity on KG(X, Y; A) for Z Y. PROOF. In the following computations we will use several of the properties from Theorem 1.3.1. P ¦ P ¦ P ¦ We ﬁrst prove associativity: For all x KG(X, Y; A), y KG(X, Z; A), z KG(X, W; A) we have X X ¦ ¢ ¦ bp bp (x y) z = ∆ (∆ id(X,W)) (x y z) ¦ = (x ÞÑ (x, x, x)) (x bp y bp z) ¦ ¢ ¦ bp bp = ∆ (id(X,Y) ∆) (x y z) = x X (y X z). Graded commutativity is of course inherited from the exterior Kasparov product. We will now z ¢ z P ¦ show that right cup product with [(X Z) C] = [C(X Z)] is the identity: For all x KG(X, Y; A) we have ¦ x Y [C(XzZ)] = ∆ (x bp [C(XzZ)]) p p ¦ = x b z [C (XzY) b C(XzZ)] b z b z [ ] C0(X Y) 0 C0(X Y) C(X Z) ∆ p ¦ = x b z [( )¦(C (XzY) b C(XzZ))] C0(X Y) ∆ 0 . ¦ z b z Ñ z Note that ∆ is the multiplication morphism C0(X Y) C(X Z) C0(X Y). It is essential, even z surjective. Thus by Lemma 1.2.9 the right-hand side pushforward bimodule is simply C0(X Y) 1 z considered as a bimodule over itself. In other words, it represents the unit element C0(X Y) and thus x Y [(XzZ) ¢ C] = x. 1 1 1 1 1.7.3 LEMMA. Let Y, Z X and Y , Z X be closed, G-invariant subspaces. If ϕ : X Ñ X is a proper 1 1 G-map sending Y and Z to Y and Z , then ¦ Y ¦ Y ¦ @ P ¦ 1 1 P ¦ 1 1 ϕ (x y) = (ϕ x) (ϕ y)( x KG(X , Y ; A), y KG(X , Z ; B)). 1 1 ¦ 1 1 Moreover, if XzZ and X zZ are compact then f [(X zZ ) ¢ C] = [(XzZ) ¢ C].

PROOF. The ﬁrst assertion is proved by the computation ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ϕ (x Y y) = ϕ ∆ (x bp y) = ∆ (ϕ ¢ ϕ) (x bp y) = ∆ (ϕ x bp ϕ y) = (ϕ x) Y (ϕ y). ¦ 1 1 Now observe that the pullback morphism ϕ : C(X zZ ) Ñ C(XzZ) is essential (in fact, unital). The second assertion thus follows by another application of Lemma 1.2.9. ¦ 1.7.4 COROLLARY. KG(X, Y) is a graded commutative ring with respect to the cup product, and unital if z ¦ ¦ X Y is compact. KG(X, Y; A) is a two-sided graded KG(X, Z)-module for Z Y. Moreover, pullbacks induce (unital) ring and module homomorphisms, respectively. ¦ ¡ In particular, we can consider KG( ; C) as a functor from G-pairs to graded commutative rings.

There is also a pairing between K-theory and K-homology which plays the role of the cap product in ordinary (co)homology theory:

1.7.5 DEFINITION (CAPPRODUCT). Let Y, Z X be closed, G-invariant subspaces and A, B be ¦ graded G-C -algebras. We deﬁne the cap product by # ¦ G G p K (X, Y; A) b K¦ (X, Y Y Z; B) Ñ K¦ (X, Z; A b B) X G : ¦ p p x X y = (x b 1 z ) b z Y y : ∆ C0(X Z) C0(X (Y Z)) . 1.7. PRODUCTS 27

It can be more conceptually deﬁned in terms of Kasparov’s representable K-theory but we will not make this precise (cf. Remark 1.5.17). In the following we will mostly need the version X ¦ b G Ñ G : KG(X; A) K¦ (X, Y) K¦ (X, Y; A) which we remark has no interpretation as a module multiplication except in the case of trivial coefﬁcients. However we have the following familiar “naturality” statements which we will prove in the general case.

1.7.6 LEMMA. Cap and cup products are compatible in the sense that Y X X X @ P ¦ P ¦ P G Y Y (x y) z = x (y z)( x KG(X, Y; A), y KG(X, Z; B), z K¦ (X, Y Z W; C)). z z ¢ P 0 G If X Z is compact then cap product with [(X Z) C] KG(X, Z) is the identity on K¦ (X, Y; A) for Z Y. ¦ G In particular, the cap product induces the structure of a left KG(X, Y)-module on K¦ (X, Y; A). PROOF. The ﬁrst assertion is proved by the following computation: ¦ ¦ p p p (x Y y) X z = ( (x b y) b 1 z ) b z Y Y z ∆ ∆ C0(X W) C0(X (Y Z W)) ¦ ¦ p p p = ( ¢ ) (x b y b 1 z ) b z Y Y z ∆ ∆ id(X,W) C0(X W) C0(X (Y Z W)) ¦ p p p = (x ÞÑ (x x x)) (x b y b 1 z ) b z Y Y z , , C0(X W) C0(X (Y Z W)) ¦ p p p = (x ÞÑ (x x x)) (x b 1 z b y) b z Y Y z , , C0(X W) C0(X (Y Z W)) ¦ ¦ p p p = ( ¢ ) (x b 1 z b y) b z Y Y z ∆ ∆ id(X,Z) C0(X W) C0(X (Y Z W)) ¦ ¦ p p p = ( (x b 1 z ) b y) b z Y Y z ∆ ∆ C0(X W) C0(X (Y Z W)) ¦ ¦ p p p p = ( (x b 1 z ) b z Y (y b 1 z Y )) b z Y Y z ∆ ∆ C0(X W) C0(X (Y W)) C0(X (Y W)) C0(X (Y Z W)) ¦ p p ¦ p p = (x b 1 z ) b z Y (y b 1 z Y ) b z Y Y z ∆ C0(X W) C0(X (Y W))∆ C0(X (Y W)) C0(X (Y Z W)) = x X (y X z). Note that we have mainly used functoriality of pullbacks and commutativity of the exterior Kas- parov product, similar to the proof of associativity (but more heavy on notation). z ¢ z The second assertion is another consequence of the fact that the pushforward of C0(X Y X Z) ¦ G z z along the multiplication morphism ∆ induces the unit element of KK0 (C0(X Y), C0(X Y)). We have seen that this is true in the proof of Lemma 1.7.2.

We also have the following version of the projection formula. 1 1 1 1 1.7.7 LEMMA. Let Y, Z X and Y , Z X be closed, G-invariant subspaces. If ϕ : X Ñ X is a proper 1 1 G-map sending Y and Z to Y and Z , then ¦ X X @ P ¦ 1 1 P G Y ϕ¦(ϕ x y) = x ϕ¦y ( x KG(X , Y ; A), y K¦ (X, Y Z; B)). ¦ 1z 1 z PROOF. From Lemma 1.2.12 we know that the Hilbert bimodules ϕ (C0(X Z )) and ϕ¦(C0(X Z)) are isomorphic. Thus ¦ ¦ ¦ p p ¦( x X y) = ¦( ( x b 1 z ) b z Y y) ϕ ϕ ϕ ∆ ϕ C0(X Z) C0(X (Y Z)) ¦ ¦ p p = ( x b ¦1 z ) b z Y y ∆ ϕ ϕ C0(X Z) C0(X (Y Z)) ¦ ¦ p p = ( ¢ ) (x b 1 1z 1 ) b z Y y ∆ ϕ ϕ C0(X Z ) C0(X (Y Z)) ¦ ¦ p p = (x b 1 1z 1 ) b z Y y ϕ ∆ C0(X Z ) C0(X (Y Z)) ¦ p p = (x b 1 1z 1 ) b z Y ¦y ∆ C0(X Z ) C0(X (Y Z)) ϕ = x X ϕ¦y. 28 1. ANALYTIC K-THEORY AND K-HOMOLOGY

Let us also note that cup and cap products are natural with respect to coefﬁcients. 1 1 1.7.8 LEMMA. Let Y, Z X be closed, G-invariant subspaces and Φ : A Ñ A , Ψ : B Ñ B be morphisms ¦ of graded G-C -algebras. Then

(Φ bp Ψ)¦(x Y y) = Φ¦(x) Y Ψ¦(y), (Φ bp Ψ)¦(x X z) = Φ¦(x) X Ψ¦(z)

P ¦ P ¦ P G Y for all x KG(X, Y; A), y KG(X, Z; B), z K¦ (X, Y Z; B)). PROOF. This is clear from the deﬁnition of cup and cap product and exterior commutativity.

We will also need the following compatibility result (which will later allow us to restrict Poincaré duality to connected components).

1.7.9 LEMMA. Cap product for disjoint unions is a direct sum of the cap product of its summands. > > PROOF. Let (X, Y) = (X1, Y1) (X2, Y2) and (X, Z) = (X1, Z1) (X2, Z2), and let A, B be graded ¦ G-C -algebras. It sufﬁces to show that the cap product for (X, Y) restricts to the cap product of its summands. In view of Examples 1.4.4 and 1.6.4, this amounts to verifying the identities X X @ P ¦ P G Y (incl1)¦(x y) = (incl1)!x (incl1)¦y ( x KG(X1, Y1; A), y K¦ (X1, Y1 Z1; B)) and X @ P ¦ P G Y (incl1)!x (incl2)¦z = 0 ( x KG(X1, Y1; A), z K¦ (X2, Y2 Z2; B)). Here, inclk denotes the inclusion Xk X which of course sends Yk to Y and Zk to Z (k = 1, 2). The ﬁrst identity follows directly from the projection formula:

X ¦ X 1.7.7 X (incl1)¦(x y) = (incl1)¦(incl1 (incl1)!x y) = (incl1)!x (incl1)¦x. For the second identity, note that X X ! X ¦ ¥ ¦ (incl1)!x (incl2)¦z = (incl1)¦(x incl1(incl2)¦z) = (incl1)¦(x (incl2 (incl1)!) z) = 0.

We conclude with the following lemma which is useful for computing cap products with actual vector bundles.

1.7.10 LEMMA. Let (X, Y) be a compact G-pair and E a graded Hilbert G-vector bundle over X. Then

p G G [E] X x = [Γ (E )] b z x P K¦ (X, Y)(@x P K¦ (X, Y)). 0 XzY C0(X Y) ( ) ( z ) ( z ) Here, we consider Γ0 E XzY as a Hilbert G-C0 X Y -C0 X Y -bimodule.

PROOF. We claim that ¦ ( )¦( ( ) bp ( )) = ( ( ) bp ( )) bp ¦ ( z ) Ñ ( ) ( b ) b ÞÑ ∆ Γ E C X Γ E C X ∆ C0 X Y Γ0 E XzY , u f g f gu z z deﬁnes an isometric isomorphism of graded Hilbert G-C0(X Z)-C0(X Y)-bimodules. Indeed, the computation x b b b b y x ¦ x y b ¦ y ¦x y ¦ x y P z (u f ) g, (u f ) g = g, ∆ ( u, u f f )g = g u, u f f g = f gu, f gu C0(X Y) shows that the right-hand side assignment deﬁnes an isometry on the algebraic tensor product. This isometry evidently has dense range, and the assertion follows at once. 1.8. ELLIPTIC OPERATORS AND ANALYTIC K-HOMOLOGY 29

1.8. Elliptic operators and analytic K-homology

In [Ati70] Atiyah observed that elliptic first-order differential operators should determine cycles in K-homology and defined a theory Ell(X) surjecting onto the K-homology of X (which was defined purely homotopically by Whitehead in [Whi62]) by axiomatizing two of the main properties of elliptic operators, namely being Fredholm and commuting modulo compact operators with multiplication. It was among Kasparov’s main achievements in [Kas75] to find the correct equivalence relation such that this map becomes an isomorphism (as well as to generalize the ¦ theory to C -algebras), and today Kasparov’s analytic K-homology theory which we have defined in Section 1.6 is usually taken as the correct definition of K-homology. In this section we will describe precisely how elliptic operators define cycles in K-homology.

1.8.1 DEFINITION (G-MANIFOLD). A smooth G-manifold is a smooth oriented Riemannian manifold M which is equipped with a continuous action of a compact Lie group G by orientation- preserving isometries (i.e. isometric diffeomorphisms). In particular, a smooth manifold will always be a smooth oriented Riemannian manifold, i.e. a G-manifold for G = 1. Observe that if M is a smooth G-manifold with boundary (which we will always state explicitly) then the group action automatically preserves its boundary and interior. Both diffeomorphisms and embeddings ϕ : M Ñ N of G-manifolds will always be required to be G- equivariant. If we write ϕ : (M, BM) Ñ (N, BN) for an embedding of G-manifolds with boundaries then we will also require that ϕ(BM) = ϕ(M) XBN and that ϕ(M) is transverse to BN (this is called a neat embedding in [Hir76] and an embedding of type 1 in [tD00]). These conditions ensure that ϕ(M) always has a G-equivariant tubular neighborhood in N (cf. [tD87, Exercise 5 in I.5.18] and [Hir76, Section 4.6]). For now, let us assume that all G-manifolds are non-empty and that their connected components have positive dimension so as to avoid pathologies. We will later see how to remove these assumptions.

1.8.2 REMARK. Although the additional restriction to compact Lie groups provides us with powerful technical tools (such as equivariant tubular neighborhoods, the Slice Theorem or the Mos- tow-Palais Embedding Theorem), it is in a sense a rather mild one. Indeed, since the isometry group of a smooth Riemannian manifold is always a ﬁnite-dimensional Lie group, every continuous action of a compact topological group by isometries factors over a compact Lie group.

1.8.3 DEFINITION (DIFFERENTIAL OPERATOR). Let E be a smooth complex vector bundle over a smooth manifold M. A (first-order linear) differential operator acting on sections of E is a linear 8 8 operator D : Γ (E) Ñ Γ (E) which locally looks like a first-order linear differential operator on some Rn with smooth coefficients. ¦ Its (principal) symbol is the vector bundle morphism σD : T M Ñ End(E) which is uniquely P 8 P 8 defined by the formula σD(d f )u := i[D, mul f ]u for all f C (M) and u Γ (E)) (cf. [HR00, discussion before Definition 10.1.3]). The operator D is called elliptic if σD(ξ) is invertible for all non-zero cotangent vectors ξ. ³ If E is Hermitian then we can introduce an inner product by the formula xu, vy := xu, vydM 8 on the subspace of compactly supported smooth sections in Γ (E). Its closure is the space of square-integrable sections L2(E). Every differential operator can be considered as a densely defined unbounded operator on L2(E) and it is well-known that as such it is closable (see [HR00, Lemma 10.2.1]). We say that D is symmetric if xDu, vy = xu, Dvy for all smooth sections u, v of E. It is an interesting question whether D is essentially self-adjoint, that is, if its closure is self-adjoint, because in this case we have the powerful Borel functional calculus at our disposal. We recall the following criteria from the monograph of Higson and Roe. 30 1. ANALYTIC K-THEORY AND K-HOMOLOGY

1.8.4 LEMMA ([HR00, Corollary 10.2.6]). Every compactly supported, symmetric differential operator is essentially self-adjoint.

1.8.5 LEMMA ([HR00, Proposition 10.2.11]). Every symmetric differential operator over a complete manifold is essentially self-adjoint if the operator has ﬁnite propagation speed, i.e.

¦ suptkσD(ξ)k : ξ P T M, kξk = 1u 8.

The following technical results are the key ingredients for the construction of K-homology classes.

1.8.6 LEMMA ([HR00, Proposition 10.5.1]). Let D be an essentially self-adjoint elliptic differential op- P P erator on a smooth manifold M. Then the operator f ψ(D) is compact for all f C0(M) and ψ C0(R). Here and in the following, we identify differential operators with their closure and ψ(D) is meant in the sense of functional calculus.

1.8.7 LEMMA ([HR00, Lemma 10.6.4]). Let D be an essentially self-adjoint elliptic differential operator P on a smooth manifold M. Then the commutator [χ(D), f ] is compact for all f C0(M). Here, χ is a so-called normalizing function, i.e. any smooth odd function on R which is positive on (0, 8) and tends to 1 for x ÞÑ 8. 1 1.8.8 LEMMA ([HR00, Lemma 10.8.4]). Let D, D be essentially self-adjoint elliptic differential operators 1 on a smooth manifold M which agree on an open subset U M. Then f χ(D) and f χ(D ) differ by a P compact operator for all f C0(U) and normalizing functions χ.

Now suppose that M is a smooth G-manifold and E a smooth Hilbert G-vector bundle, i.e. a smooth G-vector bundle with isometric G-action (cf. Remark 1.5.5). We can then deﬁne an ac- ¡ tion of G on L2(E) by the formula³ g(u) := g(u(g 1(¡))). This action is unitary because by our assumptions the integral f ÞÑ f dM is G-invariant and the action on E is unitary, and we con- 2 clude that L (E) is a Hilbert G-C-module. It is in fact a Hilbert G-C0(M)-C-bimodule if equipped with the obvious left multiplication action by C0(M). Finally note that every grading of E into orthogonal subspaces induces a corresponding grading on L2(E) (cf. Deﬁnition 1.5.18). In this situation every symmetric elliptic differential operator compatible with the structure determines a canonical K-homology class:

1.8.9 PROPOSITION. Every G-equivariant grading-reversing essentially self-adjoint symmetric elliptic differential operator acting on sections of a smooth graded Hilbert G-vector bundle E over a smooth G- manifold M determines a class 2 P G [D] := [L (E), χ(D)] K0 (M). The class [D] does not depend on the choice of normalizing function χ. In fact, [D] only depends on the symbol of D in the sense that any two operators with the same symbol determine the same class in K-homology.

SKETCHOFPROOF. From the preceding discussion we know that L2(E) is a graded Hilbert G- C(M)-C-bimodule. We will now use the theory of [HR00, Chapter 10] to verify that (L2(E), χ(D)) G satisﬁes the conditions of a cycle for KK0 (C0(M), C). By [HR00, Lemma 10.6.2], χ(D) is still grading-reversing. It is also G-equivariant by an obvious variation of this lemma since the action of G is by diffeomorphisms and hence preserves the domain of D. Of course, χ(D) is also self- adjoint. Thus it remains to verify that the following operators [χ(D), f ] and f (χ(D)2 ¡ 1) are P 2 ¡ P compact for all f C(M). This follows from Lemmas 1.8.7 and 1.8.6 with ψ = χ 1 C0(R). Moreover, any two normalizing functions differ by a function in C0(R) so the same argument also shows that [D] does not depend on the choice of χ. The last assertion is due to the fact that any two operators with the same symbol differ by a section of the endomorphism bundle and we refer to [HR00, Exercise 10.9.5]. 1.8. ELLIPTIC OPERATORS AND ANALYTIC K-HOMOLOGY 31

In fact, such a KK-element can be assigned to every symmetric elliptic operator over arbitrary open manifolds M by using a partition of unity. We will prove a more general result which abstracts from C0(M) to more general section algebras.

1.8.10 DEFINITION (ADMISSIBLEALGEBRABUNDLE). Let D be a G-equivariant grading-reversing symmetric elliptic differential operator acting on sections of a smooth graded Hilbert G-vector bundle E over a smooth G-manifold M. ¦ Furthermore, let A be a graded finite-dimensional unital G-C -algebra bundle, i.e. a graded G- ¦ vector bundle whose fibers are finite-dimensional unital C -algebras (and the local trivializations ¦ are compatible with the algebra structure). Then Γ0(A) is a graded G-C -algebra with respect to the supremum norm and pointwise multiplication and involution. We say that A is admissible for D if the following conditions hold: – E is a bundle of modules over A extending the scalar multiplication action of M ¢ C, 2 – L (E) is a Hilbert G-Γ0(A)-C-bimodule with respect to the induced action on sections, and P – for a dense subset of a Γ0(A), the graded commutator [D, a] is densely defined and extends to a bounded operator on L2(E).

1.8.11 EXAMPLE. The trivial algebra bundle M ¢ C is admissible for any operator D because the commutator [D, f ] = ¡iσD(d f ) is bounded for every compactly supported function in f P Cc(M) (which is a dense subset of C0(M)).

The two other algebra bundles we will need in the following are M ¢ Cn, whose section algebra consists of Cn-valued continuous functions on M vanishing at infinity, and the Clifford algebra bundle which we will define in Section 1.9 (this is in general a non-trivial bundle). In these cases, admissibility will not be automatic and has to be verified for the operators at hand.

1.8.12 DEFINITION (ALIGNEDCYCLE). Let D be a G-equivariant grading-reversing symmetric elliptic differential operator acting on sections of a smooth graded Hilbert G-vector bundle E over a smooth G-manifold M, and A an admissible algebra bundle for D.

2 0 We say that a cycle (L (E), T) for KG(Γ0(A)) (or, by misuse of language, T) is aligned with D if there exists an open cover (Ui) of M, essentially self-adjoint operators Di on M which agree with D over Ui and normalizing functions χi such that ¦ ¥ 2 (1.8.13) a[T, χi(Di)]a 0 (mod K(L (E))) P (A| ) for all a Γ0 Ui . 1.8.14 THEOREM. Let D be a G-equivariant grading-reversing symmetric elliptic differential operator acting on sections of a smooth graded Hilbert G-vector bundle E over a smooth G-manifold M, and A an admissible algebra bundle for D. Then there exist cycles (L2(E), T) aligned with D and the class 2 P 0 [D]A := [L (E), T] KG(Γ0(A)) 2 only depends on the symbol of D. Moreover, we have [D]A = [L (E), χ(D)] if D is essentially self-adjoint.

PROOF. We will imitate the proof of [HR00, Proposition 10.8.1]. First of all, note that Proposition 1.8.9 generalizes to our more general left action. Indeed, Lemma 1.8.6 generalizes easily using an approximate unit ( fn) C0(M), and Lemma 1.8.7 is a special case of a more general fact from the unbounded picture of KK-theory (see [Bla98, Proposition 17.11.3] for details). These two lemmas were the main ingredients of the proof of Proposition 1.8.9. We will now construct an explicit cycle (L2(T), T) which is aligned with D. To do so, choose an open covering (Ui) of M by relatively compact sets, a partition of unity (σi) subordinate to (Ui) 2 and essentially self-adjoint operators Di which agree with D on Ui. We claim that L (E) together 32 1. ANALYTIC K-THEORY AND K-HOMOLOGY with the operator ¸ ? ? T := σi Di σi i 0 P forms a cycle for KG(Γ0(A)). Indeed, let a Γc(A) be a compactly supported section. Then the sum in aT is finite and it follows from Lemma 1.8.8 (which also generalizes to our setting using 1 an approximate unit argument) that we can find an essentially self-adjoint operator D which agrees with D on a neighborhood of the support of a such that 1 (1.8.15) aT aχ(D )(mod K(L2(E))). 1 Consequently it follows from the generalization of Proposition 1.8.9 (which applies to D ) that 2 (L (E), T) is a valid cycle. Moreover, if a is supported in Ui then we have ¦ ¦ ¦ ¥ 2 a[T, χ(Di)]a a[χ(Di), χ(Di)]a = 2(aχ(D))(aχ(D)) 0 (mod K(L (E))) from which we conclude that (L2(E), T) is aligned with D. 2 1 For uniqueness, suppose that (L (E), T ) is another cycle aligned with D. Then for a P Γc(A) we have 1 ¦ 1 1 ¦ a[T, T ]a a[χ(D ), T ]a ¥ 0 (mod K(L2(E))). Again it follows that both cycles determine the same class (Lemma 1.3.6). This also shows that [D] only depends on the symbol of D. 1 For the final assertion, suppose that D is essentially self-adjoint. Then we can choose D = D in Equation (1.8.15). This shows that (L2(E), T) is a compact perturbation of (L2(E), χ(D)), hence both cycles determine the same class (Remark 1.3.5).

1.8.16 COROLLARY. Every G-equivariant grading-reversing symmetric elliptic differential operator acting on sections of a smooth graded Hilbert G-vector bundle E over an arbitrary smooth G-manifold M P G determines a class [D] K0 (M) which only depends on the symbol of D. If M is compact then [D] agrees with the class [L2(E), χ(D)] from Proposition 1.8.9.

1.8.17 DEFINITION (ISOMORPHISM). In the following we will need to consider isomorphisms of 1 smooth graded Hilbert G-vector bundles E, F over possibly different spaces M, M . These are 1 G-maps φ : E Ñ E which are ﬁberwise linear and grading-preserving such that there exists a 1 G-equivariant isometry ϕ : M Ñ M (which is of course uniquely deﬁned) making the diagram

φ 1 E / E

/ 1 M ϕ M commute. Note that we recover the notion of isomorphism from Definition 1.5.1 for ϕ = idM. If φ is also fiberwise isometric then the induced map ¦ ¦ 1 ¡ (1.8.18) (ϕ ) L2(E) Ñ L2(E ), u ÞÑ φ ¥ u ¥ ϕ 1 1 is an isometric isomorphism of graded Hilbert G-C0(M )-C-bimodules and we shall say that E 1 and E are isometrically isomorphic over ϕ. 1 1 In the presence of algebra bundles A, A which are admissible for L2(E) and L2(E ), respectively, 1 ÞÑ ¦ we shall also require that there exists a morphism Φ : Γ0(A ) Γ0(A) of graded G-C -algebras, ¦ extending the ordinary pullback ϕ , such that the above map (1.8.18) can be considered as an ¦ 2 Ñ 2 1 1 isometric isomorphism Φ L (E) L (E ) of graded Hilbert G-Γ0(A )-C-bimodules. In this case 1 1 we shall say E and E are isometrically isomorphic over (Φ, ϕ). Note that if both A and A are trivial ¦ ¢ algebra bundles (say, with fiber A) then the latter condition is automatic (use Φ = ϕ idA). 1.8. ELLIPTIC OPERATORS AND ANALYTIC K-HOMOLOGY 33

1 1 1 1 1 1.8.19 PROPOSITION. Let M, E, D, A and M ,E ,D , A be as in Theorem 1.8.14. If φ : E Ñ E is an 1 1 isometric isomorphism between E and E over (Φ, ϕ) intertwining the symbols of the operators D and D then ¦ 1 P G 1 Φ [D] = [D ] K0 (M ). PROOF. As the class determined by an operator only depends on its symbol, we may assume that φ in fact intertwines the operators. Let [D] = [L2(E), T] where T is aligned with D and denote by 1 1 T the operator on L2(E ) corresponding to T under the isometric isomorphism (1.8.18). We have 1 1 1 1 to show that T is aligned with D . For this, let U := ϕ(Ui), D the operator corresponding to 1 i 1 1 1 i Di and χ := χi. Then Equation (1.8.13) holds for U , D and χ since the isometric isomorphism i 1 1 i i i intertwines χi(Di) and χi(Di ) (this is clear for polynomials, and can be extended to arbitrary Borel functions by taking limits). 1 1 1 1.8.20 COROLLARY. Let M, E, D and M ,E ,D be as in Theorem 1.8.14. If there exists an isometric 1 1 1 isomorphism between E and E over ϕ : M Ñ M intertwining the operators D and D then 1 P G 1 ϕ¦[D] = [D ] K0 (M ).

The class [D] is compatible with Cartesian products and restriction to open subsets. Again we refer to the book of Higson and Roe for the proofs which are easily adapted to our situation since they only use properties guaranteed by admissibility and alignment. 1 1 1 1 1.8.21 PROPOSITION ([HR00, Theorem 10.7.3]). Let M, E, D, A and M ,E ,D , A be as in Theorem 1 1 1 1.8.14. Then M ¢ M is again a smooth G-manifold, the operator D ¢ D := D bp 1 + 1 bp D (defined as in the last equation of Definition 1.2.4) acting on sections of the tensor product of the (pulled back) bundles together with the tensor product of the (pulled back) algebra bundles also satisfies the requirements of the theorem, and bp 1 ¢ 1 P 0 bp 1 [D] [D ] = [D D ] KG(Γ0(A) Γ0(A )). 1 1 1 1.8.22 COROLLARY. Let M, E, D and M ,E ,D be as in Theorem 1.8.14. Then bp 1 ¢ 1 P G ¢ 1 [D] [D ] = [D D ] K0 (M M ).

1.8.23 PROPOSITION ([HR00, Theorem 10.8.8]). Let M, E, D, A be as in Theorem 1.8.14 and let U be a G-invariant open subset of M. Then D U also satisﬁes the requirements of the theorem, and [ ] = ¦[ ] P 0 ( (A| )) D U incl D KG Γ0 U . | ¦ Here, incl: Γ0(A U) Γ0(A) denotes the inclusion of G-C -algebras deﬁned by extension by zero. 1.8.24 COROLLARY. Let M, E, D be as in Theorem 1.8.14 and denote by incl: U M the inclusion of a G-invariant open subset of M. Then [ ] = ![ ] P G( ) D U incl D K0 U . ! ¦ Recall that incl denotes the inclusion of G-C -algebras C0(U) C0(M) given by extension by zero.

Finally, let us draw the connection to index theory.

1.8.25 PROPOSITION ([HR00, Corollary 10.2.6]). Every symmetric elliptic differential operator D over a smooth compact manifold is an unbounded Fredholm operator. Thus if D is also G-equivariant and grading-reversing then we can speak of its (graded) G-index which is deﬁned as in the case of bounded operators to be ¡ ¡ indexG D := [ker(D : Γ(E+) Ñ Γ(E )] ¡ [coker(D : Γ(E+) Ñ Γ(E )] + Ñ ¡ ¡ ¡ Ñ + P = [ker(D : Γ(E ) Γ(E )] [ker(D : Γ(E ) Γ(E )] R0(G). The latter identity holds by symmetry of D. 34 1. ANALYTIC K-THEORY AND K-HOMOLOGY

1.8.26 LEMMA. Let D be a G-equivariant grading-reversing symmetric elliptic differential operator acting on sections of a smooth graded Hilbert G-vector bundle E over a smooth compact G-manifold M. Then G index D = collapse¦[D] G ¦ Ñ ¦ under the identiﬁcation K0 ( ) R0(G) from Property K. Here, collapse: M is the constant map sending all of M to a point.

PROOF. By [HR00, Elliptic Regularity 10.4.8], the kernel does not change when we pass from D to its closure. Now χ is non-zero away from the origin, hence rescales all eigenvectors to non-zero eigenvalues by a non-zero scalar, and we conclude that the kernel does not change either when we pass to χ(D). The claim now follows from Corollary 1.3.17.

1.9. Dirac operators

In this section we will describe the class of generalized Dirac operators which form an important subclass of elliptic operators. These operators arise naturally in geometry. Examples are the de Rham operator and the signature operator acting on smooth forms and the Dolbeault operator acting on antiholomorphic forms.

1.9.1 DEFINITION (CLIFFORD ALGEBRA BUNDLE). The complex Clifford algebra bundle of a Rie- mannian G-vector bundle E of rank n with isometric G-action over a G-space X is deﬁned to be the associated bundle C := O ¢µ Cn SO ¢ | Cn. E E E µ SOn Here, OE denotes the orthonormal frame bundle of E, SOE the subbundle of oriented orthonormal frames, and µ is the representation On ü Cn induced by the universal property of Clifford n n algebras from the left multiplication action of On on R bR C C . It is clear how to generalize this deﬁnition to vector bundles which have different rank over different connected components.

By construction every fiber of CE is canonically isomorphic to the Clifford algebra of the com- plexification of the corresponding fiber of E (with respect to the inner product induced by the ¦ Riemannian metric and the involution given by conjugation). Thus CE is a graded C -algebra ¦ bundle and Γ0(CE), its space of sections vanishing at infinity, is itself a graded C -algebra with ¦ the supremum norm induced by the C -norm on each fiber and pointwise involution. Moreover, the isometric action of G extends to CE which thus becomes a G-vector bundle, and we know from the theory of Section 1.5 that the induced action on Γ0(CE) is strongly continuous. We ¦ conclude that the section algebra is in fact a graded G-C -algebra. In particular, the cotangent bundle of a smooth G-manifold M (with or without boundary) satisfies our assumptions and we can always form CM := CT¦ M, the Clifford algebra bundle of M. It carries a canonical smooth structure.

Clearly, we can perform the same construction using the Clifford algebras C¡n. The correspond- 2 ing bundles will be denoted by C¡E and C¡M. In fact, the entire construction can be generalized to Hilbert G-vector bundles with a ﬁberwise antilinear isometric involution so that we get a proper generalization of Example 1.1.13 (see [Kas81, 2.12]).

1.9.2 DEFINITION (DIRAC OPERATOR). Let E be a smooth graded Hilbert G-vector bundle over a smooth G-manifold M.A G-equivariant grading-reversing symmetric differential operator acting on sections of E is called a generalized Dirac operator if its symbol has the property that 2 2 2 σD(ξ) = kξk (i.e. its square D is a generalized Laplacian). Every Dirac operator is of course elliptic. ¦ Note that σD is G-equivariant with respect to the induced actions on T M and End(E) (by the G- equivariance of D) and acts by self-adjoint endomorphisms (by symmetry of D). By the universal

2 We use this notation because of its analogy to C¡n. It is unrelated to reversing the orientation of the vector bundle or manifold. 1.9. DIRAC OPERATORS 35 property of Clifford algebras it follows that σD extends to a G-equivariant morphisms CM Ñ End(E) of graded ¦-algebra bundles. That is, E is a bundle of Clifford modules. This motivates the following deﬁnition.

1.9.3 DEFINITION (DIRACBUNDLE, TWISTING).A Dirac bundle is a smooth graded Hilbert G- vector bundle E over a smooth G-manifold M together with a G-equivariant morphism

CM Ñ End(E) of graded ¦-algebra bundles, called Clifford multiplication.A Dirac operator for E then is a G- equivariant grading-reversing symmetric differential operator whose symbol extends to the Clif- ford multiplication of E. Two Dirac bundles are isometrically isomorphic if there is an isometric isomorphism of graded Hilbert G-vector bundles (not necessarily over the identity, see Deﬁni- tion 1.8.17) which intertwines Clifford multiplication. Given a Dirac bundle E and another smooth graded Hilbert G-vector bundle W, the graded tensor product W bp E is again a Dirac bundle by tensoring the Clifford multiplication action of E with the identity on W, called the twisting of E with W.

Clearly, every Dirac bundle induces an action of the section algebra Γ(CM) on the sections of E which we will also call Clifford multiplication. It is G-equivariant and preserves the involution and Ñ 2 grading. Restricting to sections vanishing at inﬁnity, we get a morphism Γ0(CM) B(L (E)) of ¦ graded C -algebras.

1.9.4 REMARK. One can prove that every Dirac bundle E in fact carries a Dirac operator, hence P G determines a canonical K-homology class [E] K0 (M) (which only depends on the Dirac bundle structure which determines the symbol of the operator). We will only need (and prove) this fact for the important subclass of twisted spinor bundles presented in the next section. Let us now present one of the most important examples of a Dirac operator.

1.9.5 EXAMPLE (DE RHAM OPERATOR). Let M be a smooth G-manifold and consider the com- plexiﬁed differential forms © © ¦ ¦ 8 ¦ ¦ 8 ¦ ¦ Ω (M) := Ω (M) b C Γ ( T M) := Γ ( T M b C) C R C R graded into forms of even and odd degree, respectively. It is a smooth graded Hilbert G-vector ¦ bundle with respect to the inner product and isometric G-action induced from T M. ¦ ¦ We claim that the de Rham operator d + d acting on Ω is a Dirac operator in the sense of the C¦ preceding deﬁnition. Indeed, it is well-known that d + d is invariant under isometries and it is obviously symmetric and grading-reversing. Let us compute its symbol. By the Leibniz rule, we have ¡ ^ @ P 8 P ¦ σd(d f )u = i[d, mul f ]u = i(d( f u) f (du)) = i(d f u)( f C (M), u ΩC(M)) so σd(ξ) is i times the operator λξ of left exterior multiplication by ξ. The symbol of the adjoint operator is the pointwise adjoint of the original symbol, hence ¦ ¦ ¡ (1.9.6) σd+d (ξ) = i(λξ λξ ). Now one can compute explicitly that its square is exterior multiplication by kξk2, and the claim is proved. The following proposition already shows the intimate connection to geometry.

1.9.7 PROPOSITION. The index of the de Rham operator over a smooth compact manifold M is equal to the Euler characteristic of M.

PROOF. Recall that the graded index of the de Rham operator over M is given by the difference ¦ ¦ ¦ ¦ ¦ ¦ dim ker(d + d : Ω2 (M) Ñ Ω2 +1(M)) ¡ dim ker(d + d : Ω2 +1(M) Ñ Ω2 (M)) P Z. 36 1. ANALYTIC K-THEORY AND K-HOMOLOGY

¦ ¦ ¦ The kernel of d + d is equal to the kernel of its square, the Hodge Laplacian ∆ = dd + d d ¦ ¦ (because if ∆u = 0 then k(d + d )uk2 = x∆u, uy = 0). Thus the index of d + d is given by the dimension of the space of even harmonic forms minus the dimension of the space of odd harmonic forms which, by Hodge theory, is precisely the Euler characteristic of the de Rham complex (e.g. [Tay96, Proposition 5.8.3]). Now apply de Rham’s theorem.

1.9.8 REMARK. A more conceptual way to understand the symbol of the de Rham operator is to note the formal similarity between Equations (1.1.15) and (1.9.6). In the same way as we have done for Clifford algebras we can define a G-equivariant action © ¦ ¦ (1.9.9) C bp C¡ Ñ End( T M) M M C of graded ¦-algebra bundles (which is also an isomorphism). Extending the map ξ ÞÑ iξ to an ¦ isomorphism CM C¡M and chasing all definitions we see that σd+d is equal to the following composition of bundle maps: © ¦ ¦ ¦ T M C ÝÑ C¡ C bp C¡ ÝÑ End( T M). M M M M C ¦ ¦ That is, the Dirac bundle structure of C T M is induced by the canonical action of C¡M. ¦ ¦ However, we also have the action of CM on C T M extending ÞÑ ¦ ξ λξ + λξ¦ , corresponding to the first factor of the left-hand side tensor product in Equation (1.9.9), and the 2 ¦ ¦ induced action of Γ0(CM) on L ( C T M). ¦ 1.9.10 LEMMA. The Clifford algebra bundle CM is admissible for d + d .

PROOF. Clearly, we have a natural inclusions

M ¢ C Ñ CM, (x, 1) ÞÑ [px, 1] where px P OE is any lift of x (this is well-deﬁned because 1 P Cn is invariant under the action of On). Moreover, the action of Γ0(CM) evidently restricts to the canonical action of C0(M). For the commutativity condition, recall from the preceding remark that the left action of CM gradedly ¦ commutes with the symbol. Consequently, the graded commutator [u, d + d ] is bounded for P compactly supported sections u Γc(CM), a dense subset of Γ0(CM).

¦ 1.9.11 COROLLARY. The de Rham operator d + d over a smooth G-manifold M determines a canonical K-homology class ¦ ¦ [[ + ]] = [ + ] P 0 ( (C )) d d : d d CM KG Γ0 M . It is called the Dirac element of M.

PROOF. This follows by Theorem 1.8.14 and the previous lemma.

1.9.12 EXAMPLE. Using that the de Rham operator is self-adjoint over the complete manifold Rn (Lemma 1.8.5) we ﬁnd the following concrete representation for the Dirac element of Rn: ¦ 2 n ¦ P G n bp [[d + d ]] = [L (R , Wn), χ(d + d )] KK0 (C0(R ) Cn, C). n n bp Here, we have identiﬁed the Clifford algebra bundle of R with C0(R ) Cn in the obvious way and Cn acts on Wn via its canonical action (cf. Remark 1.9.8).

If Rn is equipped with an isometric, orientation-preserving action of compact Lie group G (i.e. n ¦ by rotations), then in the above representation both Wn = C R and the C -algebra Cn are equipped with the induced action (cf. Example 1.1.13). 1.9. DIRAC OPERATORS 37

1.9.13 KASPAROV’S POINCARÉ DUALITY. Let M be a smooth compact G-manifold with boundary ¦ and A a G-C -algebra. Then “cap product” with the Dirac element over the interior determines an isomorphism # ¦ Ñ G KG(M; A) KK¡¦(Γ0(CM˚ ), A) ÞÑ bp ¦ ¦ x x C(M)µ [[d + d ]]. bp Ñ Here, µ is the multiplication morphism C(M) Γ0(CM˚ ) Γ0(CM˚ ). PROOF. In the proof of [Kas88, Theorem 4.10], Kasparov constructs a Dirac element (which he D = >B calls dX1 ) over the double M : M M M of M (cf. Section 1.12), restricts it to the open subset ˚ M via the inclusion Γ0(CM˚ ) Γ(CDM) and shows that the pullback of this element along µ (which he calls ζX) implements such an isomorphism. [[ + ¦]] Now, Kasparov’s Dirac element dX1 is nothing but our class d d over the double, thus it sufﬁces to show that its restriction to the interior of M is given by the corresponding class ¦ [[d + d ]] over M˚ . This is precisely the statement of Proposition 1.8.23.

We will see in the Section 1.11 how this result is related to ordinary Poincaré duality where the section algebra is replaced by C0(M˚ ). We remark that Kasparov’s Poincaré duality holds in more general situations if one replaces ordinary K-theory by representable K-theory. This indicates that representable K-theory is in a sense the more correct dual theory to K-homology (cf. Remark 1.5.17 and also [EM09b]).

1.9.14 REMARK. Note that the orthogonal group On acts unitarily on Cn with respect to its canonical inner product norm, hence we can also regard CM as a graded Hilbert G-vector bundle. As such, it is isometrically isomorphic to the complexiﬁed exterior bundle by the map induced by ^ ^ ¤ ¤ ¤ ( ) sending a monomial Ei1 ... Eik to the product Ei1 Eik , where Ei is any local orthonormal ¦ n frame, because the isomorphism C R Cn intertwines the respective On-actions (cf. Example 1.1.13).

1.9.15 EXAMPLE (CLIFFORD ALGEBRA BUNDLE). By combining the computations in the proof of Example 1.2.13 with Remark 1.9.8, we see that under the above isomorphism the Clifford multiplication action of a covector ξ corresponds to right multiplication with iξ times the grading operator and the additional action of CM is simply given by left multiplication. This is in agree- ment with [CS84, Remark 3.9].

But the Clifford algebra bundle CM is also a Dirac bundle simply by left multiplication on itself. ¦ We will now show that under a suitable identification we can also consider d + d as its Dirac ¦ operator. Indeed, if we first transform d + d using the isometric isomorphism # © © ¦ ¦ ¦ ¦ x if x is even T M Ñ T M, x ÞÑ C C ¡ix if x is odd ÞÑ ¦ then the symbol of the new operator is given by the formula ξ λξ + λξ¦ , while the additional ¦ ÞÑ ¡i( ¡ ¦ ) left action is given by the formula ξ λξ λξ (we have essentially performed a rotation). ¦ ¦ The canonical identification C T M CM then identifies the symbol with left multiplication and the additional left action of a covector ξ with right multiplication by ¡iξ times the grading ¦ ¦ operator (cf. the proof of Example 1.2.13). In particular, the Dirac bundles CM and C T M are isometrically isomorphic.

1.9.16 REMARK (CONNECTIONS). Let us brieﬂy recall the following picture of connections on principal bundles and their associated bundles. Let π : P Ñ M be a smooth principal H-bundle. A connection on P is deﬁned as a choice of H-equivariant complement of the vertical tangent space TvertP := ker(π¦) TP. This is of course equivalent to a choice of H-equivariant projection TP Ñ TvertP. Recall that identifying an 38 1. ANALYTIC K-THEORY AND K-HOMOLOGY

h = d ( ) element X in the Lie algebra with its fundamental vector field defined by Xp : dt p exp tX t=0 induces an isomorphism between the vertical tangent space and P ¢ h. Hence a connection on P is nothing but a h-valued differential form ω : TP Ñ h which satisfies the following properties: ¦ P ¦ 1. H-equivariance: h ω = Adh¡1 (ω) for all h H, where h denotes the pullback induced by the action of h on P and Ad the adjoint representation of H on the Lie algebra h, 2. Projection: ω(Xp) = X for all X P h Tp H, where Xp is the image of X in TpP. Indeed, any such connection form ω determines a connection on P by taking its kernel. Now let ρ : H Ñ GL(W) be a representation of H on a finite-dimensional vector space W. Then we can form the associated bundle P ¢H W which is a smooth vector bundle. Its sections can be considered as H-equivariant maps u : P Ñ W, and we can define a covariant derivative by the formula

(∇ ) = ( ˜ ) + ( ¦ ( ˜ )) Xu p : du X ρ ω X u p where X˜ is a lift of X|p and ρ¦ : h Ñ gl(W) = L(W) is the differential of ρ (this assignment is well-deﬁned, see e.g. [Roe98, Proposition 2.9] or [BGV92, Section 1.1]). More precisely, if ° 8 8 u = f b x P C (P, W)H C (P) b W then i i i ¸

(∇ ) = ( ˜ ) + ( )( ¦ ( ˜ ))( ) (1.9.17) Xu p d fi X xi fi p ρ ω X xi . i

This deﬁnes a connection (or covariant derivative) ∇ on the associated bundle E := P ¢H W.

1.9.18 REMARK (STRUCTURE-PRESERVINGCONNECTIONS). If the representation has good properties then these are inherited by the associated bundle and induced connection. For instance, suppose that W is graded and that the group H acts by grading-preserving automorphisms. Then the associated bundle E is canonically graded. Moreover, the differential ρ¦ takes values in the space of grading-preserving endomorphisms and it follows from Equation (1.9.17) that the connection preserves the grading in the sense that all covariant derivatives ∇X, or 8 8 ¦ equivalently the induced homomorphism Γ (E) Ñ Γ (T M b E), are grading-preserving with respect to the induced grading on the respective spaces of sections. Similarly, if W is a complex vector space and H acts by complex-linear automorphisms, then the connection preserves the complex structure, i.e. the above homomorphisms are also complex-linear. And if the representation is orthogonal or unitary then E is canonically a Riemannian or Hermit- ian vector bundle and ∇ preserves the metric in the sense that 8 8 8 x∇Xu, vy + xu, ∇Xvy = X(xu, vy) P C (M)(@X P Γ (TM), u, v P Γ (E)).

1.9.19 REMARK (EQUIVARIANT CONNECTIONS). Now suppose that M is a G-manifold and the left G-action lifts to an action on P commuting with the right H-action. In this case, any associate bundle E = P ¢H W is of course canonically a G-vector bundle. If we identify its sections with ¡ H-equivariant maps u : P Ñ W then the G-action is given by the formula g(u)(p) := u(g 1(p)). We will often be interested in G-equivariant connections. In the case of the principal bundle P, this means that the splitting is G-equivariant, i.e. the projection TP Ñ ker(π¦) is G-equivariant.

In the case of the vector bundle E this means that ∇g¦(X)g(u) = g(∇Xu) for all g, X, u. In other 8 8 ¦ words, the induced homomorphism Γ (E) Ñ Γ (T M b E) is G-equivariant with respect to the induced G-actions. Note that since G is assumed to be a compact Lie group we can always ﬁnd equivariant connections by averaging over G.

1.9.20 LEMMA. Suppose that the structure group H is a compact Lie group. Then the following conditions are equivalent: 1. the connection on P is G-equivariant, 2. the connection form ω : TP Ñ h is G-invariant, 3. all induced connections on its associated bundles are G-equivariant, and 1.9. DIRAC OPERATORS 39

4. the induced connection on a bundle associated to a faithful representation is G-equivariant.

PROOF. Equivalence of the ﬁrst two conditions follows from commutativity of the diagram

Φp h / TpP AA AA AA g¦ Φgp AA TgpP where Φp is the map sending an element X of the Lie algebra to Xp, the value of the fundamental vector at p. This is immediate by the chain rule.

ñ ˜ ¡ ¦( ˜ ) ¦( ) For (2) (3), note that if X is a lift of X g 1(p) then g X is a lift of g X p. Thus

(∇ ( )) = ( ( ))( ¦( ˜ )) + ( ¦ ( ¦( ˜ ))) ( ) g¦(X)g u p d g u g X ρ ω g X g u p

= ( ˜ ) + ( ¦ ( ˜ )) ¡ du X ρ ω X u g 1(p)

= (∇ ) ¡ Xu g 1(p) = ( (∇ )) g Xu p, where we have again used the chain rule and G-invariance of ω. It is clear that (3) implies (4). For (4) ñ (2), let E be the G-vector bundle associated to a faithful representation ρ of H. Then ρ¦ is injective, and from the above computation it is clear that the induced connection on E can only be G-equivariant if the connection form ω is G-invariant (construct suitable “test sections” using local triviality).

1.9.21 EXAMPLE (LEVI-CIVITA CONNECTION). The Levi-Civita connection of a smooth G-manifold is the unique connection ∇LC on its tangent bundle which preserves the metric and is torsion- LC ¡ LC free, i.e. ∇X Y ∇Y X = [X, Y]. It is automatically G-equivariant. Indeed, G acts by isometries ¡1 and it is easy to see that the “transformed” connection g (∇g¦(X)g(Y)) is also torsion-free and preserves the metric, hence by uniqueness agrees with ∇LC.

1.9.22 PROPOSITION. Let M be a smooth G-manifold. There exists a canonical G-equivariant connection ∇ on the Clifford algebra bundle which preserves the metric, the complex structure, the grading and is compatible with the algebra multiplication in the following sense: 8 8 ∇X(uv) = (∇Xu)v + u(∇Xv)(@X P Γ (TM), u, v P Γ (CM)). ¦ ¦ Moreover, under the canonical isometrical isomorphism CM C T M this connection corresponds to the canonical extension of the Levi-Civita connection.

¦ Ñ PROOF. Let ω : SOT M son be the connection 1-form associated to the Levi-Civita connection ¦ on T M (identiﬁed with TM using the G-invariant Riemannian metric). Because SOn acts by ¦ n grading-preserving unitaries on both Cn and C R , ω induces canonical connections ∇ on both ¦ ¦ CM and C T M which preserve the metric, complex structure and grading (Remark 1.9.18). Moreover, we have just seen that the canonical isomorphism between those bundles intertwines these representations, thus the connections agree under this identiﬁcation. Since the Levi-Civita connection is invariant under isometries and G acts by (orientation-preserving) isometries, the preceding lemma shows that the connection form ω on SO ¦ as well as the ¦ T M induced connections on CM and C are all G-equivariant.

Finally, note that SOn acts on Cn by algebra automorphisms, hence the differential ρ¦ takes values in the space of derivations of W. Thus we see from Equation (1.9.17) that ∇ is compatible with the algebra multiplication (use the Leibniz rule for the exterior derivative d). Of course, the same argument also works for the connection on the complexiﬁed exterior bundle. 40 1. ANALYTIC K-THEORY AND K-HOMOLOGY

1.10. Spinc-structure and spinor bundles

In this section we will present a rather geometrical source of Dirac operators: To every G-manifold M with boundary carrying a G-equivariant Spinc-structure, the K-theory analogon of an orientation, we can associate a family of vector bundles, so-called spinor bundles, which carry Dirac operators and hence determine canonical classes in analytic K-homology. We quickly recall the deﬁnition of the complex Pin and Spin groups.

1.10.1 DEFINITION (PINC ,SPINC ). The complex Pinc and Spinc groups of Rn are deﬁned as follows: c t ¤ ¤ ¤ P P n u ¢ Pinn := λx1 xk : λ U1, xi R , kxik = 1 Cn , c t ¤ ¤ ¤ P P n u c X Spinn := λx1 xk : λ U1, xi R , kxik = 1, k even = Pinn Cn,+.

Here, Cn,+ denotes the even part of the complex Clifford algebra. Note that the inverse of an P c ¦ element g Pinn is simply g .

1.10.2 REMARK (SPIN-STRUCTURE). These groups can be more conceptually understood by considering the subgroups Pinn and Spinn of the invertible elements of the real Clifford algebra Rn of Rn. These are defined the same way except that they miss the phase factor λ, i.e. t ¤ ¤ ¤ P n u ¢ Pinn := x1 xk : xi R , kxik = 1 Rn , t ¤ ¤ ¤ P n u X Spinn := x1 xk : xi R , kxik = 1, k even = Pinn Rn,+. It is easy to see that the complex Pinc and Spinc groups are given by the quotients c ¢ Pinn Pinn ¨1 U1, c ¢ Spinn Spinn ¨1 U1. ¦ n The twisted adjoint action of Pinn on Rn, induced by the formula Ad(x)y := ¡xyx for x P R , leaves Rn invariant and one computes that Ad(x) is the reflection over the hyperplane with normal vector x. Consequently, Ad surjects onto On, and it turns out that Pinn and Spinn fit into the following short exact sequences:

Ad 0 / Z/2 / Pinn / On / 0

Ad=Ad 0 / Z/2 / Spinn / SOn / 0

Here, Ad is just the ordinary adjoint action. In other words, Spinn is a two-fold covering of SOn, ¥ and in fact a universal covering if n 3. In particular, Spinn is a compact Lie group, and we remark that we get an isomorphic group when starting from R¡n instead of Rn.

The group Spinn is geometrically interesting for the following reason: If we think of an orientation of a Riemannian manifold as a reduction of the structure group of its orthonormal frame bundle from On to SOn (which is connected) then the next step in this program would be to lift the ¥ structure group further to Spinn (which is simply connected at least if n 3). Such a reduction is called a Spin-structure. We refer to the monograph [LM89] for a comprehensive treatment of the geometry and index theory connected to Spin-structures. One important feature for us is that every Spin-manifold determines a canonical Dirac operator. However, requiring a manifold to be Spin turns out to impose much more restrictions on it than what is really needed to deﬁne Dirac operators (see Remark 1.10.30). We will show in the following that it sufﬁces to lift the structure group of the bundle of oriented orthonormal frames c to Spinn along the adjoint homomorphism in order that the manifold determines an (almost) canonical Dirac operator. 1.10. SPINc-STRUCTURE AND SPINOR BUNDLES 41

The correct deﬁnition in the equivariant situation and generalized to arbitrary Riemannian G- vector bundles is the following:

1.10.3 DEFINITION (G-SPINC -STRUCTURE). Let W be a Riemannian G-vector bundle of rank n with orientation-preserving orthogonal G-action over a G-space X.A G-Spinc-structure on W consists of a G-equivariant lift of the structure group of the oriented orthonormal frame bundle c c SOW of W to Spinn under the adjoint homomorphism. That is, there exists a principal Spinn- c bundle SpinW along with a bundle morphism c Ñ ξ : SpinW SOW such that ξ(pg) = ξ(p) Ad(g) for all p, g, and we also require that the principal bundle is a G- space with commuting left and right action and that the projection map is G-equivariant with respect to the induced action on SOW. If W is a smooth vector bundle over a smooth G-manifold c then we will also require that the principal bundle SpinW and the morphism ξ are smooth. This will always be the case except in the general version of the Thom isomorphism. It is clear how to generalize this definition to vector bundles of non-constant rank. In the follow- c ing we will somewhat sloppily talk about principal Spinn-bundles and so on when we should really be considering one connected component at a time. Our assertions however will always transfer to the general case. We say that W is G-Spinc if it admits any such structure, and W is called a G-Spinc-vector bundle if it is equipped with a fixed G-Spinc-structure. An isometric isomorphism of G-Spinc-vector bundles is a G-equivariant isometric isomorphism of vector bundles (not necessarily over the identity, cf. Definition 1.8.17), which is orientation-preserving, together with a lift of the induced c isomorphism of the oriented orthonormal frame bundles to the principal Spinn-bundles. A G-Spinc-structure on a smooth G-Spinc-manifold M then is a G-Spinc-structure on its tangent c c c bundle. In this case, we write SpinM := SpinTM for the corresponding principal Spinn-bundle. We shall say that M is a G-Spinc-manifold if it is equipped with a fixed G-Spinc-structure and that M is G-Spinc if it admits any such structure. An isometry of G-Spinc-manifolds is a G- equivariant orientation-preserving isometry whose differential is an isometric isomorphism of G-Spinc-vector bundles. In the case of G-Spinc-manifolds with boundary we will require that the boundary is mapped into the boundary.

1.10.4 REMARK (K-ORIENTATION). G-Spinc-structures on manifolds and vector bundles are some- times also called K-orientations as they play a similar role in K-theory as ordinary orientations do in ordinary cohomology. Indeed, just as an orientation on a connected smooth manifold amounts B to a choice of generator of the top homology group Hdim M(M, M), we shall see that every c G B connected smooth G-Spin -manifold determines a (non-zero) fundamental class in Kdim M(M, M) implementing Poincaré duality (Section 1.11). And just as an orientation of a vector bundle corresponds to a choice of Thom class, we will see that every G-Spinc-structure bundle similarly determines a KK-element which implements a Thom isomorphism theorem (Section 1.14). c 1.10.5 REMARK. In the case of Spinn, the adjoint homomorphism is not a covering map, but it implements the U1-ﬁbration

/ / c Ad / / 0 U1 Spinn SOn 0. However, together with the homomorphism 2 c ¢ Ñ ÞÑ 2 λ : Spinn Spinn ¨1 U1 U1, [x, λ] λ we get a double covering

(Ad,λ2) / / c / ¢ / 0 Z/2 Spinn SOn U1 0. c c ` In particular, Spinn is a compact Lie group with Lie algebra spinn := son u1. 42 1. ANALYTIC K-THEORY AND K-HOMOLOGY

c We identify the principal U1-bundle constructed by postcomposing the transition maps of SpinW 2 with λ with the complex line bundle it determines and call it the determinant line bundle ΛW of c W (or ΛM in the case of a G-Spin -manifold M).

1.10.6 LEMMA. Let W be a Riemannian G-vector bundle of rank n with orientation-preserving orthogonal c G-action. A G-Spin -structure on W is equivalent to a choice of G-equivariant principal U1-bundle ΛW ¢ c together with a G-equivariant double covering of SOW ΛW which on each fiber restricts to Spinn. The last condition is more precisely formulated as follows: Recall that a double covering is nothing else than a principal Z/2-bundle. These are classified by the first Cechˇ cohomology group with Z/2-coefficients. Now the inclusion of a fiber induces a restriction homomorphism 1 ¢ Ñ 1 ¢ Hˇ (SOW ΛW; Z/2) Hˇ (SOn U1; Z/2), c Ñ and we require that the image of the covering agrees with the class determined by the covering Spinn ¢ SOn U1. PROOF. From the discussion in the previous remark it is clear that every G-Spinc-structure determines a G-equivariant double covering of the correct kind and a principal U1-bundle, its determinant line bundle. Ñ Conversely, suppose we are given a line bundle ΛW and a G-equivariant double covering p : P ¢ c SOW ΛW which is fiberwise of type Spin . We claim that P is a locally trivial fiber bundle with c fiber Spinn. Indeed, choose a contractible open subset U M which is trivializing for W and ΛW. ¢ ¢ ¢ Then SOW ΛW U U SOn U1 and since U is contractible, its connected double coverings ¢ ¢ are in bijection with those of SOn U1 (they are all of the form U C where C is a double covering ¢ c of SOn U1). Since the covering is of type Spin over a single fiber this shows that P is indeed c locally trivial with fiber Spinn, i.e. / ¢ c P U U Spinn

¢ 2 p idU (Ad,λ ) ¢ / ¢ ¢ SOW ΛW U U SOn U1. c We will now argue that P is a principal Spinn-bundle. For this, it sufﬁces to show that the transi- c tion functions are implemented by Spinn-valued cocycles. Consider the following commutative diagram where the horizontal arrows are the respective transition functions from U to V:

(id X ,g ) X ¢ c U V U,V / X ¢ c (U V) Spinn (U V) Spinn

¢ 2 ¢ 2 idUXV (Ad,λ ) idUXV (Ad,λ ) X ¢ ¢ X ¢ ¢ (U V) SOn U1 / (U V) SOn U1 (idUXV , fU,V ) X Ñ ¢ The bottom transition function is implemented by a cocycle φU,V : U V SOn U1 so that 2 2 ¤ 2 (Ad, λ )gU,V (x, g) = fU,V (x, (Ad, λ )(g)) = φU,V (x) (Ad, λ )(g) P X P c for all x U V and g Spinn. Consequently, 2 2 ¤ 2 (Ad, λ )gU,V (x, g) = (Ad, λ )gU,V (x, 1) (Ad, λ )(g) ¤ ¨ so that gU,V (x, g) = gU,V (x, 1) ( g). By continuity, the sign is constant, and comparing at g = 1 shows that it is positive for all x P U X V. Consequently, the transition functions of P c are implemented by the Spinn-valued cocycle γU,V := gU,V (x, 1), and we conclude that P is c a principal Spinn-bundle. Moreover, the G-action on the double covering commutes with the c Z/2-action, hence with the Spinn-action. We have thus proved that P determines a canonical G-Spinc-structure on W. It is clear that both constructions are inverse to each other. 1.10. SPINc-STRUCTURE AND SPINOR BUNDLES 43

1.10.7 REMARK. We can use the preceding lemma to characterize G-Spinc-structures in a way that does not use the Riemannian structure. For this, recall that the inclusion of SOW into the + oriented frame bundle GLW is a G-equivariant homotopy equivalence so that we can identify G-equivariant double coverings over both spaces. In particular, this applies to the inclusion + ˇ 1 + ¢ SOn GLn so that there exists a canonical element in H (GLn U1; Z/2) corresponding to c ¥ the Spinn-covering. For example, if n 3 then it is easy to see that this is precisely the element corresponding to the covering + ¢ Ñ + ¢ ÞÑ 2 GLn Z/2 U1 GLn U1, [x, λ] (p(x), λ ) + Ñ + + where p : GLn GLn denotes the universal covering of GLn with kernel Z/2.

1.10.8 COROLLARY. Let W be a Riemannian G-vector bundle of rank n with orientation-preserving or- c thogonal G-action. A G-Spin -structure on W is equivalent to a choice of G-equivariant principal U1- + ¢ bundle ΛW together with a G-equivariant double covering of GLW ΛW which on each ﬁber restricts to Spinc (in the sense just discussed). In particular, every G-Spinc-structure for one G-invariant Riemannian metric canonically determines a G-Spinc-structure for every other metric. We can use this discussion to deﬁne a less restrictive notion of equality.

1.10.9 DEFINITION (G-SPINC -STRUCTURE-PRESERVING ISOMORPHISM AND DIFFEOMORPHISM). Let φ : V Ñ W be a G-equivariant orientation-preserving isomorphism of G-Spinc-vector bundles (not necessarily over the identity) and denote by φ¦ the induced isomorphism between the oriented frame bundles. We say that φ preserves the G-Spinc-structure if there is a G-equivariant Ñ ¢ c isomorphism ψ : ΛV ΛW of principal U1-bundles such that φ¦ ψ identiﬁes the G-Spin - structures of M and N in the sense of G-equivariant double coverings. If ϕ : M Ñ N is an G-equivariant orientation-preserving diffeomorphism of smooth G-Spinc- manifolds then we shall similarly say that ϕ preserves the G-Spinc-structure if its differential does so in the sense just deﬁned. In the case of manifolds with boundary we shall also require that the boundary is mapped into the boundary.

We will now present a number of important examples of G-Spinc-structure.

1.10.10 EXAMPLE (TRIVIALVECTORBUNDLES, PARALLELIZABLEMANIFOLDS). Every trivial vector bundle, and hence every parallelizable manifold, carries a canonical Spinc-structure for every choice of Riemannian metric and orientation. We will always consider Rn and intervals I R equipped with the trivial G-action and the canonical orientation, metric and Spinc-structure. Note that every orientation-preserving diffeomorphism automatically preserves the G-Spinc-structure (this follows immediately from uniqueness).

1.10.11 EXAMPLE (SPINORACTIONSON Rn). In particular, the canonical Spinc-structure of Rn n ¢ c with its usual orientation and Riemannian metric is given by the principal bundle R Spinn. If H is a group acting spinorly on Rn, i.e. the action factors as

/ c Ad / H Spinn SOn, then this structure is in fact an H-Spinc-structure if we let H act on the ﬁber by left multiplication.

1.10.12 EXAMPLE (G-SPIN-STRUCTURE). In the same way as above we can deﬁne G-Spin-structure, G-Spin-vector bundles and G-Spin-manifolds. Every G-Spin-structure determines a canon- c c ical G-Spin -structure by associating to its principal Spinn-bundle a principal Spinn-bundle via the group homomorphism Spin ãÑ Spinc, x ÞÑ [x, 1]. 44 1. ANALYTIC K-THEORY AND K-HOMOLOGY

Consequently, every G-Spin-vector bundle is canonically a G-Spinc-vector bundle, and every G- Spin-manifold is canonically a G-Spinc-manifold.

1.10.13 EXAMPLE (HERMITIANVECTORBUNDLES, ALMOST HERMITIANMANIFOLD). There ex- Ñ ¢ ÞÑ ists a unique lift of the map Uk SO2k U1, g (g, det(g)) to a group homomorphism ãÑ c c Uk Spin2k. Thus, every Hermitian vector bundle W determines a canonical Spin -structure associated to its unitary frame bundle UW via this homomorphism. Note that in this case the ¢ determinant line bundle is indeed UW det U1. In particular, every almost Hermitian manifold carries a canonical Spinc-structure. Moreover, if W is equipped with a unitary G-action, i.e. a Hilbert G-vector bundle, then the same construction yields a G-Spinc-structure. In particular, every unitary representation of G on a ﬁnite-dimensional Hilbert space carries a canonical G-Spinc-structure. For details as well as the cohomological classiﬁcation of Spinc-structures we refer to [LM89, Ap- pendix D]. Let us also remark that every smooth compact orientable manifold of dimension less or equal to 4 is Spinc (see [TV94]).

There are certain operations by which we can construct new Spinc-structures out of old ones. We will ﬁrst describe a general construction for vector bundles and then specialize to the case of manifolds.

1.10.14 DEFINITION (DIRECTSUM). Let V, W be (smooth) G-Spinc-vector bundles. Then the direct sum V ` W is again a (smooth) G-Spinc-vector bundle in the following canonical way: Observe that we have a commutative diagram c ¢ c / c Spinm Spinn Spinm+n

Ad ¢ Ad Ad SOm ¢ SOn / SOm+n p of group homomorphisms, where the top arrow is induced by the inclusion Cm b Cn Ñ Cm+n and the bottom arrow is given by sending a pair of matrices to the corresponding block ma- ¢ ¢ trix. Moreover, the pushforward of the principal SOm SOn-bundle SOV SOW along the bottom arrow is precisely the oriented orthonormal frame bundle of V ` W with the usual product orientation. We thus conclude from commutativity of the diagram that the pushforward of c ¢ c c ` SpinV SpinW along the top arrow determines a G-Spin -structure on V F. 1.10.15 TWOOUTOFTHREE PRINCIPLE. Let V and W be (smooth) Riemannian G-vector bundles with orientation-preserving orthogonal G-action. If two out of three of the vector bundles V, W and V ` W are equipped with a G-Spinc-structure then there is a unique G-Spinc-structure on the third such that V ` W carries the canonical direct sum G-Spinc-structure.

PROOF. This is proved similar to [LM89, Proposition II.1.15].

We now specialize to the case of manifolds.

1.10.16 DEFINITION (CARTESIANPRODUCT). Let M, N be smooth G-manifolds such that at most one of them has a boundary. Then the Cartesian product M ¢ N is again a smooth G-Spinc- manifold with boundary if we equip T(M ¢ N) TM ` TN with the direct sum G-Spinc- structure.

1.10.17 TWOOUTOFTHREE PRINCIPLE. Let M, N be smooth G-Spinc-manifolds such that at most one of them has a boundary. If two out of three of the manifolds M, N and M ¢ N are equipped with a G-Spinc-structure then there is a unique G-Spinc-structure on the third such that M ¢ N carries the canonical Cartesian product G-Spinc-structure. 1.10. SPINc-STRUCTURE AND SPINOR BUNDLES 45

PROOF. This follows of course directly from the more general Two out of Three Principle 1.10.15.

1.10.18 DEFINITION (BOUNDARY). The boundary of a smooth G-Spinc-manifold M with boundary is canonically a G-Spinc-manifold in the following way: Denote by ν the normal bundle of BM in M. Then G acts trivially on ν (since G preserves the boundary) and we get a G-equivariant decomposition ` (B ) TM BM ν T M Since ν is trivial, it carries a canonical G-Spinc-structure compatible with the orientation deﬁned by requiring outward-pointing normal vectors to be positively oriented (Example 1.10.10). We can then use the Two out of Three Principle 1.10.15 to ﬁnd a unique G-Spinc-structure on BM such that the above decomposition respects the G-Spinc-structure. The boundary G-Spinc-structure can be described rather explicitly: Consider the commutative diagram c / c Spinn¡1 Spinn

Ad Ad SOn¡1 / SOn ¡ where the top arrow is induced by the inclusion Rn 1 Rn, x ÞÑ (0, x) and the bottom arrow c is the lower-right corner embedding of SOn¡1 in SOn. We can thus restrict the action on SpinM c and SOM to Spinn¡1 and SOn¡1 in a consistent way. Using the boundary orientation we have constructed above, we get a diagram

c SpinM BM

(1.10.19) ξ

B ¢ / ( ¢ ) SO M ΛM BM SOM ΛM BM where the bottom arrow is given by inserting the outward-pointing normal vector in front of ori- ¢ ented orthonormal frames. This map is easily seen to be G- and SOn¡1 U1-equivariant. After c c completing the pullback diagram we get a ﬁber bundle SpinBM with a right Spinn¡1-action. Cov- ering theory shows that this is a two-fold covering, and as the projection to SOBM is compatible c c with Ad we see that SpinBM is a principal Spinn¡1-bundle. It is clear that this is precisely the boundary G-Spinc-structure on BM.

1.10.20 DEFINITION (RESTRICTION). The restriction of a smooth G-Spinc-manifold M with boundary to a G-invariant open subset U M is again a smooth G-Spinc-manifold (restrict the principal c Spinn-bundle to U). Of course we can also restrict to more general embedded submanifolds of codimension 0. We will always use these canonical G-Spinc-structures unless explicitly stated otherwise. The following lemmas sketch some useful relations between these constructions: c c 1.10.21 LEMMA. Let M be a smooth G-Spin -manifold with boundary and N a smooth G-Spin -manifold without boundary. Then B(M ¢ N) is isometric to (BM) ¢ N.

PROOF. Both assertions are evident from the respective constructions.

c 1.10.22 LEMMA. Let M be a smooth compact G-Spin -manifold with boundary. Then there exists an open neighborhood U of the boundary in M and a G-Spinc-structure preserving diffeomorphism (0, 1] ¢ BM U. 46 1. ANALYTIC K-THEORY AND K-HOMOLOGY

PROOF. We can always identify (0, 1] ¢ BMG-equivariantly with an open neighborhood U M of the boundary, but we have to argue that any such diffeomorphism preserves the G-Spinc- structure. Since (0, 1] is contractible, it sufﬁces to verify this for the identity map of the boundary which is in fact an isometry of G-Spinc-structures (by deﬁnition of the boundary G-Spinc- structure).

1.10.23 DEFINITION (OPPOSITE). Let M be a smooth G-Spinc-manifold. We denote by ¡M the same manifold equipped with opposite orientation and opposite G-Spinc-structure which is defined by identifying M with the subset t0u ¢ M B([0, 1] ¢ M). If M is a smooth G-Spinc-manifold with boundary we can still define the opposite G-Spinc-structure 1 as follows: Embed M into a smooth G-Spinc-manifold M without boundary (e.g. use the collar neighborhood from the preceding lemma to adjoin an open cylinder to its boundary) and denote by ¡M the manifold M equipped with opposite orientation and G-Spinc-structure inherited from 1 ¡M (but equipped with the original Riemannian metric; use Corollary 1.10.8). Observe that this definition is in fact independent of the choice of embedding. There are more explicit ways to define the opposite G-Spinc-structure but we will not need that in the following (see e.g. [BHS08, p. 7]). However, we note the following simple consequence of Lemma 1.10.21. c 1.10.24 LEMMA. Let M and N be smooth G-Spin -manifolds, at least one of them with empty boundary. Then ¡(M ¢ N) is isometric to (¡M) ¢ N.

We will now introduce the associated vector bundles on which our Dirac operators will act.

1.10.25 DEFINITION (SPINORBUNDLE). Let M be a smooth G-Spinc-manifold and W a finite- dimensional graded Hilbert Cn-C-bimodule (i.e. a finite-dimensional ¦-representation of Cn on a graded Hilbert space). The associated bundle c Spin ¢ c W M Spinn is called the spinor bundle of M associated to W. Note that Spinc and in fact Pinc always acts unitarily on W — this follows from the computation ¦ xxv, xwy = xx xv, wy = kxk2xv, wy (@x P Rn, v, w P W). Consequently, every spinor bundle is a smooth graded Hilbert G-vector bundle over M with respect to the inner product defined by the formula x[p, v], [p, w]y := xv, wy.

1.10.26 COUNTEREXAMPLE (CLIFFORD ALGEBRA BUNDLE). Let M be a smooth G-Spinc-manifold. We can identify the tangent and cotangent bundle via the G-invariant Riemannian metric, and ¦ this induces an isometric isomorphism of the orthonormal frame bundles OM OT M. Conse- quently, we have canonical isometric isomorphisms c C = O ¦ ¢µ Cn O ¢µ Cn SO ¢ | Cn Spin ¢ Cn M T M M M µ SOn M Ad where µ is induced by the left multiplication representation of On. Clearly, all these representations are unitary with respect to the canonical inner product of Cn (cf. Remark 1.9.14), hence the above bundles are all graded Hilbert G-vector bundles isometrically isomorphic to each other. c Note that CM is not a spinor bundle because the adjoint representation of Spinn is not the restriction of a representation of Cn. In general, it is only a Dirac bundle, but we will see soon that if M is even-dimensional then CM is at least a twisted spinor bundle.

1.10.27 REMARK (SPINORBUNDLESARE DIRACBUNDLES). We can use the above identification c to equip every spinor bundle S = Spin ¢ c W with the structure of a Dirac bundle: Define M Spinn the Clifford multiplication action of CM by the formula [p, x] ¤ [p, v] := [p, xv] (note that this is well-defined because (Ad(g)x)gv = gxv). Two spinor bundles are isometrically isomorphic if they 1.10. SPINc-STRUCTURE AND SPINOR BUNDLES 47 are isometrically isomorphic as Dirac bundles. It is now also clear what we mean by a twisted spinor bundle.

We will soon be able to construct to every twisted spinor bundle a Dirac operator whose symbol is given by Clifford multiplication.

1.10.28 EXAMPLE (FULL, TAUTOLOGICALANDREDUCEDSPINORBUNDLE). Let M be a smooth c G-Spin -manifold. By Example 1.2.15, we always have the representation Wn of Cn,n on the com- c plexified exterior algebra. The full spinor bundle is the bundle S := Spin ¢ c W associated M M Spinn n to the representation of Cn defined by the composition Cn Ñ C¡n Ñ L(Wn) where the first arrow is induced by the map x ÞÑ ix on vectors x P Rn. taut c ¢ We will also need the tautological spinor bundle SM := SpinM mul Cn associated to the tautological representation of Cn on itself by left multiplication. It is isometrically isomorphic to the full spinor bundle, but we will not need that here (apply a similar transformation as in Example 1.9.15). If M has even dimension n = 2k then we can also consider the representation of C2k Ck,k on c the complex conjugate W . The associated bundle /SM := Spin ¢ c W is called the reduced k M Spin2k k spinor bundle of M. We have seen in Example 1.2.15 that Ck,k L(Wk). Consequently, Clifford multiplication induces a chain of canonical isometrical isomorphisms of Dirac bundles bp ¦ bp CM End(/SM) /SM /SM /SM S/ M.

Note that left and right multiplication of CM on itself correspond to Clifford multiplication on /SM and S/ M (with the transpose) in the right-hand side picture.

1.10.29 REMARK (CONVENTIONS). At this point we should give some background for our seem- ingly curious deﬁnitions.

Let us point out that the identification Cn C¡n we have used for the definition of the full spinor bundle is not equal to the composition of canonical isomorphisms Cn = Cn,0 C0,n C¡n as ¡ k the latter sends a basis vector ek to ( 1) iek. However, the discussion in Remark 1.9.8 shows that our choice is in fact more in the spirit of Kasparov’s Dirac element. This point of view will be profitable when we try to relate our fundamental class (which will be defined in terms of the full spinor bundle) to Kasparov’s cycle.

We also note that using the dual Wk of the standard irreducible representation is in fact the natural choice. Indeed, under the canonical identiﬁcation Ck,k C0,2k this is precisely the irreducible k ¤ ¤ ¤ representation of C0,2k where i e1 e2k implements the grading which is used in [LM89, Section II.6] and [HR00, Section 11.3].

We will see that our deﬁnitions are consistent in the sense that the induced K-homology classes correspond to each other under Clifford periodicity (Lemma 1.11.16). Moreover, they will naturally lead us to Kasparov’s classical Bott element (Section 1.14).

1.10.30 REMARK. Let us now sketch why a Spinc-structure (as opposed to a Spin-structure) should be sufﬁcient in order to deﬁne complex spinor bundles and hence Dirac operators.

Suppose that SpinM is the principal Spinn-bundle associated to a Spin-manifold M. Then one can ¦ also construct complex spinor bundles by associating to SpinM graded -representations of Rn on some ﬁnite-dimensional graded (complex) Hilbert space W. But the unitary representation ü c Spinn W together with multiplication by scalars in U1 descends to a representation of Spinn ¢ Spinn ¨1 U1, and we see that we get the same spinor bundle if we use this latter representation to deﬁne a spinor bundle with respect to the induced Spinc-structure of M. 48 1. ANALYTIC K-THEORY AND K-HOMOLOGY

c 1.10.31 PROPOSITION. Let M be a smooth G-Spin -manifold. Then every choice of G-equivariant connection on the determinant line bundle ΛM canonically determines a G-equivariant connection on the c principal bundle SpinM. Moreover, the induced connection ∇S on every spinor bundle S is G-equivariant, preserves the metric, the complex structure, the grading and is compatible with Clifford multiplication in the sense that S S @ P 8 P 8 P 8 ∇X(uv) = (∇Xu)v + u(∇Xv)( X Γ (TM), u Γ (CM), v Γ (S)).

Here, ∇ is the canonical connection on CM from Proposition 1.9.22.

PROOF. The Levi-Civita connection on SOM (which is G-equivariant, see Example 1.9.21) together with any G-equivariant connection on the determinant line bundle ΛM determines a G- equivariant connection on the product SOM ¢ ΛM. The corresponding connection form ω takes ` c values in the Lie algebra son u1 = spinn, hence its pullback along the covering map is also a c S G-equivariant connection form for SpinM. We thus get a G-equivariant connection ∇ on every spinor bundle S. As in the proof of Proposition 1.9.22 we see from the discussion in Remark 1.9.18 that ∇S preserves the metric, complex structure and grading. We claim that this connection is compatible with Clifford multiplication. To see this, ﬁrst note that the above process determines a connection on CW, considered as the spinor bundle corresponding to the adjoint representation (Example 1.10.26). It is clear that this connection is precisely the connection ∇ from Proposition 1.9.22 (the differential Ad¦ agrees on son with the differential induced by the rotation action and is trivial on u1 because U1 is Abelian). Equation (1.9.17) now shows that it sufﬁces to prove that @ P c P P ρ¦(X)(xv) = Ad¦(X)(x)v + ρ¦(X)v ( X spinn, x Cn, v S) where ρ denotes the representation of Cn corresponding to the spinor bundle S. This follows from “deriving” the identity

ρ(g)(xv) = (Ad(g)(x))(ρ(g)(v)).

p 1.10.32 COROLLARY. Every twisted spinor bundle S b W carries a G-equivariant connection which preserves the metric, complex structure and grading and which is compatible with Clifford multiplication.

PROOF. Choose any connection on the determinant line bundle, average over G in order to make it G-equivariant and denote by ∇S the connection induced on S by the preceding proposition. Furthermore, choose any connection ∇W on W which preserves the metric, complex structure and grading, and average over G in order to also make it G-equivariant. It is straightforward to verify that the tensor product connection ∇W b 1 + 1 b ∇S satisﬁes our requirements.

c 1.10.33 THEOREM. Let M be a smooth G-Spin -manifold and S a twisted spinor bundle. Then there exists a Dirac operator acting on sections of S whose symbol is given by Clifford multiplication.

PROOF. Choose a connection ∇S on S using the corollary and denote by ∇ the canonical connection on CM provided by Proposition 1.9.22. Let us deﬁne an operator D by the composition

8 ∇S 8 ¦ ¡i 8 ¦ 8 Γ (S) / Γ (T M bp S) / Γ (T M bp S) / Γ (S) where the last arrow is Clifford multiplication3. Clearly, D is a G-equivariant grading-reversing differential operator.

3 The extra factor ¡i comes from our using the Clifford algebra bundle CM instead of C¡M which we always identify by ξ ÞÑ iξ. 1.10. SPINc-STRUCTURE AND SPINOR BUNDLES 49

Let us compute an explicit representation of D. Around any point p P M, we may choose an orthonormal tangent frame (Ei); denote by (ξi) its dual. Then the operator D is given by the map ¸n ¸n ¸n (1.10.34) u ÞÑ ∇Su = ξ b ∇S u ÞÑ ¡i ξ b ∇S u ÞÑ ¡i ξ ¤ ∇S u j Ej j Ej j Ej j=1 j=1 j=1 where the right-hand side dot is Clifford multiplication. Consequently, ¸n ¸n σ (d f )u = ξ ¤ [∇S , mul ]u = ξ ¤ E ( f )u = d f ¤ u, D j Ej f j j j=1 j=1 and we conclude that the symbol of D is indeed given by Clifford multiplication.

We still have to show that D is symmetric. For this, let us work with an orthonormal frame (Ei) which is synchronous at p, i.e. ∇LCE = 0 for all i, j. Then also ∇ ξ = 0, since the connection Ei j p Ei j p on CM agrees with the Levi-Civita connection (Proposition 1.9.22), and ¸n ¸n

xDu, vy = x¡iξ ¤ ∇S u, vy = ¡x∇S u, ¡iξ ¤ vy p j Ej p Ej j p j=1 j=1 ¸n

= xu, ¡i∇S (ξ ¤ v)y ¡ E (xu, ¡iξ ¤ vy) Ej j p j j p j=1 ¸n = x ¡ ¤ ∇ ( )y ¡ (x ¡ ¤ y) u, iξj Ej v p Ej u, iξj v p = j 1 ¸ = x y + (x ¤ y) u, Dv p i Ej u, ξj v p. j We will now show that the right-hand side is the divergence at p of the vector field X defined by x y x x¡ y ¤ y the condition X, Y = u, , Y v . Indeed, again using that (Ei) is synchronous at p we get ¸n ¸n ¸n = x∇ y = (x y) = (x ¤ y) div X p Ej X, Ej p Ej X, Ej p Ej u, ξj v p. j=1 j=1 j=1 P 8 Thus for any two compactly supported sections u, v Γc (S) we find that » xDu, vy = xu, Dvy + div(X)dM = xu, Dvy by Stokes’ theorem.

c 1.10.35 COROLLARY. Every twisted spinor bundle S on a smooth G-Spin -manifold M determines a canonical K-homology class [S] P KG(M). Moreover, if S is isometrically isomorphic to another twisted 1 0 1 spinor bundle S over ϕ : M Ñ N then ϕ¦[S] = [S ].

PROOF. Set [S] equal to the class [D] determined by any Dirac operator D associated to S via Corollary 1.8.16. This class only depends on the symbol (which is always Clifford multiplication), hence the assignment is well-deﬁned. The second assertion holds by the same argument and Corollary 1.8.20.

1.10.36 EXAMPLE. We can thus rephrase Example 1.9.15 using the discussion at the end of Exam- ple 1.10.28 in the following way: For every smooth even-dimensional G-Spinc-manifold M we ¦ G have [CM] = [d + d ] in K0 (M). Twistings can also be expressed in terms of Kasparov products: 50 1. ANALYTIC K-THEORY AND K-HOMOLOGY

p c 1.10.37 PROPOSITION. Let W b S be a twisted spinor bundle on a smooth G-Spin -manifold M. Then [W bp S] = [ (W)] bp [S] P KG(M) Γ0 C0(M) 0 .

Here, Γ0(W) is considered as a graded Hilbert G-C0(M)-C0(M)-bimodule. 2 bp 2 bp PROOF. Write [S] = [L (S), T] and [W S] = [L (W S), TW ] as cycles aligned with respective Dirac operators D and DW. We will use the characterization of Kasparov products in terms of connections due to Connes and Skandalis (since this is the only place where this is needed, we will not repeat the theory but instead refer to [Bla98, Chapter 18] for details). Clearly, we can identify (W) bp L2(S) Ñ L2(W bp S) u b v ÞÑ (x ÞÑ u(x) b v(x)) Γ0 C0(M) , (the right-hand side assignment deﬁnes an isometry with dense range on the algebraic tensor product). As the operator in the cycle [Γ0(W)] is zero we only have to verify that TW is a T- connection. Moreover, since both T and TW are self-adjoint it in fact sufﬁces to verify that the P following operators are compact for all w Γ0(W): 2 Ñ 2 bp ÞÑ b ¡ ¡ Bw b L (S) L (W S), s w Ts ( 1) TW (w s).

Working with compactly supported sections w P Γc(W) and using the alignment property (1.8.13), 1 1 we can first replace T and TW by operators of the form χ(D ) where D is an essentially self- adjoint operator agreeing with D and DW, respectively, on an open neighborhood of the support of w (as in Equation (1.8.15)). From the unbounded theory from [Bla98, Proposition 17.11.3] we then see that it in fact suffices to argue that the operators 2 Ñ 2 bp ÞÑ b ¡ ¡ Bw b Dw : L (S) L (W S), s w Ds ( 1) DW (w s) are densely defined and extend to a bounded operator for all compactly supported sections P w Γc(W). But this is indeed the case since the symbols of D and DW are given by Clifford multiplication on the respective bundles, thus related by b ¡ Bw b w σD(ξ)s = ( 1) σDW (ξ)(w s) for all covector fields ξ (here, we have identified the symbol with its induced action on sections), and consequently the operators Dw are compactly supported bundle endomorphisms, hence bounded. bp 1.10.38 COROLLARY. Let DW be a Dirac operator for a twisted spinor bundle W S on a smooth compact G-Spinc-manifold M. Then G bp P index DW = [W] C(M)[S] R(G).

PROOF. Since collapse¦[Γ0(W)] = [W] this follows from combining the preceding proposition with Lemma 1.8.26.

1.10.39 COROLLARY. Let W be a smooth trivially-graded Hilbert G-vector bundle on a smooth compact G-Spinc-manifold M with boundary and let S be a spinor bundle on the interior of M. Then [ bp ] = [ ] X [ ] P G( B ) W M˚ S W S K0 M, M . PROOF. Combine Lemma 1.7.10 and the preceding proposition.

Note in particular that the Hermitian structure of W has no impact on the K-homology class of the twisted operator because the K-theory class determined by W only depends on its G-vector bundle structure (Lemma 1.5.4).

1.10.40 REMARK. The notion of an elliptic differential operator can be generalized to A-vector bundles in the sense of Section 1.5, and the index theory of these operators has been studied in the literature starting with the work of Mišˇcenkoand Fomenko in [MF79]. Every such operator should determine a class in K-homology with coefficients in A and most of the above operations should still make sense in this more general context. We will not try to develop this theory as it 1.11. FUNDAMENTAL CLASSES AND POINCARÉ DUALITY 51 is not needed in the following (but see [Sch05, Section 6.3] and [Bla98, Section 24.5]). However, we can (and will) still form the cap product X 0 b G B Ñ G B : KG(M; A) K0 (M, M) K0 (M, M; A) and think of it as twisting an ordinary elliptic differential operator over the interior of M with an A-vector bundle over M. We will see in the following section that under suitable assumptions this map surjects so that every class in K-homology with coefficients arises by twisting a class in K-homology without coefficients.

1.11. Fundamental classes and Poincaré duality

In this section we will show that every smooth G-Spinc-manifold M with boundary determines a B P G B fundamental class [M, M] Kdim M(M, M). If M is compact, this fundamental class implements Poincaré duality as in ordinary (co)homology theory. c 1.11.1 LEMMA. Let M be a smooth G-Spin -manifold of dimension n. Then the canonical action of Cn on 2 Wn induces an action of the algebra bundle M ¢ Cn on the full spinor bundle SM and further on L (SM) such that M ¢ Cn is admissible for any Dirac operator associated to SM.

PROOF. We have the following situation:

p Cn / C¡n Cn b C¡n / L(Wn) The horizontal representation is the one used for the construction of the full spinor bundle (and hence for Clifford multiplication), and the vertical composition is the canonical one we would like to consider in addition. Since the actions commute gradedly (Example 1.2.15), the horizontal c action of Spinn Cn,+ commutes with the vertical action, and we conclude that the latter indeed induces a well-deﬁned action of Cn on SM.

We will now verify the conditions for admissibility. Clearly, M ¢ C is a subbundle of M ¢ Cn 2 bp and L (SM) is a Hilbert G-C0(M) Cn-C-bimodule in the way required for admissibility. For the last condition, consider a Dirac operator D for the full spinor bundle and a compactly supported function u P Cc(M, Cn). Since the symbol of D is given by Clifford multiplication (horizontal action) and u acts by the vertical action, we ﬁnd as in the proof of Proposition 1.10.37 that the graded commutator [u, D] is a compactly supported bundle endomorphism, hence bounded.

1.11.2 DEFINITION (FUNDAMENTAL CLASS). The fundamental class of a smooth G-Spinc-manifold M of dimension n is the K-homology class

¢ P G G bp C C [M] := [D]M Cn Kdim M(M) = KK0 (C0(M) n, ) determined using Theorem 1.8.14 by any Dirac operator D for the full spinor bundle with respect c to the admissible algebra bundle M ¢ Cn from the preceding lemma. If M is a smooth G-Spin - B ˚ P G B manifold with boundary then we deﬁne [M, M] := [M] Kdim M(M, M). In the following we will always assume that all manifolds have a well-deﬁned dimension.

1.11.3 REMARK (CONVENTIONS). Let us compare our definition to the one of Higson and Roe (see [HR00, Definition 11.2.3]). They define the fundamental class in terms of the tautological spinor bundle constructed from the Clifford algebra C0,n. Consequently there is a natural right action of C0,n and one can show that the associated Dirac operator is C0,n-linear and determines G a class in KK0 (C0(M), C0,n). G bp While this group is isomorphic to KK0 (C0(M) Cn,0, C), the isomorphism changes the right action of C0,n which commutes with the operator to a left action of Cn,0 which commutes with 52 1. ANALYTIC K-THEORY AND K-HOMOLOGY the operator only gradedly. This explains why we cannot hope to find a description of our full spinor bundle which is as elegant as the one of Higson and Roe if we would still like it to directly determine an element in the latter KK-group (while we could use the tautological spinor bundle, the additional action of Cn would not be given simply by right multiplication; cf. Example 1.9.15). At any rate, one can show that Higson and Roe’s fundamental class corresponds to our fundamental class under this isomorphism (maybe up to a sign), and while the approach of Higson and Roe might be more intuitive from the point of view of spin geometry, it will be easier for us to work with our definition which is closer to Kasparov’s Dirac element.

1.11.4 EXAMPLE (SPINORACTIONSON Rn). Suppose that Rn is equipped with the spinor action of a compact Lie group H and its canonical H-Spinc-structure (see Example 1.10.11). Chasing the relevant deﬁnitions, it is easy to see that its fundamental class is given by n 2 n ¦ P G n H n bp [R ] = [L (R , Wn), χ(d + d )] Kn (R ) = KK0 (C0(R ) Cn, C). ¦ This element almost agrees with the cycle [[d + d ]] determined by the de Rham operator over n R (Example 1.9.12), except that now the module Wn is equipped with the action of H induced by the equivariant Spinc-structure, i.e. by the composition Ñ c Ñ Ñ H Spinn Cn C¡n L(Wn), ¦ and the C -algebra Cn is equipped with the trivial H-action. We will later see that both elements are related by a KK-equivalence, and in fact this observation will be the key to proving Poincaré duality for compact G-Spinc-manifolds. c 1.11.5 LEMMA. Let ϕ : (M, BM) Ñ (N, BN) be an isometry of G-Spin -manifolds with boundary. Then ϕ¦[M, BM] = [N, BN].

PROOF. It is clear from the deﬁnition that every such isometry of G-Spinc-manifolds with bound- Ñ ary induces an isometrical isomorphism Φ : SM˚ SN˚ of Dirac bundles intertwining the action ¢ of the algebra bundle M Cn. The claim thus follows from Proposition 1.8.19.

The fundamental class is compatible with products, submanifolds, boundaries and opposites: c 1.11.6 PROPOSITION. If M, N are smooth G-Spin -manifolds such that at most one of them has non- empty boundary then [M, BM] bp [N, BN] = [(M, BM) ¢ (N, BN)].

PROOF. For simplicity of notation we shall assume that M and N have empty boundary (otherwise replace them by their interiors). From the construction of the product G-Spinc-structure (Deﬁnition 1.10.16) we get an isometric isomorphism φ of spinor bundles over the identity map given by the composition c S ¢ = Spin ¢ ¢ c W + M N M N Spinm+n m n 1.2.17 c p = Spin ¢ ¢ c (W b W ) M N Spinm+n m n c p c Ñ (Spin ¢ c W ) b (Spin ¢ c W ) M Spinm m N Spinn n p = SM b SN. By using the representation Ad everywhere, we get a similar isomorphism between the Clif- p ford algebra bundles CM¢N and CM b CN, and from the above description it is easy to see that φ intertwines Clifford multiplication. We also note that φ intertwines the additional action of p Cm+n Cm b Cn.

Now choose Dirac operators DM and DN for the full spinor bundle of M and N. The symbol of p p the product operator DM ¢ DN = DM b 1 + 1 b DN is given by Clifford multiplication in the last 1.11. FUNDAMENTAL CLASSES AND POINCARÉ DUALITY 53 picture. Consequently it follows from Proposition 1.8.21 and the preceding lemma that bp ¢ ¢ [M] [N] = [DM DN] = [M N].

c 1.11.7 PROPOSITION. If U M is a G-invariant open subset of a smooth G-Spin -manifold M with boundary then incl![M, BM] = [U, BU]. Here, incl denotes the inclusion of the submanifold with boundary (U, BU) (M, BM). PROOF. This follows similarly from combining Proposition 1.8.23 and the preceding lemma.

We refer to [HR00, Proposition 9.6.6 and 11.2.5] for the proof of the next result.

1.11.8 LEMMA. For every G-space X we have p B([( )] b ¡) = G 0, 1 idK¦(X) . c ¢ c B Here, (0, 1] is endowed with its canonical orientation and G-Spin -structure (0, 1] Spin1, and is the boundary map in K-homology associated to the pair ((0, 1], t1u) ¢ X. In particular, we can use this lemma to show that our notion of G-Spinc-structure-preserving diffeomorphisms is indeed the correct one in the context of K-homology. This result will be quite useful in many of the geometrical constructions in the following. For example, it allows us to safely identify tubular neighborhoods with open subsets of a manifold, intervals of different lengths or spheres of different sizes. c 1.11.9 PROPOSITION. Let ϕ : (M, BM) Ñ (N, BN) be a G-Spin -structure-preserving diffeomorphism of smooth G-Spinc-manifolds with boundary. Then ϕ¦[M, BM] = [N, BN].

PROOF. Let us assume that M has empty boundary (otherwise replace M by M˚ ). Using Lemma 1.11.5 we can reduce to the case where ϕ is the identity. In other words, if M is a smooth G- 1 Spinc-manifold and if M is the same manifold equipped with the G-Spinc-structure determined by changing the Riemannian metric (in the sense of Corollary 1.10.8) then we need to show that 1 [M] = [M ]. c + ¢ By construction, both G-Spin -structures agree if considered as double coverings of GLM ∇M = + ¢ 1 ¢ GLM1 ∇M . Moreover, if we equip the Cartesian product N := R M with its canonical G- Spinc-structure and change the metric so that it smoothly interpolates between the original metric 1 over (0, 1) ¢ M and with the product metric with respect to M over (2, 3) ¢ M then we can 1 1 isometrically identify (0, 1) ¢ M and (0, 1) ¢ M (2, 3) ¢ M with respective subsets of N. As the ¢ ¢ 1 corresponding inclusions incl1 : (0, 1) M N and incl2 : (0, 1) M N are G-equivariantly homotopic we ﬁnd that ¢ 1.11.6 ¢ 1.11.7 ! ! 1.11.7 ¢ 1 1.11.6 ¢ 1 [(0, 1)] [M] = [(0, 1) M] = incl1[N] = incl2[N] = [(0, 1) M ] = [(0, 1)] [M ]. The claim now follows by applying the preceding lemma.

Instead of appealing to Lemma 1.11.8 we could probably just as well have constructed directly a KK-homotopy corresponding to a homotopy between the Riemannian metrics. Our next result is a more general statement relating submanifolds of the boundary with the boundary map of K-homology. c 1.11.10 PROPOSITION ([BOOSW10, Lemma 3.8]). Let N be a smooth compact G-Spin -manifold with boundary and let M BN be a G-Spinc-submanifold with boundary (of codimension 0). Then B[N, BN] = [M, BM] where B is the boundary map in K-homology for the G-pair (N˚ Y M˚ , M˚ ). 54 1. ANALYTIC K-THEORY AND K-HOMOLOGY

PROOF. Using Lemma 1.10.22 we can identify (0, 1] ¢ M˚ with an open neighborhood of t1u ¢ M˚ M˚ in M˚ Y N˚ . We have the following commutative diagram of short exact sequences ¢ ¢ 0 / C0((0, 1) M˚ ) / C0((0, 1] M˚ ) / C0(M˚ ) / 0

incl! incl! Y 0 / C0(N˚ ) / C0(N˚ M˚ ) / C0(M˚ ) / 0. The vertical inclusions are all given by extension by zero. We denote the boundary map associ- 1 ated to the upper sequence by B and recall that B is the boundary map associated to the lower sequence. Using naturality of boundary maps in KK-theory and the preceding two results we get

1 ! 1.11.7 1 1.11.8 B[N, BN] = B[N˚ ] = B incl [N˚ ] = B [(0, 1) ¢ M˚ ] = [M˚ ] = [M, BM].

We can use the preceding proposition to show that the boundary of a fundamental class is just the fundamental class of the boundary. c 1.11.11 COROLLARY. If M is a smooth compact G-Spin -manifold with boundary then [BM] = B[M, BM] where B is the boundary map in K-homology of the pair (M, BM). c 1.11.12 PROPOSITION. The opposite G-Spin -structure induces the negative fundamental class, i.e. [¡(M, BM)] = ¡[M, BM].

PROOF. By deﬁnition of the opposite G-Spinc-structure we may identify M˚ > ¡M˚ with the boundary of the smooth compact G-Spinc-manifold [0, 1] ¢ M˚ . Using naturality of boundaries with respect to the morphism t u ¢ ˚ Ñ ˚ ˚ proj2 : ([0, 1], 0, 1 ) M (M, M) given by projection onto the second component, we ﬁnd that

¦ ˚ > ¡ ˚ B ¦ ¢ ˚ P G ˚ (proj2) [M M] = (proj2) [(0, 1) M] = 0 K¦ (M) because we have factorized over K¦G(H) = 0. On the other hand,

¦ ˚ > ¡ ˚ ¦ ˚ ` ¡ ˚ ˚ ¡ ˚ (proj2) [M M] = (proj2) ([M] [ M]) = [M] + [ M] ¦ ˚ ¡ ˚ since, given our identiﬁcations, the restriction of (proj2) to the K-homology of both M and M is the identity (cf. Example 1.6.4).

1.11.13 REMARK (TWISTEDFUNDAMENTALCLASS). We can also consider twisted fundamental classes [M, BM; W]. These are deﬁned as above by considering twistings of the full spinor bundle of M˚ with the restriction of an auxiliary G-vector bundle W on M. Proposition 1.10.37 is easily adapted to this situation so that we have p G (1.11.14) [M, BM; W] = [Γ (W )] b [M, BM] = [W] X [M, BM] P K (M, BM). 0 M˚ C0(M˚ ) dim M From this representation it is easy to see that twisted fundamental classes are similarly compatible with products and boundaries. For example, we have B[ B ] = [B ] M, M; W M; W BM . 1.11.15 REMARK (REDUCEDFUNDAMENTALCLASS). If M is a smooth compact G-Spinc-manifold with boundary of even dimension n = 2k then we can also consider its reduced fundamental class which is deﬁned to be the class P G B [/SM˚ ] K0 (M, M) determined by the reduced spinor bundle over the interior of M. 1.11. FUNDAMENTAL CLASSES AND POINCARÉ DUALITY 55

G G 1.11.16 LEMMA. The image of the reduced fundamental class under Clifford periodicity K0 K2k is precisely the fundamental class of M.

PROOF. We may assume that M has empty boundary for simplicity of notation. Recall from Sketch 1.3.13 that Clifford periodicity is given by tensoring with the imprimitivity bimodule Wk. Consequently, we have to compare the fundamental class of M with the element [W ] bp [S ] = [ (M ¢ W )] bp [S ] P KKG(C (M) bp C C) k /M Γ0 k C0(M) /M 0 2k, . ¢ We claim that the full spinor bundle SM and the reduced spinor bundle /SM twisted with M Wk are isometrically isomorphic Dirac bundles. To see this, note that p c p (M ¢ W ) b /SM Spin ¢ c (W b W ). k M Spin2k k k c Here, the bundle is associated via the action of Spin2k C2k Ck,k on the second factor, while the additional left action of C2k Ck,k acts on the ﬁrst. Considering the right-hand side repre- bp 2k ¤ ¤ ¤ sentation as a representation of C4k C2k C2k, the action of the orientation element i e1 e4k on even elements is by (¡1)k times the identity (cf. proof of Lemma 1.2.17). On the other hand, c SM = Spin ¢ c W . M Spin2k 2k c ÞÑ P n Here, Spin2k acts via C2k C¡2k (induced by x ix for x R ) on W2k and the additional action of C2k is the canonical one. If we consider W2k as a representation of C4k, it is straightforward to verify that the action of the orientation element agrees with the above (use Remark 1.10.29). Consequently, it follows by the representation theory of complex Clifford algebras that both Dirac bundles are isometrically isomorphic and that the additional action of C2k is intertwined by the isomorphism. The assertion follows now by an easy variation of the proof of Proposition 1.10.37 taking the additional left action into account.

It is clear from the representation in Equation (1.11.14) that the analogue holds for twisted fundamental classes.

We will now prove Poincaré duality for compact G-Spinc-manifolds with boundary. We start with two technical computations which will turn out to be ﬁberwise at the heart of the matter.

1.11.17 LEMMA. The Clifford algebra Cn, considered as a bimodule over itself and equipped with the action c c of Spinn by left multiplication with the conjugate (which we denote by mul), is a Spinn-imprimitivity c c bimodule between Cn, equipped with the adjoint action of Spinn, and Cn, equipped with the trivial Spinn- action. c PROOF. This is clear except for equivariance with regards to the respective Spinn-actions. Indeed, the left action is equivariant since ¡1 ¡1 ¡1 @ P P c Ad(g)(x)y = (gxg )(y) = (gxg )y = g(xg (y))( x, y Cn, g Spinn), the right action is equivariant as ¡1 @ P P c yx = g(g (y)x)( x, y Cn, g Spinn), and the inner product is invariant because x y ¦ ¦ ¦ x y @ P P c g(x), g(y) = x g gx = x x = x, y ( x, y Cn, g Spinn).

1.11.18 LEMMA. We can consider the graded Hilbert Cn-C-bimodule Wn (equipped with the canonical c left action of Cn) as a Spinn-equivariant bimodule in each of the following ways: ¦ c 1. equip the C -algebra Cn with the trivial action and the bimodule Wn with the left action of Spinn n via Cn C¡n (induced by x ÞÑ ix for x P R ), or ¦ c 2. equip the C -algebra with the adjoint action and the bimodule with the rotation action of Spinn. 56 1. ANALYTIC K-THEORY AND K-HOMOLOGY

c Moreover, the interior tensor product of the Spinn-imprimitivity bimodule from the preceding lemma and c the ﬁrst option is isometrically isomorphic to the second option as graded Hilbert Spinn-Cn-C-bimodules (via the map x b y ÞÑ xy).

PROOF. Both options are special cases of the much more general construction of fundamental classes and de Rham classes (see Examples 1.11.4 and 1.9.12). Regarding the last assertion, it is of course clear that the module action implements an isometrical isomorphism bp Ñ b ÞÑ Cn Cn Wn Wn, x w xw of graded Hilbert Cn-C-bimodules. We have to show that gx(g y) = g (gxy) is equal to the P c P P image of xy under the rotation action of an element g Spinn for all x Cn and y Wn. One way P n ¡ ¦ to see this is the following: Recall that the action for a vector x C is given by i(λx λx¦ ). ¦ n Now under the canonical identiﬁcation of Wn = C with Cn (Equation (1.1.14)), the rotation c action of Spinn corresponds to its adjoint action. But in the proof of Example 1.2.13 we have c derived concrete expressions for the left and right multiplication action of Spinn on itself, and it is clear from these that g (gxy) P Wn indeed corresponds to t ¦ gxyg = gxyg = Ad(g)(xy) P Cn. This proves the claim.

c 1.11.19 REMARK. Conceptually, it makes in fact sense to use the conjugate Spinn-action on the imprimitivity bimodule. This is because the imprimitivity bimodule is supposed to “transform” c an action on Wn where scalars in U1 Spinn act by scalar multiplication (first option) to an action where scalars act as the identity (second option, since Ad(U1) 1). Note that the conjugate of the tautological vector bundle can be identified with the bundle asso- c ciated to the representation of Spin2k by left multiplication with the conjugate via the isometrical isomorphism taut c ¢ C c ¢ C SM = Spin mul n Spin mul n induced by complex conjugation. We define a left action of the Clifford algebra bundle in this last picture by the formula [p, x] ¤ [p, y] := [p, xy] (this is well-defined by Lemma 1.11.17). Note that this action has no interpretation as a Clifford multiplication. Of course we also have a canonical right action of Cn by right multiplication in the fiber. In the following we will always consider the conjugate bundle in this picture and equipped with these actions. c bp 1.11.20 PROPOSITION. Let M be a smooth G-Spin -manifold. The Hilbert G-Γ0(CM)-C0(M) Cn- bimodule of sections of the conjugate tautological vector bundle determines a KK-equivalence taut P G bp [Γ0(SM )] KK0 (Γ0(CM), C0(M) Cn). taut PROOF. From the preceding discussion we see that the KK-element [Γ0(SM )] is the image under the induction homomorphism c Spinn Ñ G bp KK0 (Cn, Cn) KK0 (Γ0(CM), C0(M) Cn) c of the class determined by the Spinn-imprimitivity bimodule Cn from Lemma 1.11.17 (see Prop- erty M and its Sketch 1.3.12). This element is a KK-equivalence because induction is compatible with Kasparov products and units.

c 1.11.21 PROPOSITION. Let M be a smooth G-Spin -manifold. Then taut p ¦ 0 [Γ (S )] b bp [M] = [[d + d ]] P K (Γ (CM), C). 0 M C0(M) Cm G 0 PROOF. By Lemma 1.11.18, we have the following isometrical isomorphism of graded Hilbert ¢ c c G Spinn-C0(SpinM, Cn)-C-bimodules: c p p 2 c 2 c C (Spin b Cn) b c L (Spin , Wn) Ñ L (Spin , Wn), u b v ÞÑ (p ÞÑ u(p)v(p)). 0 M C0(SpinM,Cn) M M 1.11. FUNDAMENTAL CLASSES AND POINCARÉ DUALITY 57

c Here, Spinn acts as follows on the individual objects: ¦ – trivially on the interior C -algebra Cn, ¦ – by the adjoint action on the C -algebra Cn whose action is intertwined by the isomorphism, – by left multiplication with the conjugate on the left-hand side bimodule Cn, – by left multiplication on the bimodule Wn in the middle, and – by rotations on the right-hand side bimodule Wn. We will now use the fact that interior tensor products are compatible with passing to the invariants ([Kas88, Lemma 3.3]). We can thus identify the graded Hilbert G-Γ0(CM)-bimodules via the induced isomorphism ©¦ taut p 2 2 ¦ Γ (S ) b bp L (SM) L ( T M) 0 M C0(M) Cn C (cf. Examples 1.10.26 and 1.10.28), but we still have to worry about the operators. Observe however that we are almost in the situation of a twisting. It is easy to see that under the above identiﬁcation the symbols of the Dirac operator D for the full spinor bundle and of the de Rham operator (or rather their induced actions on sections) are related by B bp ¡ u ¦ bp u σD(ξ)v = ( 1) σd+d (ξ)(u v)

(since in each ﬁber, the actions of Cn and C¡n on Wn commute gradedly). One uses this as in the proof of Proposition 1.10.37 to show that the conditions of a Kasparov product are satisﬁed.

We need one last ingredient before we can deduce Poincaré duality. c 1.11.22 LEMMA. For every smooth compact G-Spin -manifold M with boundary we have an isomorphism bp taut bp ˚ Ñ 1 ¦ taut b b ÞÑ (C(M) Γ0(SM˚ )) µC0(M) (µ ) Γ0(SM˚ ), ( f u) g f ug bp bp 1 of Hilbert G-C(M) Γ0(CM˚ )-C0(M) Cn-bimodules. Here, µ and µ denote the multiplication mor- bp ˚ Ñ ˚ bp Ñ phisms C(M) C0(M) C0(M) and C(M) Γ0(CM˚ ) Γ0(CM˚ ), respectively. PROOF. On the algebraic tensor product, the above assignment deﬁnes an isometry with dense range.

1.11.23 POINCARÉ DUALITY. Let M be a smooth compact G-Spinc-manifold with boundary and ¦ A a graded G-C -algebra. Then cap product with the fundamental class implements an isomorphism ¡X B ¦ ÝÑ G B [M, M] =: PDM : KG(M; A) Kdim M¡¦(M, M; A) which is natural in the coefﬁcient variable. taut PROOF. We claim that left Kasparov multiplication with [Γ0(SM˚ )] intertwines PDM with Kas- parov’s Poincaré Duality 1.9.13. As this element is a KK-equivalence this implies the main assertion of the theorem (Proposition 1.11.20). Let us use the notation of the preceding lemma. Then taut p [Γ (S ˚ )] b bp (x X [M, BM]) 0 M C0(M˚ ) Cn taut p p ¦ = [Γ (S ˚ )] b bp x b µ [M, BM] 0 M C0(M˚ ) Cn C(M) p taut p ¦ = x b [Γ (S ˚ )] b bp µ [M, BM] C(M) 0 M C0(M˚ ) Cn p p taut p ¦ = x b [C(M) b Γ (S ˚ )] b bp bp µ [M, BM] C(M) 0 M C(M) C0(M˚ ) Cn p p taut p = x b µ¦[C(M) b Γ (S ˚ )] b bp [M, BM] C(M) 0 M C0(M˚ ) Cn 1.11.22 p 1 ¦ taut p = x b (µ ) [Γ (S ˚ )] b bp [M, BM] C(M) 0 M C0(M˚ ) Cn 1.11.21 bp 1 ¦ ¦ = x C(M)(µ ) [[d + d ]]. 58 1. ANALYTIC K-THEORY AND K-HOMOLOGY

The last assertion, naturality in the coefﬁcient variable, is a direct consequence of the corresponding property of the cap product (Lemma 1.7.8).

1.11.24 REMARK. Although we believe that our comparison of Kasparov’s Poincaré isomorphism with its G-Spinc-counterpart might be of independent interest, this approach to proving Poincaré duality is rather indirect. It should also be possible to give a more concrete proof using a Mayer-Vietoris argument as n in [HR00, Exercise 11.8.11] if one replaces the domains R by associated vector bundles G ¢H W, since the Slice Theorem asserts that every G-orbit in a smooth G-manifold M has an open neighborhood equivariantly diffeomorphic to such a bundle (here, H is simply the stabilizer of any point of the orbit, and W is normal to the tangent space of the orbit at p and naturally an H-representation, see [GGK02, Theorem B.24] or [tD87, Theorem I.5.6], and also [CW92, Section 7] for a similar approach).

1.11.25 REMARK. Note that Poincaré duality together with the topological description of K-theory G B (Section 1.5) already gives a geometric description of Kp (M, M; A) in terms of G-A-vector bundles over M — at least if M is a compact G-Spinc-manifold with boundary and p dim M. In the second chapter we will show how to use this basic idea to deﬁne a geometric version of equivariant K-homology with coefﬁcients.

We will now describe how to relax some of our current restrictions so that we can deﬁne fundamental classes in more general situations.

1.11.26 REMARK (POINTSANDTHEEMPTYMANIFOLD). So far we have always assumed that manifolds are non-empty and of positive dimension in every connected component. Let us sketch how to remove these assumptions. H P G H It is clear that the fundamental class of the empty G-manifold ought to be [ ] := 0 K0 ( ) = t0u. Moreover, analogy with parallelizable manifolds of higher dimension indicates that every point should automatically be Spinc and that it should sufﬁce to specify an orientation (see Ex- ample 1.10.10). Consequently, we shall deﬁne the fundamental class of a point to be ¦ ¨ ¨ P ¦ [ ] := 1 = [C] K0( ) = KK0(C, C) where the sign is given by the orientation, and similarly for arbitrary zero-dimensional G-manifolds (by taking one copy of C for each point and letting the G-action permute accordingly). It is straightforward to verify that all results of this section, including Poincaré duality, are still true in the presence of empty and zero-dimensional G-manifolds. This will be useful as to sim- plify our geometric theory in the second chapter.

1.11.27 REMARK (Z/2-GRADEDTHEORY). If we consider analytic K-homology as a Z/2-graded theory then it in fact suffices to require throughout this section that the parity n of the dimension is the same for all connected components (and that only a finite number of dimensions occurs). In this case, the fundamental class is defined to be B ` B P G B [M, M] := i[Mi, Mi] Kn (M, M). Here, Mi denotes the union of those connected components of M which have dimension i n (mod 2), and we have identified the K-homology of (M, BM) with the direct sum of the K- B homology of the G-pairs (Mi, Mi) (see Example 1.6.4). Observe that this makes sense because the group action cannot permute components of different dimension. It is clear from Examples 1.4.4 and 1.6.4 and Lemma 1.7.9 that all statements, including Poincaré duality, still hold in this setting. 1.11. FUNDAMENTAL CLASSES AND POINCARÉ DUALITY 59

We conclude this section by collecting some important properties of the Poincaré duality isomorphism. c ¦ 1.11.28 LEMMA. Let M be a smooth compact G-Spin -manifold with boundary and A, B graded G-C - algebras. Then Y X @ P ¦ P ¦ PDM(x y) = x PDMy ( x KG(M; A), y KG(M; B)). PROOF. This follows from the computation Y Y X B 1.7.6 X X B X PDM(x y) = (x y) [M, M] = x (y [M, M]) = x PDMy.

Poincaré duality also intertwines restrictions with the generalized boundary map from Proposi- tion 1.11.10: c 1.11.29 LEMMA ([BOOSW10, Corollary 3.9]). Let N be a smooth compact G-Spin -manifold with boundary and let M BN be a G-Spinc-submanifold with boundary (of codimension 0). Then BPD [ ] = PD [ ] N E M E M for all G-A-vector bundles E over N. Here, B is the boundary map in K-homology for the G-pair (N˚ Y M˚ , M˚ ).

PROOF. In the following computations we will also denote the corresponding boundary element by B. We have

BPDN[E] = B([E] X [N, BN]) p p p p = ([E] b 1 ) b ¢ (1 b B) b ¢ ∆¦[N, BN]. C0(M˚ ) C(N M˚ ) C(N) C0(N N˚ ) bp B ¢ By the last remark in Property L, 1C(N) is the boundary element associated to the pair N (N˚ Y M˚ , M˚ ), i.e. to the ﬁrst extension in the commutative diagram ¢ ¢ Y ¢ 0 / C0(N N˚ ) / C0(N (N˚ M˚ )) / C0(N M˚ ) / 0

¦ ¦ ¢ ¦ ∆ ∆ (incl idM˚ ) Y 0 / C0(N˚ ) / C0(N˚ M˚ ) / C0(M˚ ) / 0. Here, incl denotes the inclusion M N. We now ﬁnd by naturality of boundaries that the above expression is in fact equal to p p ([E] b 1 ) b ¢ (incl ¢ id )¦∆¦B[N, BN] C0(M˚ ) C(N M˚ ) M˚ ¦ p p = (incl [E] b 1 ) b ¢ ∆¦B[N, BN] C0(M˚ ) C(M M˚ ) ¦ = incl [E] XB[N, BN]

= [E|M] X [M, BM]. For the last identity, we have used Proposition 1.11.10 and the fact that pulling back along the inclusion of a closed subspace corresponds to restricting vector bundles to this subspace (see Deﬁnition 1.5.13). 60 1. ANALYTIC K-THEORY AND K-HOMOLOGY

1.12. The double of a manifold

The double of a smooth G-manifold M with boundary, which we recall is defined to be the G- manifold > DM := M+ BM M¡ we get by glueing two copies of M along their boundaries (and using the opposite orientation on M¡), is an important tool that will allow us to reduce problems involving boundaries to the boundaryless case, without sacrificing important properties (such as compactness) or structure (such as G-Spinc-structures). For example, note that if M is a G-Spinc-manifold then its double carries a canonical G-Spinc-structure defined by using the opposite G-Spinc-structure on the second copy (this is clear from Definition 1.10.23 and Lemma 1.10.22). In the following we will relate the K-theory and K-homology of the double to the original space and then show that this is compatible with all important operations.

The ﬁrst key observation is to note that the associated inclusions incl¨ : M¨ ãÑ DM are sections of the respective projections proj¨ : DM Ñ M¨ which are deﬁned by identifying both copies of M. Consequently, the short exact sequence associated to the pair (DM, M) is split and we get the following direct sum decomposition:

1.12.1 LEMMA. We have a direct sum decomposition ¦ ¦ ` ¦ B KG(DM; A) KG(M; A) KG(M¡, M¡; A). ¦ ¦ Here, inclusion of and projection onto the ﬁrst summand are given by proj+ and incl+, respectively, and ¦ inclusion of the second summand is induced by the inclusion of G-C -algebras C0(M˚ ¡) C(DM) given by extension by zero.

Reversing the roles of M+ and M¡ and passing to the exact sequence of K-homology instead, we get the following “dual” result:

1.12.2 LEMMA. We have a direct sum decomposition

G G G K¦ (DM; A) K¦ (M, BM; A) ` K¦ (M¡; A). ¦ Here, projection onto the ﬁrst direct summand is induced by the inclusion of G-C -algebras C0(M˚ ) C(DM) given by extension by zero, and inclusion of and projection onto the second direct summand are given by (incl¡)¦ and (proj¡)¦, respectively. Pullback and pushforwards respect these direct sum decompositions:

1.12.3 LEMMA. Let f : (M, BM) Ñ (N, BN) be a continuous map between smooth G-manifolds and ¦ denote by F : (DM, M) Ñ (DN, N) the obvious extension to the double. Then f and f¦ are direct ¦ summands of F and F¦, respectively.

ROOF = P . Since F M f this follows readily from the naturality of the exact sequences of K-theory and K-homology.

As Poincaré duality is not a natural isomorphism, it is not obvious that it is compatible with the direct sum decomposition of K-theory and K-homology of a double. However, the following result shows that this is indeed the case. c 1.12.4 LEMMA (CF.[BOOSW10, Lemma B.2]). Poincaré duality for a compact Spin -manifold M is a direct summand of Poincaré duality for its double DM.

PROOF. We will use the notation of Lemmas 1.12.1 and 1.12.2. We also denote by j the inclusion B C0(M˚ ) C(DM) which implements the projection onto the K-homology of (M, M) and by s 1.12. THE DOUBLE OF A MANIFOLD 61 the corresponding inclusion. We have to show that the following diagram commutes:

¦ ¦ proj incl ¦ + / ¦ + / ¦ KG(M; A) KG(DM; A) KG(M; A)

PDM PDDM PDM ¦ j G B s / G / G B Kdim M¡¦(M, M; A) Kdim DM¡¦(DM; A) Kdim M¡¦(M, M; A)

Indeed, from the left square it follows that PDDM restricts to PDM and then the right square implies that PDM is indeed a direct summand. ¦ Right square: For all x P K (DM; A) we have ¦ ¦ X B PDM incl+(x) = incl+(x) [M, M] ¦ X ¦ = incl+(x) j [DM] ¦ ¦ bp b ¦ = ∆M(incl+(x) 1C(M)) C(M) j [DM] bp b ¥ ¦ ¥ ¢ ¦ bp = (x 1C(M)) C(DM¢M) [j ∆M (incl+ idM) ] C(DM)[DM] bp b ¦ ¥ bp bp = (x 1C(M)) C(DM¢M) [∆DM (idC(DM) j)] C(DM)[DM] bp b ¦ bp = (x [j]) C(DM¢DM) [∆DM] C(DM)[DM] ¦ ¦ bp bp = j ∆DM(x 1C(DM)) C(DM)[DM] ¦ = j (x X [DM]) ¦ = j PDDM(x).

Left square: Observe that s is characterized by the identities ¦ j s = 1, ran s = ker(proj¡)¦. Thus, by what we have just proved, ¦ ¦ ¦ ¦ ¦ j sPDM = PDM = PDM incl+ proj+ = j PDDM proj+ . ¦ As j is injective on the range of s, i.e. on the kernel of (proj¡)¦, it suffices to show that ¦ @ P ¦ 0 = (proj¡)¦PDDM proj+(x)( x KG(M; A)). Since Poincaré duality intertwines cap and cup product (Corollary 1.11.28) we only need to prove this for the trivial vector bundle. That is, we have to show that (proj¡)¦[DM] vanishes. To see this, recall that DM can be realized as the boundary of the manifold ([¡1, 1] ¢ M)/ where for every x PBM the equivalence relation identifies all (t, x) for t P [¡1, 1] (the classical construction still works in the situation of G-Spinc-manifolds, use Lemma 1.10.22). Identifying M¡ with (t¡1u ¢ M)/, we can define a G-map # ((M ¢ [¡1, 1])/, DM) Ñ ((M ¢ [¡1, 0])/, M¡) P : [t, x] ÞÑ [¡|t|, x] extending proj¡. Then, using naturality of boundary maps, we find that

1.11.11 (proj¡)¦[DM] = (proj¡)¦B[(M ¢ [¡1, 1])/, DM] = BP¦[(M ¢ [¡1, 0])/, M¡] = BP¦[M˚ ¢ (¡1, 0]]. ¦ Observe that we have factorized over the K-homology group of M˚ ¢ [0, 1). But the G-C -algebra ¢ C0(M˚ [0, 1)) is contractible (it is the cone of C0(M˚ )), hence its K-homology is trivial. This shows that the expression (proj¡)¦[DM] is indeed equal to zero. 62 1. ANALYTIC K-THEORY AND K-HOMOLOGY

1.13. Equivariant Spinc-structure of the spheres

c c In this section we will analyze the canonical Spinn+1-equivariant Spin -structure on the spheres Sn in more detail. This will be crucial for our topological derivation of Bott periodicity and the Thom isomorphism as presented in the next section. A short version of this has already appeared in [BOOSW10, Appendix A].

n+1 c The closed ball D is canonically a Spinn+1-manifold with boundary with respect to the rota- c Ñ tion action induced by the adjoint homomorphism Ad: Spinn+1 SOn+1. Clearly, the canonical Spinc-structure on Dn+1 which is given by the trivial bundle n+1 ¢ c Ñ n+1 D Spinn+1 D c c is Spinn+1-equivariant if we let Spinn+1 act by rotations on the base and by left multiplication on c the fiber (cf. Example 1.10.11). The boundary construction then induces a Spinn+1-equivariant Spinc-structure on Sn = BDn+1 (Definition 1.10.18). Since we are using the “outward-pointing c c ÞÑ P n normal vector first” convention and embed Spinn Spinn+1 by ek ek+1, the north pole e1 S c is stabilized by the rotation action of Spinn and we get the following lemma: c c n 1.13.1 LEMMA. The Spinn+1-equivariant principal Spinn-bundle of S is given by c Ñ Ad: Spinn+1 SOn+1. n Here, we identify SOn+1 with the oriented orthonormal frame bundle of S in such a way that the first column corresponds to the base point and the other columns specify the frame, and the left and right actions are given by multiplication.

PROOF. Consider the following diagram:

c / c Sn ¢ c Spinn+1 SpinDn+1 Sn Spinn+1 ¢ Ad idSn Ad / n ¢ SOn+1 SODn+1 Sn S SOn+1

n Here, the bottom arrow is the canonical embedding SOS SODn+1 Sn , which in our picture is given by sending a matrix x to the pair (xe1, x), and the top arrow is defined similarly by sending an element g to the pair (Ad(x)e1, x). It is evident that the diagram thus defined commutes, and the claim follows at once by comparison with the construction in Definition 1.10.18.

1.13.2 EXAMPLE (ONE-POINT COMPACTIFICATION OF UNITARY REPRESENTATIONS). Let W = Ck be a finite-dimensional unitary G-representation which we identify with R2k in the usual way. By Lemma 1.1.6, the action of G extends trivially to the one-point compactification W+ which thus becomes a smooth compact G-manifold. On the other hand, W+ S2k+1 carries the canonical Spinc-structure we have described above. Since G acts through the unitary group, we can use ãÑ c the canonical homomorphism Uk Spin2k from Example 1.10.13 and the above embedding c c c Spin2k Spin2k+1 in order to equip its principal Spin2k-bundle with a G-action. We thus get a G-Spinc-structure. Using this construction we can equip the one-point compactification of every finite-dimensional, unitary G-representation with a compatible G-Spinc-structure.

c c We now restrict the left action to Spinn Spinn+1 (under the above embedding). Then both n t¨ ¥ u hemispheres S¨ := x1 0 are invariant subsets and a variation of the argument in [ABS64, Section 13] yields the following representation: 1.13. EQUIVARIANT SPINc-STRUCTURE OF THE SPHERES 63

c c n 1.13.3 LEMMA. The Spinn-equivariant principal Spinn-bundle of S is (equivariantly) isomorphic to the one obtained by glueing the two bundles n ¢ c Ñ n S¨ Pinn,¨ S¨ t u ÞÑ c c c along the equator x1 = 0 via the identification ((0, x), g) (x, ixg). Here Pinn,+ = Spinn and Pinn,¡ c c is the other component of Pinn. The right action of Spinn is given by right multiplication in the fiber, and the left action is given by the adjoint action on the base point and the left multiplication in the fiber.

PROOF. We parametrize the hemispheres S¨ by use of polar coordinates in the following way:

¨ π ¢ t u ¢ n Ñ n ÞÑ ¨ p : [0, 2 ] ( 0 R ) S¨, (t, x) cos(t)e1 + sin(t)x. n n+1 Recall that our standard inclusion Cn Cn+1 was induced by the identification R R , ÞÑ x (0, x). But we also have an algebra isomorphism Cn Cn+1,+ given by sending a vector P n n+1 x R R (identified as above) to the element ixe1. It agrees with the canonical inclusion c for even elements. Let us write H¨ for the image of Pinn,¨ under this isomorphism. Now define candidate trivializations by the formula n ¢ Ñ c ÞÑ ¨ t t β¨ : S¨ H¨ Spin n , (p¨(t, x), h¨) ( cos( ) + sin( )xe )h. n+1 S¨ 2 2 1 Indeed, the maps are bundle morphisms since

t t ¡1 t t ¡1 Ad(β+(p+(t, x), h+))e1 = (cos( 2 ) + sin( 2 )xe1)h+e1h+ (cos( 2 ) + sin( 2 )xe1) t t t t = (cos( 2 ) + sin( 2 )xe1)e1(cos( 2 ) + sin( 2 )e1x) t t 2 = (cos( 2 ) + sin( 2 )xe1) e1 = (cos(t) + sin(t)xe1)e1 = p+(t, x) and ¡ ¡ ¡ ¡ ¡ ¡ t t ¡ 1 ¡ t t 1 Ad(β (p (t, x), h ))e1 = ( cos( 2 ) + sin( 2 )xe1)h e1h¡ ( cos( 2 ) + sin( 2 )xe1) ¡ ¡ t t ¡ t t = ( cos( 2 ) + sin( 2 )xe1)e1( cos( 2 ) + sin( 2 )e1x) ¡ ¡ t t 2 = ( cos( 2 ) + sin( 2 )xe1) e1 ¡ ¡ = (cos(t) sin(t)xe1)e1 = p¡(t, x). c Of course, both maps are equivariant with respect to the right action of Spinn, thus they are in fact local trivializations of the principal bundle. Moreover, β¨ are equivariant with respect to the c left action of Spinn since ¡ ¨ ¨ ¨ ¨ t t 1 ¨ ¨ ¨ β (Ad(g)p (t, x), gh ) = g( cos( 2 ) + sin( 2 )xe1)g gh = gβ (p (t, x), h ). We still need to compute the clutching function. To do so, note that both parametrizations agree π for points on the equator (t = 2 ). There, we have ¨ ¨ π ¨ ¨ π π ¨ β (p ( 2 , x), h ) = ( cos( 4 ) + sin( 4 )xe1)h , so the clutching function h+ ÞÑ h¡ is determined by the requirement that ¡ ¡ ¡ π π 1 π π h = ( cos( 4 ) + sin( 4 )xe1) (cos( 4 ) + sin( 4 )xe1)h+ ¡ π π 2 = (cos( 4 ) + sin( 4 )xe1) h+ ¡ π π ¡ = (cos( 2 ) + sin( 2 )xe1)h+ = xe1h+. c ÞÑ Replacing H¨ by Pinn,¨ we arrive at the clutching function g ixg.

1.13.4 REMARK. The additional factor i in our clutching function in contrast to [ABS64, Proposi- tion 13.2] or [BOOSW10, Lemma A.2] comes from our use of the Clifford algebra Cn = Cn,0. 64 1. ANALYTIC K-THEORY AND K-HOMOLOGY

Note that for every graded Cn-module W = W+ ` W¡, the module action induces a canonical isomorphism c Pin ¡ ¢ c W+ W¡. n, Spinn Consequently, we can describe the even parts of all spinor bundles on Sn by analogous clutching constructions: c n 1.13.5 COROLLARY. The Spinn-equivariant spinor bundle of S associated to a ﬁnite-dimensional graded Hilbert Cn-C-bimodule W is given by glueing the bundles n n S¨ ¢ W¨ Ñ S¨ t u ÞÑ along the equator x1 = 0 via the identiﬁcation ((0, x), w) ((0, x), ixw).

Let us now turn to computing equivariant indices of the associated Dirac operators in the case of 2k c even-dimensional spheres S . We consider again H := Spin2k+1. On the one hand, note that from the computation

2k 1.11.11 2k+1 2k 2k+1 2k collapse¦[S ] = collapse¦ B[D , S ] = B collapse¦[D , S ] = 0

(where using naturality of boundary maps we have factorized over R2k+1(H) = 0) it follows that the equivariant index of the Dirac operator associated to the reduced spinor bundle vanishes (Lemma 1.11.16). In fact, this argument only depends on the spinor bundle over the sphere being induced by the boundary construction from a spinor bundle over the ball, so the index still vanishes if we twist the reduced spinor bundle with any H-vector bundle that extends over the full ball (Remark 1.11.13). On the other hand, recall that for every even-dimensional Spinc-manifold M, Clifford multipli- p cation induces isometrical isomorphisms of Dirac bundles CM End(/SM) /SM b S/ M (Exam- ple 1.10.28). Also recall from Example 1.9.15 that we can consider the de Rham operator as a Dirac operator for CM. Its kernel consists precisely of the harmonic forms (proof of Proposition 1.9.7). Hence in the case M = S2k, de Rham’s theorem and the cohomology of S2k show that it is spanned by a 0-form and a 2k-form, both of which are rotation-invariant. In other words, ¦ indexH(d + d ) = 2 P R(H).

Now there is a canonical involution on CM induced by right Clifford multiplication with γ := k ¤ ¤ ¤ i E1 E2k where (Ei) is any oriented local orthonormal frame. Let us denote its positive eigen- 1/2 1/2 b + bundle by CM . The above isomorphism restricts to CM /SM S/ M , and the eigenbundle is left invariant both by the group action (since it leaves the element γ invariant) and by the de Rham operator (cf. [HR00, Example 11.1.4]). In the case M = S2k, the involution evidently interchanges 0-forms and 2k-forms and we conclude from the above discussion that + bp 1.10.38 P [S/S2k ] C(S2k)[/SS2k ] = 1 R(H). We summarize our ﬁndings in the following proposition:

1.13.6 PROPOSITION. We have + ¡ b P c ([S/S2k ] [E]) C(S2k) [/SS2k ] = 1 R(Spinn+1) 2k c for every equivariant vector bundle E over S pulled back from a representation in R(Spinn+1). 1.14. BOTT PERIODICITY AND THOM ISOMORPHISM 65

1.14. Bott periodicity and Thom isomorphism

Bott periodicity and the Thom isomorphism are classical results of K-theory, and it is well-known that they can be implemented by Kasparov multiplication with certain KK-equivalences called the Bott element and Thom element, respectively. As the relationship between their topological and analytical descriptions is crucial to a proper understanding of a certain equivalence relation in geometric K-homology (vector bundle modification), we will describe their construction in this section. P 0 2k We start with the topological construction of the equivariant reduced Bott element b2k KH(R ) which is a simple consequence of the analysis of the previous section. Here, H is any group acting 2k Ñ c spinorly on R , i.e. the action factors via a continuous group homomorphism H Spin2k over c the canonical rotation action of Spin2k (see Example 1.10.11). Clearly, the above results still hold 2k c c for S equipped with the induced action of H (via the canonical embedding Spin2k Spin2k+1). Using stereographic projection from the south pole, we can identify R2k H-equivariantly with an open subset of its one-point compactification S2k whose complement is the south pole. We get an induced identification of the respective K-theory and K-homology groups.

1.14.1 LEMMA. Under these identiﬁcations we have 2k 2kzt¡ u ! 2k [R ] = [S e1 ] = incl [S ] where incl denotes the inclusion R2k S2k.

PROOF. The second identity is of course the content of Proposition 1.11.7. For the ﬁrst identity, we note that every orientation-preserving self-diffeomorphism of Rn automatically preserves the c G-Spin -structure (Example 1.10.10) and apply Proposition 1.11.9.

Now consider the split short exact sequence in K-theory

incl / 0 2k ! / 0 2k / 0 ¦ / (1.14.2) 0 KH(R ) KH(S ) o KH( ) = R(H) 0 + (cf. Remark 1.4.5). We can use it to pull the south pole fiber of the Bott bundle F2k := S/S2k back to a 8 ¡ 8 bundle F2k over the entire sphere, so that by exactness we find a unique preimage of [F2k] [F2k ] P 0 2k which we call the reduced Bott element b2k KH(R ). By the above, it satisfies the following relation: p 1.14.1 p ! p 1.13.6 b b 2k [/S 2k ] = b b 2k incl [/S 2k ] = incl b b 2k [/S 2k ] = 1 P R(H) 2k C0(R ) R 2k C0(R ) S ! 2k C(S ) S

We can now use the following version of Atiyah’s rotation trick from [Ati68] to establish that b2k is in fact a KK-equivalence: 0 n H n 1.14.3 EMMA Let b P K (R ) and D P K (R ) satisfy b b n D = . Then they are in fact L . H 0 C0(R ) 1 KK-equivalences inverse to each other.

PROOF. As the rotation (x, y) ÞÑ (y, ¡x) is H-equivariantly homotopic to the identity of R2n, we have p p H n 2n 1 n b b = b b P KK (C (R ) C (R )) C0(R ) Θ 0 0 , 0 where Θ is the KK-involution corresponding to x ÞÑ ¡x (cf. [HR00, p. 112]). Consequently, we ﬁnd that p p p p D b b = (1 n b b) b 2n (1 n b D) C0(R ) C0(R ) C0(R ) p p p = (b b Θ) b 2n (1 n b D) C0(R ) C0(R ) p p = (b b n D) b C0(R ) Θ = Θ. This shows that b also has a left KK-inverse (and, in fact, Θ = 1). 66 1. ANALYTIC K-THEORY AND K-HOMOLOGY

1.14.4 TOPOLOGICAL BOTT PERIODICITY. Let H be a compact Lie group acting spinorly on R2k. P 0 2k Then the associated reduced Bott element b2k KH(R ) and the reduced fundamental class P H 2k [/SR2k ] K0 (R ) are KK-equivalences inverse to each other. ¦ In particular, Kasparov multiplication by these elements induces natural isomorphisms KH ¦ ¡ ¢ 2k 2k KH( R ) with respect to the trivial action of H on R . This is the famous Bott Periodicity Theorem in K-theory.

We will now derive an analytic cycle for the reduced Bott element. Clearly, the difference bundle ¡ 8 0 2k [F2k] [F2k ] is represented by the KH(S )-cycle ` 8 op (Γ(F2k) Γ(F2k ) , 0). + 8 Recall the description of F2k = S/S2k from Corollary 1.13.5. Evidently, F2k can be described by a similar clutching construction given by using the southern hemisphere representation over 1 both hemispheres and glueing using the identity. We can thus define an operator T acting on 1 even/odd sections in the following way: On the upper hemisphere, T acts by pointwise Clifford multiplication by ï times the vector containing the last n coordinates of the base point, and on 1 the lower hemisphere it acts by the identity. Evidently, the operator T is H-equivariant, self- adjoint and squares to the identity, and we get another cycle ` 8 op 1 (Γ(F2k) Γ(F2k ) , T ) which is homotopic to the above using the linear path. Now we can restrict to the open upper hemisphere via the homotopy given by the Hilbert C([0, 1], C(S2k))-module t P ` 8 op P ` 8 opu f C([0, 1], Γ(F2k) Γ(F ) ) : f (0) Γ0(F2k 2k ) Γ0(F 2k ) 2k S˚+ 2k S˚+ and thea obvious operator. Identifying R2k with the open upper hemisphere via the map x ÞÑ (1, x)/ 1 + kxk2 (which is equivariantly homotopic to our previous identification) we conclude 0 2k that the reduced Bott element can also be represented by the KH(R )-cycle 2 (C (R2k, W ), T ) 0 k a 2k 2 ¨ 2 with the obvious Hilbert C0(R )-module structure and where T acts by ix/ 1 + kxk on the standard irreducible representation Wk of C2k Ck,k. Note that this is indeed the correct representation since the reduced spinor bundle uses the conjugate and F2k is the even part of its dual (cf. Example 1.10.28). After transforming the cycle via the isometric isomorphism of Hilbert modules induced by # w if w is even W Ñ W , w ÞÑ k k ¡iw if w is odd we arrive at the following analytical representation of the reduced Bott element: b = [C (R2k, W ), T] P K0 (R2k) 2k 0 k H a Here, T is the operator given by pointwise Clifford multiplication by x/ 1 + kxk2. Now observe that bp ¦ bp 2k bp P (b k [W ]) 2k bp [R ] = b k 2k [/S 2k ] = 1 R(H) 2 k C0(R ) C2k 2 C0(R ) R since the image of the reduced fundamental class under Clifford periodicity is the fundamental class (Lemma 1.11.16). Using the canonical isomorphism C2k L(Wk) induced by the module action we get the following representation of the Bott element β2k: ¦ 1.3.15 β := b bp [W ] = [C (R2k, C ), T ] P KKH(C, C (R2k) bp C ) K2k(R2k) 2k 2k k 0 2k 2k 0 a0 2k H 2 c Here, T2k acts by pointwise Clifford multiplication by x/ 1 + kxk , and H (via Spin2k) acts by 2k ¦ rotations on R , by left multiplication on the Hilbert module C2k and trivially on the C -algebra 1.14. BOTT PERIODICITY AND THOM ISOMORPHISM 67

C2k. Observe that T2k is equivariant with respect to that action. This is the familiar cycle for the Bott element due to Kasparov [Kas81, Section 5]. We have proved the following result for n = 2k:

1.14.5 ANALYTICAL BOTT PERIODICITY. Let H be a compact group acting spinorly on Rn. Then the Bott element n P n n βn := [C0(R , Cn), Tn] KH(R ) (where Tn is the Clifford multiplication operator deﬁned as above) and the fundamental class n P H n [R ] Kn (R ) are KK-equivalences inverse to each other. A purely analytic argument can be used to show that it holds in arbitrary dimensions (see e.g. n [Kas81, 5.7]). Note that we have βn = (β1) (for appropriate group actions); this follows readily from the product formula for fundamental classes (Proposition 1.11.6).

1.14.6 REMARK. The same cycle make sense for arbitrary orthogonal actions on Rn if the Clifford algebra is equipped with the induced action. In this case one also gets a KK-equivalence, but the inverse is the equivariant Dirac element from Example 1.9.12 (see [Kas81, p. 546] or [Bla98, Theorem 20.3.2]).

1.14.7 REMARK. It also follows from Bott periodicity that [M] never vanishes for any non-empty smooth Spinc-manifold M. This is because restricting to the domain of a coordinate chart sends [M] to some [Rn]. But we have just seen that this is a KK-equivalence and thus in particular cannot be equal to zero.

We will now use Kasparov’s induction machinery in order to define Thom elements. Let G be an Ñ c arbitrary compact Lie group and πW : W X a G-Spin -vector bundle of rank n over a G-space c c c n X. Then Spin /Spin X and Spin ¢ c R W. We define the Thom element β as the W n W Spinn W image of the Bott element βn under the induction homomorphism c n n Spinn n ÝÑ G K c (R ) = KK¡ (C, C0(R )) KK¡ (C0(X), C0(W)). Spinn n n As in the case of Poincaré duality, compatibility with Kasparov products and units shows that βW is a KK-equivalence (Property M). Its inverse which we will call the fiberwise Dirac element [W/M] is given by the image of the fundamental class [Rn] under the analogous composition (see Proposition 1.14.11 below for the terminology). From Sketch 1.3.12 we see that the Thom element can be represented in the following way: c c ¢ n Spinn ¥ ¡ βW = [C0(SpinW R , Cn) , Tn ] This cycle can be represented in a more intuitive way using the isometric isomorphism c ¦ c ¢ n Spinn Ñ ÞÑ ÞÑ C0(SpinW R , Cn) Γ0(πW (SW )), f ([p, x] ([p, x], [p, f (p, x)])) taut c ¢ of graded Hilbert G-C0(W)-modules. Here, SW/M := SpinW mul Cn is called the tautological spinor bundle of W. Chasing the operator through this identification we get the following result: Ñ c 1.14.8 ANALYTICAL THOM ISOMORPHISM. Let πW : W X be a G-Spin -vector bundle of constant rank over a G-space X. Then the Thom element ¦ taut P G βW = [Γ0(πW (SW/M)), TW/M] KK¡ rank W (C0(X), C0(W)) is a KK-equivalence. Here, TW/M is the operator of pointwise Clifford multiplication defined by ? w ¤ (TW/Ms)(w) := s(w). 1+kwk2

In the Z/2-graded picture of KK-theory, the corresponding assertion also holds for G-Spinc- vector bundles with different rank over each connected component as long as their parity is the same. This is easily seen by working with one connected component at a time. The same argument applies to the results which follow. 68 1. ANALYTIC K-THEORY AND K-HOMOLOGY

In the even case, rank W = 2k, we can perform the same construction starting with the reduced Bott element b2k. The resulting reduced Thom element bW is just the image of βW under Clifford periodicity (as the latter is compatible with induction). Clearly, bW is given by the obvious cycle c using the reduced spinor bundle /S := Spin ¢ c W . But we can also start from the W/M W Spinn k topological representation ¡ 8 ` 8 op P 0 2k incl! b2k = [F2k] [F ] = [Γ(F2k) Γ(F ) ] K c (S ). 2k 2k Spin2k c 2k c c Write Xˆ := Spin ¢ c S with respect to the canonical rotation action of Spin Spin W Spin2k 2k 2k+1 (i.e. the one we used above). This is a fibration of spheres (fiberwise one-point compactification of c the bundle W) and in fact the unit sphere bundle of (X ¢ R) ` W. If we think of F := Spin ¢ c W W Spin2k 8 c 8 F and F := Spin ¢ c F as G-vector bundles over Xˆ then a straightforward computation 2k W W Spin2k 2k shows that the image of the above cycle is given by ` 8 op P G ˆ [Γ0(FW ) Γ0(FW ) ] KK0 (C0(X), C0(X)). By naturality of induction, this is equal to incl!(bW ), the pushforward of the reduced Bott element along the second arrow in the short exact sequence associated to the pair (Xˆ , X), i.e.

incl! c 2k / / C (W) / C (Spin ¢ c S ) o C (X) / 0 0 0 W Spin2k 0 0.

c c ¡ Spin2k This is indeed the image of the short exact sequence (1.14.2) under the operation C0(SpinW, ) , and the right split is given by extending the values of a function in C0(X) over each sphere. More- over, exactness of the above sequence shows that bW is uniquely characterized by this property.

1.14.9 TOPOLOGICAL THOM ISOMORPHISM. Let W be a G-Spinc-vector bundle of even rank over each connected component over a G-space X. Then the reduced Thom element bW, which is given by the cycle ` 8 op P G G ˆ [Γ0(FW ) Γ0(FW ) ] KK0 (C0(X), C0(W)) KK0 (C0(X), C0(X)) under the identiﬁcation induced by the split short exact sequence associated to the pair (Xˆ , X), ¢ ` is a KK-equivalence. Here, Xˆ is the unit sphere bundle of (X R) W, FW is the Thom bundle ˆ 8 over X which agrees with some F2k over each sphere and FW is the pullback of its south pole restriction back to Xˆ .

1.14.10 REMARK. We have seen before in Corollary 1.13.5 that F2k can be described (equivariantly) using a clutching construction. Consequently, the associated bundle F also arises from a clutching construction, and it is easy to see that it is precisely the one used for vector bundle modiﬁcation in [BD82, BHS07, BHS08, BOOSW10].

The intuition that the inverse of the Thom element is a ﬁberwise Dirac operator can be made precise in the following way: Ñ c 1.14.11 PROPOSITION. Let πW : W M be a smooth G-Spin -vector bundle of constant rank over a ¦ smooth compact G-Spinc-manifold M. Every choice of G-equivariant decomposition TW π (W) ` ¦ W πW (TM) determines a G-invariant Riemannian metric on the total space W which thus becomes a G- Spinc-manifold by the Two out of Three Principle 1.10.15. Clearly, such decompositions are equivalent to G-invariant connections on W which we have already seen to exists, and every choice leads to isomorphic G-Spinc-structures. Then [W M] bp [M] = [W] P KG (W) / C0(M) dim W . PROOF. In the course of the proof, we will annotate all entities related to W considered as a vector bundle by the subscript W/M. For example, we write CW/M for its Clifford algebra bundle, c c c Spin for its principal Spin -bundle and S := Spin ¢ c W for its full spinor bundle. W/M n W/M W/M Spinn n 1.14. BOTT PERIODICITY AND THOM ISOMORPHISM 69

c The corresponding objects for the total space are denoted by CW, SpinW and SW, and similarly for the base space. The canonical decomposition of the tangent space induces an isomorphism ¦ bp ¦ πW (CW/M) πW (CM) CW of algebra bundles, and by construction of the Spinc-structure the respective full spinor bundles are similarly related by an isomorphism ¦ bp ¦ (1.14.12) πW (SW/M) πW (SM) SW.

Note that this latter isomorphism intertwines the respective CW-actions (defined on the first bundle via the above isomorphism). We have seen before that [W/M] is the image of the fundamental class [Rn] under the induction homomorphism. Hence we find that, similar to the case of the Thom element, c ¦ ¦ c 2 n Spinn ¥ ¡ 2 [W/M] = [(C0(SpinW/M, L (R , Wn)) , χ(d + d ) ] = [L (πW (SW/M)), TW/M].

Here, TW/M is aligned with a differential operator DW/M whose symbol is given by the Clifford multiplication of CW/M. Observe that Equation (1.14.12) induces an isometric isomorphism ¦ L2( (S )) bp L2(S ) L2(S ) πW W/M C0(M) M W of Hilbert G-C0(W)-C-bimodules so that the desired identity holds on the level of bimodules. We still have to show that it holds on the level of operators. An easy but indirect argument goes as follows: Since the condition of being a Kasparov product can be veriﬁed locally, we can P ¡1 always work relative to functions f C0(W) supported in the preimage πW (U) of an open set U over which the vector bundle is trivial. The G-Spinc-structure on the total space is constructed ¡1 n ¢ precisely in such a way that we can identify πW (U) with the Cartesian product R U in a c G-Spin -structure-preserving fashion. Under this identiﬁcation, the differential operator DW/M ¦ restricts to a compact perturbation of (d + d ) bp 1, and any Dirac operator for W restricts to a Dirac operator for Rn ¢ U. Of course, any Dirac operator for M restricts to a Dirac operator for U. The claim thus follows from the product formula for fundamental classes (Proposition 1.11.6).

We conclude this section by proving that the Thom element is natural with respect to restriction to closed subsets. Ñ c 1.14.13 LEMMA. Let πW : W X a G-Spin -vector bundle of constant rank over a G-space X. Denote by incl the inclusion of a closed subspace A X and by inclW the corresponding inclusion W A W. Then ¦ G incl β = incl¦ β | P KK (C (X), C (W )). W W W A ¡ rank W 0 0 A

ROOF ( ) ( ) P . There exists an obvious isometric isomorphism of Hilbert G-C0 X -C0 W A -bimodules ¦ ¦ taut bp ¦ Ñ taut b ÞÑ Γ0(πW (SW/M)) incl C0(W ) Γ0(πW (SW )), u f f u ¡1 W A A A πW (A) intertwining Clifford multiplication. The first bimodule is the one in the first cycle, and the second bimodule can be identified with the one in the second cycle. 70 1. ANALYTIC K-THEORY AND K-HOMOLOGY

1.15. Gysin maps

Every proper G-map f : M Ñ N induces a pullback morphism in equivariant K-theory. In the case of compact G-Spinc-manifolds with boundary, Poincaré duality can be used to deﬁne a pushforward morphism which goes in the other way.

1.15.1 DEFINITION (GYSINMAP). Let f : (M, BM) Ñ (N, BN) be a G-map between smooth com- c pact G-Spin -manifolds with boundary. The Gysin map f! is deﬁned to be the unique morphism making the following diagram commute:

PD ¦ M / G B KG(M; A) K ¡¦(M, M; A) dim M

f! f¦ dim N¡dim M+¦ / G B K (N; A) Kdim M¡¦(N, N; A) G PDN

¥ ¥ c Clearly, Gysin maps are functorial in the sense that (g f )! = g! f! if the G-Spin -structure of the intermediate manifold is chosen to be the same.

1.15.2 EXAMPLE. In particular, the inclusion incl: (M, BM) (N, BN) of a union of connected components of a smooth compact G-Spinc-manifold M with boundary determines a Gysin map. We claim that it agrees with the pushforward map incl! we have deﬁned in Section 1.4. Recall that the latter was deﬁned to be the pushforward associated to the morphism C(M) C(N) of ¦ G-C -algebras given by extending functions by zero. Thus

incl¦ PDM(x) = incl¦(x X [M, BM]) 1.11.7 = incl¦(x X incl![N, BN]) 1.7.9 X B = incl!(x) [N, N]

= PDN incl!(x).

1.15.3 REMARK (GYSINELEMENTS). It was an important discovery by Connes and Skandalis that these maps can be implemented in terms of right Kasparov multiplication by functorial Gysin P G elements f! KKdim M¡dim N(C(M), C(N)). For details on their approach we refer the reader to the landmark article [CS84], where the construction is in fact done for more general K-oriented maps, and to [KS91] for the equivariant version.

Another way to see this is to observe that pushforwards as well as Poincaré duality and its inverse can be written in terms of Kasparov products. We have seen this for the ﬁrst two operations, and for the latter one can use the dual Dirac element from [Kas88] as in our proof of Poincaré duality.

In the following we will prove descriptions of Gysin maps in terms of Kasparov products with certain KK-elements, but we will not show that these elements are in fact functorial.

c 1.15.4 PROPOSITION. Let M, N be smooth closed G-Spin -manifolds. The Gysin map associated to the ¢ Ñ projection proj2 : M N N is implemented by right Kasparov multiplication by the fundamental class of M, i.e.

¡ bp ¦ ¢ Ñ ¦¡dim M (proj2)! = C(M)[M] : KG(M N; A) KG (N; A). 1.15. GYSIN MAPS 71

PROOF. We will show that this implementation ﬁts into the above diagram. Note that the diagonal maps of M, N and M ¢ N are related by the formula ∆M¢N = ∆M ¢ ∆N. Consequently, ¦ ¦ X ¢ (proj2) PDM¢N(x) = (proj2) (x [M N]) ¦ bp ¦ bp ¢ = ∆M¢N(x [proj2 ]) C(M¢N)[M N] ¢ ¦ bp ¦ bp bp ¢ = (∆M ∆N) (x [collapseM] 1N) C(M¢N)[M N] ¦ bp bp ¢ = (∆N) (x 1N) C(M¢N)[M N] 1.11.6 ¦ bp bp bp = (∆N) (x 1N) C(M) bp C(N)([M] [N]) ¦ bp bp bp = (∆N) (x C(M)[M] 1N) C(N)[N] bp X = (x C(M)[M]) [N] bp = PDN(x C(M)[M]).

1.15.5 DEFINITION (EMBEDDING). Let us also define the appropriate notion of an embedding of smooth G-Spinc-manifolds (with or without boundary). This will be a G-equivariant embedding as in Definition 1.8.1 (preserving the boundaries and transversal to the boundary) which preserves the G-Spinc-structure on connected components of codimension 0. This latter requirement ensures that we can always equip the normal bundle with a canonical G- Spinc-structure via the Two out of Three Principle 1.10.15 if we consider M to be equipped with the induced Riemannian metric. c 1.15.6 PROPOSITION. Let f : (M, BM) ãÑ (N, BN) be an embedding of smooth compact G-Spin -manifolds with boundaries and consider the normal bundle νM as an open tubular neighborhood of M in N (cf. Definition 1.8.1). The Gysin map associated to f is then given by = (¡ bp ) ¦ ( ) Ñ dim N¡dim M+¦( ) f! incl! C(M)βνM : KG M; A KG N; A . c Here, βν is the Thom element of the normal bundle (equipped with its canonical G-Spin -structure we M ¦ have just defined) and incl denotes the inclusion νM N of G-C -algebras.

PROOF. First of all, note that by Proposition 1.11.9 we may indeed assume that M is equipped with the Riemannian metric induced from N. For closed manifolds: Let us for now assume that both M and N are closed manifolds. Then the assertion follows from the computation

f¦PDM(x) = f¦(x X [M]) ¦ bp bp = f¦∆M(x 1C(M)) C(M)[M] ¦ = f¦ (x bp 1 ) bp bp [ M] bp [M] ∆M C(M) C(M)βνM C0(νM) νM/ C(M) 1.14.11 ¦ = f¦ (x bp 1 ) bp bp [ ] ∆M C(M) C(M)βνM C0(νM) νM 1.11.7 ¦ = f¦ (x bp 1 ) bp bp ![N] ∆M C(M) C(M)βνM C0(νM) incl ¦ ¦ bp bp bp = f ∆M(x 1C(M)) C(M) incl! βνM C(N)[N] ! ¦ bp bp bp = ∆N(x C(M) incl! βνM 1C(N)) C(N)[N] bp X = (x C(M) incl! βνM ) [N] bp = PDN incl!(x C(M)βνM ) ¦ if we manage to prove the identity marked with an exclamation mark for all x P K (M; A). We will in fact show that ¦ ¦ ¦ bp bp bp bp (1.15.7) f ∆M(x 1C(M)) C(M) incl! βνM = ∆N(x C(M) incl! βνM 1C(N)) 72 1. ANALYTIC K-THEORY AND K-HOMOLOGY by carefully analyzing the KK-cycles involved. For this, represent x by some cycle (E0, T0). Then using the analytical description of the Thom isomorphism 1.14.8 we can write ¦ ¦ ∆ (x bp 1 ) bp incl β = [E bp (A bp Γ (π (Staut ))), T] M C(M) C(M) ! νM loooooooooooooooooooooomoooooooooooooooooooooon0 C(M,A) 0 νM νM/M =:E for some operator T. Here, we can think of the left action of a function g P C(M) to be given by pointwise multiplication of sections of the pullback bundle by the pullback of g along πνM . The pullback of this action gives a concrete description of the left-hand side of Equation (1.15.7) in terms of the cycle (E, T) with left action ü ¤ b b b b ¥ ¥ b b ÞÑ C(N) E, g e a u := e a (g f πνM )u = e a (v g(0v)u(v)). On the other hand, ¦ bp bp ¦ bp ¦ bp collapse¦ ∆M(x 1C(M)) C(M) incl! βνM = ∆M(x [collapse ]) C(M) incl! βνM bp = x C(M) incl! βνM . Consequently, the right-hand side of Equation (1.15.7) is given by ¦ bp bp bp ¦ bp bp ∆ ([E, T] 1 ( )) = [(E C(N)) C(N, A), T 1 1] = [E, T] N C N ∆N where in the last cycle functions in C(N) act on sections of the pullback bundle simply by pointwise multiplication by their restriction to νM N. It is intuitively clear that both left actions should result in homotopic cycles since they differ by a ﬁberwise contraction of the vector bundle. The Hilbert bimodule ¦ E bp (A bp C([0, 1], Γ (π (Staut )))) 0 C(M,A) 0 νM νM/M with the obvious operator and action C(N) ü C([0, 1], E), g ¤ e b a b h := e b a b ((t, v) ÞÑ g(tv)h(t, v)) implements this idea. It is clear that this deﬁnes a homotopy between both cycles. General case: Denote by F : (DM, M) Ñ (DN, N) the extension of f to the doubles (cf. Section 1.12). Let us also embed the normal bundles consistently into the destination spaces so that | νM = (νDM) N. By the above, we know that F! is implemented by right Kasparov multiplication by the element P (inclD)!βνDM KKdim M¡dim N(C(DM), C(DN)) where inclD denotes the inclusion νDM DN. On the other hand, Lemmas 1.12.3 and 1.12.4 show that f! is a direct summand of F!. Consequently, we only have to prove that the following diagram commutes: ¦ incl ¦ M / ¦ KG(DM; A) KG(M; A)

¡ bp ( ) ¡ bp C(DM) inclD ! βνDM C(M) incl! βνM ¦ incl ¦ N / ¦ KG(DN; A) KG(N; A)

Here, inclM and inclN denote the inclusions of M and N into their respective double, and we will also need the inclusion inclν of the normal bundle of M into the normal bundle of its double. Naturality of the Thom element with respect to restriction to closed subspaces as used in the following computation shows that the above diagram is indeed commutative: ¦ bp bp ¦ inclM(x) C(M) incl! βνM = x C(DM) incl!(inclM) βνM 1.14.13 bp ¦ = x C(DM) incl!(inclν) βνDM bp ¦ = x C(DM) inclN(inclD)!βνDM ¦ bp = inclN(x C(DM)(inclD)!βνDM ) 1.15. GYSIN MAPS 73

1.15.8 EXAMPLE. Let M be a smooth compact G-Spinc-manifold with boundary and W a smooth G-Spinc-vector bundle over M. Consider the unit sphere bundle Mˆ of (M ¢ R) ` W. Its base point projection π : Mˆ Ñ M determines a short exact sequence ¦ (1.15.9) 0 / ker(π¦) / TMˆ / π TM / 0. ¦ Both ker(π¦) and π TM carry canonical G-invariant Riemannian metrics and G-Spinc-structures determined by the vector bundles (M ¢ R) ` W and TM. This is clear for the latter bundle. For the former, observe that there exists a trivial G-vector bundle ρ over Mˆ such that

` ( ¦) vert(( ¢ R) ` ) ρ ker π T M W Mˆ (ρ is the bundle spanned by the radially-pointing vertical tangent vectors) and apply the Two out of Three Principle 1.10.15. This essentially amounts to performing the boundary construction of Section 1.13 in each ﬁber (note that we cannot apply the boundary construction to the unit disk bundle because it is in general a manifold with corners). Consequently, any choice of a G-equivariant splitting of the short exact sequence (1.15.9) automatically determines a G-invariant Riemannian metric on Mˆ as well as a G-Spinc-structure on Mˆ (by the Two out of Three Principle 1.10.15). Such splittings certainly exist because we can always average over the compact group G. For example, we could choose a G-equivariant splitting for the tangent bundle of the total space (that is, a G-invariant connection, cf. Proposition 1.14.11) and then “restrict” the Riemannian metric to the unit sphere bundle). At any rate, it is clear that any two choices of such a splitting lead to isomorphic G-Spinc-structures (over the identity). Observe that we can identify the normal bundle of the canonical north pole section ( B ) Ñ ( ˆ ˆ ) ÞÑ ( ) ` s : M, M M, M BM , x x, 1 0Wx with W (in the sense of G-Spinc-vector bundles), so that from the preceding proposition we ﬁnd that ¡ bp ¦ Ñ rank W+¦ ˆ s! = incl!( C(M)βW ) : KG(M; A) KG (M; A). Here, incl denotes the inclusion W Mˆ . We can now use the topological description of the Thom isomorphism in order to get the following geometric description of the Gysin map induced by the north pole section.

1.15.10 PROPOSITION. In the situation of the preceding example, we have ¦ ¡ Y ¡ 8 ¦ Ñ rank W+¦ ˆ s! = π ( ) ([FW ] [FW ]) : KG(M; A) KG (M; A). ˆ Ñ 8 Here, π is the projection M M and FW and FW are the vector bundles from the topological description of the Thom isomorphism 1.14.9.

PROOF. We claim that bp ¦ Y x C(M)[Γ(F)] = π (x) [F] P ¦ ˆ for all x KG(M; A) and every Hilbert G-vector bundle F over M (where we interpret Γ(F) as a Hilbert G-C(M)-C(Mˆ )-bimodule). In view of the preceding discussion and the topological description of the Thom isomorphism, the assertion of the proposition follows at once from this. Indeed, comparing bp bp bp x C(M)[Γ(F)] = x C(M,A)[Γ(F) A] with the result of the computation ¦ Y ¦ ¢ ¦ bp π (x) [F] = ∆ (π idMˆ ) (x [F]) = µ¦(x bp [F]) bp bp bp ˆ = x C(M,A)[(C(M, A) Γ(F)) µC(M, A)] 74 1. ANALYTIC K-THEORY AND K-HOMOLOGY

(where µ is the multiplication morphism C(M, A) bp C(Mˆ ) Ñ C(Mˆ , A)) we see that it suffices to establish that p p p p p 1 1 (C(M) b A b Γ(F)) b µ(C(Mˆ ) b A) Ñ Γ(F) b A, ( f b a b u) b (g b a ) = ( f ¥ π)gu b aa defines an isometric isomorphism of Hilbert G-C(M, A)-C(Mˆ , A)-bimodules. This is clear because again the right-hand side assignment defines an isometry with dense range on the algebraic tensor product.

We conclude the ﬁrst chapter with the following compatibility result.

1.15.11 LEMMA. For every pullback diagram

f M / N

g h O / P j of G-equivariant embeddings of smooth compact Spinc-manifolds with boundaries we have ¦ ¦ g! f = j h!

PROOF. Denote by νM and νN the normal bundles of g and h. By our assumptions, we can ¦ identify the restriction f νN with νM. We can even consider νM as an open neighborhood of | g(M) in O and νN as an open neighborhood of h(N) in P in such a way that j(νM) = (νN) f (M). Denote by inclM and inclN the corresponding inclusions C0(νM) C(O) and C0(νN) C(P). P ¦ Then we ﬁnd for all x KG(N; A) that ¦ 1.15.6 ¦ bp ¦ g! f (x) = f x C(M)(inclM) βνM bp ¦ ¦ = x C(N)(inclM) f βνM 1.14.13 ¦ bp ¦ = x C(N)(inclM) j βνN ¦ bp ¦ = x C(N) j (inclN) βνN 1.15.6 ¦ = j h!(x). CHAPTER 2

Geometric K-homology

In this chapter we define the geometric K-homology groups of an equivariant pair. Since all definitions have a suitable analytic representation it is easy to show that the obvious natural transformation to analytic K-homology is well-defined. We then proceed to show that this map is in fact a natural isomorphism of generalized homology theories, generalizing the results of [BHS07, BOOSW10]. As our geometric theory will be naturally Z/2-graded, we shall from now on consider KK-theory as a Z/2-graded theory. This can be elegantly realized by taking the direct limit of the systems defined by the even and odd KK-groups, respectively, together with the natural Clifford periodicity isomorphisms from Property N: G G KK[¦](A, B) := ÝÑlim KK¦+2k(A, B) kPZ All operations defined in the first chapter are compatibly with this identification (see Property N and Remark 1.11.27). Moreover, in this picture the ordinary Bott and Thom elements are identified with their reduced counterparts (see the discussion in Section 1.14).

2.1. Cycles and equivalence relations

In [BD82], Baum and Douglas proposed a geometric deﬁnition of K-homology groups inspired by oriented bordism theory [tD00, VIII.13]. Instead of being singularly oriented, the manifolds in their cycles are equipped with a K-orientation, i.e. a Spinc-structure. Furthermore, they carry a vector bundle and additional equivalence relations are introduced to deal with the direct sum of vector bundles and to explicitly incorporate Bott periodicity. The obvious generalization of this idea to the equivariant situation and coefﬁcients is the following.

2.1.1 DEFINITION (GEOMETRICCYCLES).A geometric cycle for the K-homology of a G-pair (X, Y) ¦ with coefﬁcients in a coefﬁcient algebra, i.e. a trivially graded, unital G-C -algebra A, is a triple (M, E, f ) containing the following data:

– a compact smooth G-Spinc-manifold M with boundary,1 –a G-A-vector bundle2 E Ñ M, and – a (continuous) G-map f : (M, BM) Ñ (X, Y). A cycle will be called even if all connected components of M have even dimension, and odd if all of them have odd dimension. In any such homogeneous cycle, the parity of the dimension of M is well-deﬁned in Z/2 and will be denoted by [dim M]. 1 1 1 We say that two cycles (M, E, f ) and (M , E , f ) are isomorphic if there exists a G-Spinc-structure- 1 1 preserving diffeomorphism ϕ : (M, BM) Ñ (M , BM ) (in the sense of Deﬁnition 1.10.9) which 1 1 pulls back E to E and f to f .

1This manifold may not be connected, and different connected components may have different dimensions (including zero, cf. Remark 1.11.26). 2One could also work with smooth vector bundles only (see Remark 1.5.5).

75 76 2. GEOMETRIC K-HOMOLOGY

2.1.2 DEFINITION (OPERATIONS). The opposite of a cycle (M, E, f ) is defined to be (¡M, E, f ) where ¡M denotes the manifold M equipped with its opposite G-Spinc-structure (see Definition 1.10.23). The disjoint union of two cycles is defined to be 1 1 1 1 1 1 (M, E, f ) + (M , E , f ) := (M > M , E > E , f > f ). The empty cycle where M = H is the empty manifold represents the identity element with respect to the induced operation on the set of isomorphism classes of cycles.

In analogy with the singular case, every homogeneous cycle (M, E, f ) determines an analytic K- homology class ¦ X B ¦ P G f ([E] [M, M]) = f PDM[E] K[dim M](X, Y; A).

2.1.3 LEMMA. Isomorphic cycles determine the same analytic K-homology class.

PROOF. Suppose we are given isomorphic cycles as in the deﬁnition. Since ϕ preserves the Spinc- 1 1 structure, ϕ¦[M, BM] = [M , BM ] (Proposition 1.11.9). Hence using the Projection Formula 1.7.7 we get ¦ 1 1.7.7 1 1 1 f¦([E] X [M, BM]) = f¦ ϕ¦(ϕ [E ] X [M, BM]) = f¦([E ] X [M , BM ]).

2.1.4 LEMMA. The assignment of analytic K-homology classes to cycles (for a fixed G-pair and coefficient algebra) determines a homomorphism from the semigroup of isomorphism classes of cycles, equipped with the operation of disjoint union, to analytic K-homology. 1 1 1 PROOF. Let (M, E, f ) and (M , E , f ) be two cycles. We have to establish the identity 1 1 1 1 > 1 > 1 ( f f )¦PDM>M [E E ] = f¦PDM[E] + f¦PDM [E ]. 1 1 1 Let us denote by incl and incl the inclusions of M and M into M > M . Then using Example 1.4.4 we find that > 1 1 1 [E E ] = incl![E] + incl![E ]. Indeed, it is easy to see that incl![E] is nothing but the K-theory class determined by the G-A- 1 vector bundle E extended by zero to all of M > M . Consequently, 1 1 1 1 1 1 > 1 > > 1 > 1 ( f f )¦PDM>M [E E ] = ( f f )¦PDM>M incl![E] + ( f f )¦PDM>M incl![E ] 1.15.2 1 1 1 1 > > 1 = ( f f )¦ incl¦ PDM[E] + ( f f )¦ incl¦ PDM [E ] 1 1 = f¦PDM[E] + f¦PDM1 [E ]

2.1.5 REMARK (CORRESPONDENCES). In [CS84], Connes and Skandalis generalized the original Baum-Douglas picture of K-homology to bivariant KK-theory. Their cycles are correspondences, that is, diagrams E

f g X o M / Z where E is a vector bundle over a smooth manifold M, f is proper and g is K-oriented in the ¦ ¦ sense that T M ` f TY is endowed with a Spinc-structure. Clearly, in the absolute case and for trivial coefﬁcients and group actions our geometric cycles are just correspondences with Z = t¦u. Connes and Skandalis developed a way to assign functorial Gysin elements to a K-oriented map f¦([ (E)] bp g ) P KK (X Z) so that every correspondence as above determines an element Γ0 C0(M) ! 0 , (see also Remark 1.15.3). Every element in KK0(X, Z) can be described using a correspondence. There is a recent generalization of this theory to a very general equivariant setting due to Emerson and Meyer which we will brieﬂy discuss in the outlook. 2.1. CYCLES AND EQUIVALENCE RELATIONS 77

If X is a compact G-Spinc-manifold with boundary, Poincaré duality shows that every analytic K-homology class arises from a geometric cycle, although the latter is certainly not unique. In the following we will introduce three equivalence relations on the set of isomorphism classes of geometric cycles which together will make the above assignment a bijection. 1 2.1.6 DEFINITION (DIRECTSUM). The direct sum of two geometric cycles (M, E, f ) and (M, E , f ) for the equivariant K-homology of a pair (X, Y) with coefﬁcients in A which only differ in their 1 G-A-vector bundle datum is deﬁned to be the cycle (M, E ` E , f ).

2.1.7 LEMMA. The analytic K-homology class determined by a direct sum cycle is the sum of the classes determined by its summands.

PROOF. This is an immediate consequence of Lemma 2.1.4.

2.1.8 DEFINITION (BORDISM).A bordism for the equivariant K-homology of a pair (X, Y) with coefﬁcients in A consists of the following data:

– a compact smooth G-Spinc-manifold N with boundary, –a G-A-vector bundle F Ñ N, – a (continuous) G-map g : N Ñ X, and ¡ – a smooth G-invariant map h : BN Ñ R with regular values ¨1 and g(h 1[¡1, 1]) Y.

¡1 ¡1 The preimages M+ := h ([1, 8)) and M¡ := h ((¡8, ¡1]) carry the structure of compact smooth G-Spinc-manifolds with boundary (cf. [tD00, Satz VIII.5.2]), and we say that the cycle | | | | (M+, F M+ , g M+ ) and the opposite of (M¡, F M¡ , g M¡ ) are bordant.

2.1.9 REMARK. A more geometric way to think about our version of bordism is the following (cf. [tD00, VIII.13]): The function h partitions the boundary of N into three G-submanifolds M+, ¡1 ¡ X B M¡ and M0 := h [ 1, 1] with boundary (of codimension 0) such that M0 M¨ = M¨ and B B + > B ¡ M0 = M M , and g(M0) Y. It will be easier in the following to work with the function h instead of directly constructing partitions of the boundary. However, we will still need to deal with the problem that most bordisms are naturally manifolds with corners.

2.1.10 LEMMA. Bordism deﬁnes an equivalence relation on the set of isomorphism classes of cycles. More- over, the disjoint union of a cycle and its opposite is bordant to the empty cycle.

PROOF. The main content of the ﬁrst assertion is that ordinary bordism for manifolds with boundaries is an equivalence relation. This well-known fact is proved by the usual glueing arguments, see e.g. [tD00, Satz VIII.13.1 and discussion after Satz VIII.13.11]. The last assertion is a consequence of reﬂexivity.

2.1.11 LEMMA. Bordant cycles determine the same analytic K-homology class.

PROOF. Suppose we are given a bordism as in the deﬁnition. As the opposite of a geometric cycle determines the inverse analytic class (Proposition 1.11.12) we have to establish the following identity:

(g )¦PD [F ] + (g )¦PD ¡ [F ] = 0 (M+,BM+) M+ M+ (M¡,BM¡) M M¡ c Consider the compact G-Spin -manifold M := M+ > M¡ with boundary. Using ﬁrst Lemma 2.1.4 and then Lemma 1.11.29, we can rewrite the left-hand side to

( )¦PD [ ] = ( )¦BPD [ ] g (M,BM) M F M g (M,BM) N F 78 2. GEOMETRIC K-HOMOLOGY where B is the boundary map associated to the bottom row in the following commutative diagram: z z 0 / 0 / C0(X Y) / C0(X Y) / 0 ¦ ¦ g ¡ g (N,h 1([¡1,1])) (M,BM) z ¡1 ¡ 0 / C0(N˚ ) / C0(N h ([ 1, 1])) / C0(M˚ ) / 0 Naturality of boundaries now shows that this last expression factors over 0, thus vanishes.

So far, we have only related cycles with manifolds of equal dimensions. However, periodicity in analytic K-homology implies that there are additional cycles we have to identify:

2.1.12 EXAMPLE. Recall from Section 1.14 that Bott periodicity is a simple consequence of the fact 2k that the Dirac operator D of an even-dimensional sphere S twisted with the Bott bundle F2k has index 1, i.e. 2k 1.13.6 collapse¦ PD 2k [F ] = [F ] b 2k [S ] = 1 P K (¦) Z. S 2k 2k C0(S ) [0] 2k Consequently, all the geometric cycles (S , F2k, collapse) determine the same class in the analytic K-homology of a point. In order to deal with more general situations, Bott periodicity needs to be replaced by its parame- terized version, the Thom isomorphism, which relates the total space of a Spinc-vector bundle W to its base space. In fact, since geometric cycles may only contain compact manifolds, the proper generalization will rather use the unit sphere bundle Mˆ of (M ¢ R) ` W which is the ﬁbration of spheres we have already seen in the topological description of the Thom isomorphism 1.14.9.

2.1.13 DEFINITION (VECTOR BUNDLE MODIFICATION). Let (M, E, f ) be a geometric cycle for the equivariant K-homology of the pair (X, Y) with coefficients in A and let W be a G-Spinc-vector bundle over M of even rank over each connected component. The vector bundle modification of (M, E, f ) along W is defined to be the cycle ¦ b ¥ (Mˆ , π (E) FW, f π) where, as in the topological description of the Thom Isomorphism 1.14.9, Mˆ denotes the unit ¢ ` Ñ sphere bundle of (M R) W, π : Mˆ M the base point projection and FW the Thom bundle, and where b is the tensor product of Hilbert bundles from Lemma 1.5.20. It follows from the discussion in Example 1.15.8 that this is a well-defined cycle. 8 2.1.14 LEMMA. In the situation of the definition, let FW denote the trivial vector bundle pulled back from ˆ ¦ b 8 ¥ the south pole fiber of the Thom bundle. Then (M, π (E) FW , f π) is bordant to the empty cycle and, in particular, determines the zero class in analytic K-homology.

PROOF. If M is a closed manifold then Mˆ is the boundary of the unit disk bundle of (M ¢ R) ` W. In the general case, the unit disk bundle is a manifold with corners (in the sense of e.g. [Lee06, p. 363]), but it can be smoothed in such a way that we get a homeomorphic G-Spinc- manifold D together with a smooth G-invariant map h : BW Ñ R with regular values ¨1 such ¡ ¡ that h 1([1, 8)) Mˆ and h 1((¡8, ¡1]) is empty.

Let us sketch this smoothing procedure in some detail: Suppose D is the disk bundle of a rank n-vector bundle over a connected G-manifold M with boundary (imagine D as the solid cylinder in Figure 1 and M as its center line) and consider the associated sphere bundle S as a subspace of D (its lateral surface). Apart from S BM (the edge of the cylinder), D carries a canonical smooth structure because it locally looks like the product of a manifold with boundary and a manifold without Figure 1: Disk bundle. ¢ ¢ ¢ boundary. Around S B and after choosing collars, D locally looks like W := (0, 1] U (0, 1] ¡ M V where U Sn 1 and V BM are open subsets (Lemma 1.10.22). We can thus perform the 2.2. GEOMETRIC K-HOMOLOGY GROUPS 79 usual smoothing procedure which amounts to using polar coordinates to identify W with R ¢ [0, 8) ¢ U ¢ V (or the opposite). The latter space carries a canonical smooth structure (see e.g. [tD00, VIII.11.2]), and these smooth structures fit together so that we can consider D as a compact smooth G-Spinc-manifold with boundary (the G-Spinc-structure does not depend on the smooth B Ñ R structure). The function h : W is now constructed as follows: On D BW (the caps of the cylinder), set h to the radial coordinate. Using the collar for M, the side parts of the cylinder close ( ] ¢ [ ) to the edges look like 0, 1 S BM and we define h by mapping the first coordinate onto 1, 2 in a strictly decreasing way. On the remaining side parts of the cylinder, set h equal to 2. It is clear that this can be done in a smooth way and such that +1 is a regular value. Finally, average over G to obtain a smooth G-invariant map with the desired properties. It is clear that the map f ¥ π extends to the entire smoothed disk bundle D (it only depends on ¦ 8 the base point). Finally, the tensor product of π (E) and FW extends to a G-A-vector bundle F over D because both factors are pulled back from M. We have produced the desired bordism (D, F, f ¥ π, h). For the last assertion, apply Lemma 2.1.11.

2.1.15 LEMMA. A cycle and its modiﬁcations determine the same analytic K-homology class.

PROOF. Suppose we are in the situation of the deﬁnition. By the preceding lemma, ¦ ¦ 8 ¥ ¦ b ¥ ¦ Y ¡ ( f π) PDMˆ [π (E) FW ] = ( f π) PDMˆ (π [E] ([FW ] [FW ])) . ãÑ The right-hand K-theory element is just s![E] for the north pole section s : M Mˆ (Proposition 1.15.10). Hence we can rewrite the right-hand side further to ¥ ¦ ¥ ¦ ¦ ¦ ( f π) PDMˆ s![E] = ( f π) s PDM[E] = f PDM[E].

2.2. Geometric K-homology groups

We can ﬁnally deﬁne geometric K-homology.

2.2.1 DEFINITION (GEOMETRIC K-HOMOLOGY). We define the G-equivariant geometric K-homolo- G,geom gy of a G-pair (X, Y) with coefficients in a coefficient algebra A to be the set K¦ (X, Y; A) of isomorphism classes of geometric cycles modulo the equivalence relation generated by bordism, vector bundle modification and the relation 1 1 (2.2.2) (M, E ` E , f ) (M, E, f ) + (M, E , f ) identifying the direct sum of two cycles with their disjoint union. G,geom 2.2.3 PROPOSITION. The set K¦ (X, Y; A) is an Abelian group with respect to the operation + of disjoint union, Z/2-graded into classes represented by even cycles and odd cycles, respectively. The identity element of this group is represented by the empty cycle, and the inverse of a class [M, E, f ] is given by [¡M, E, f ].

PROOF. We have already seen that the set of isomorphism classes is a semigroup with respect to disjoint unions. Clearly, this is compatible with the deﬁning equivalence relations and thus G,geom induces an operation on K¦ (X, Y; A). The last assertion in Lemma 2.1.10 thus shows that the resulting semigroup is already a group. Furthermore, the equivalence relation preserves even and odd cycles. As every cycle can be uniquely decomposed into the disjoint union of an even and an odd cycle, this shows that our Z/2-grading is well-deﬁned.

2.2.4 REMARK. By not using vector bundle modiﬁcation we could also try to deﬁne a Z-graded geometric theory where cycles are graded by the dimension of the underlying manifold. How- ever it is easy to see that such a theory will not be 2-periodic and hence cannot agree with analytic K-homology. 80 2. GEOMETRIC K-HOMOLOGY

1 1 1 Every morphism (ϕ; Φ) : (X, Y; A) Ñ (X , Y ; A ) of G-pairs and coefﬁcient algebras induces a grading-preserving map G,geom G,geom G,geom 1 1 1 K¦ (ϕ; Φ) : K¦ (X, Y; A) Ñ K¦ (X , Y ; A ), [M, E, f ] ÞÑ [M, Φ¦(E), ϕ ¥ f ]. G,geom G,geom Again, we write ϕ¦ := K¦ (ϕ; idA) for pushforwards of spaces and Φ¦ := K¦ (id(X,Y); Φ) for pushforwards of coefﬁcients. The following statement summarizes our discussion.

2.2.5 PROPOSITION. Geometric K-homology is a bifunctor from G-equivariant pairs and coefﬁcient algebras to Z/2-graded Abelian groups, covariant in both variables. There is also a straightforward way to deﬁne boundary morphisms for our theory by restricting cycles to the boundary of the underlying manifold:

2.2.6 DEFINITION (BOUNDARYMAPS). The boundary map for a G-pair (X, Y) with coefﬁcients in A is deﬁned to be G,geom G,geom ¦ B ¦ ( ) Ñ ¦ ( ) [ ] ÞÑ [B ] = [B ( ) ¥ ] : K X, Y; A K Y; A , M, E, f M, E BM, f BM M, incl E , f incl where incl: BM M denotes the inclusion of the boundary.

2.2.7 PROPOSITION. The boundary maps are well-deﬁned and natural in (X, Y; A). 1 1 1 PROOF. We only verify the latter assertion. Let (ϕ; Φ) : (X, Y; A) Ñ (X , Y ; A ) be a morphism G,geom and (M, E, f ) a cycle for K¦ (X, Y; A). Then B ¥ G,geom B ¥ ¥ ( K¦ (ϕ; Φ))[M, E, f ] = [ M, VectG(incl; Φ)(E), ϕ f incl] G,geom ¥ B = (K¦ (ϕY; Φ) )[M, E, f ] where incl: BM M denotes the inclusion of the boundary.

We will also need the following notation for geometric K-homology classes: [M, x, f ] := [M, E, f ] ¡ [M, F, f ] Here, M and f are just as in the deﬁnition of a geometric cycle, but the G-A-vector bundle is ¡ P 0 replaced by a K-theory class x = [E] [F] KG(M; A). 2.2.8 LEMMA. This notation is well-deﬁned, and [M, [E], f ] = [M, E, f ].

PROOF. We need to show that the deﬁnition is independent of the representation of x as a difference bundle. Since K0 (M; A) is the Grothendieck group of G-A-vector bundles over M (Theorem G 1 1 1.5.16), for any two representations x = [E] ¡ [F] = [E ] ¡ [F ] there exists a G-A-vector bundle H 1 1 such that E ` F ` H = E ` F ` H. Relation (2.2.2) then ensures that indeed 1 1 [M, E, f ] ¡ [M, F, f ] = [M, E , f ] ¡ [M, F , f ].

Note that in this picture it makes similarly sense to talk about bordant cycles (replace the vector bundle in the bordism by an arbitrary K-theory datum).

2.2.9 EXAMPLE. In the situation of vector bundle modiﬁcation we have ¥ [M, E, f ] = [Mˆ , s![E], f π] where s : M ãÑ Mˆ is the north pole section. This is because the latter cycle is bordant to the vector bundle modiﬁcation of a cycle (M, E, f ) by Lemma 2.1.14 and Proposition 1.15.10.

Lemmas 2.1.3, 2.1.11, 2.1.15 and 2.1.7 show that the assignment of analytic cycles to geometric cycles descends to geometric classes. That is, we get morphisms G,geom G α : K¦ (X, Y; A) Ñ K¦ (X, Y; A), [M, E, f ] ÞÑ f¦PDM[E]. 2.2. GEOMETRIC K-HOMOLOGY GROUPS 81

2.2.10 PROPOSITION. The maps α define a natural transformation from equivariant geometric K-homology with coefficients to equivariant K-homology with coefficients, compatible with the boundary morphisms of both theories. G,geom PROOF. Let (M, E, f ) a cycle for K¦ (X, Y; A). For naturality, let us consider a morphism 1 1 1 (ϕ; Φ) : (X, Y; A) Ñ (X , Y ; A ). Then

G,geom αK¦ (ϕ; Φ)[M, E, f ] = ϕ¦ f¦PDMΦ¦[E] 1.11.23 = ϕ¦Φ¦ f¦PDM[E] G = K¦ (ϕ; Φ)α[M, E, f ]. We now show compatibility with the boundary morphisms. Indeed,

B[ ] = ( )¦PDB [ ] α M, E, f f BM M E BM ¦ = ( )¦PDB [ ] f BM M incl E 1.11.29 = ( )¦BPD [ ] f BM M E = B f¦PDM[E] = Bα[M, E, f ]. Here, incl still denotes the inclusion BM M and the second last identity is due to naturality of analytic boundaries with respect to the map f : (N, BN) Ñ (X, Y).

2.2.11 REMARK (INDEXTHEORY). What we have showed so far can be used for a conceptual formulation of an equivariant Mišˇcenko-Fomenkoindex theorem for Dirac operators on a G- Spinc-manifold M (cf. [BD82, Part 5]). For this, consider the following diagram:

K0(M; A) o QQ η oo QQ σ oo QQQ ooo QQQ ow oo QQ( G,geom α / G K (M; A) K0 (M; A)

collapse¦ collapse¦ G,geom ¦ / G(¦ ) = 0 ( ) K ( ; A) α K0 ; A KG A

Here, η sends a K-theory cycle represented by a G-A-vector bundle [E] to the cycle [M, E, idM], while σ sends [E] to [DE], the class represented by a Dirac operator for M twisted with E (but see Remark 1.10.40). The triangle commutes by Corollary 1.10.39, and the square is commutative by naturality of the natural transformation α. Commutativity of the diagram then implies that P 0 α[M, E, collapse] = collapse¦[DE] KG(A). This can be understood as an index theorem, since the analytical right-hand side is expressed in terms of the topological left-hand side. It is possible to derive the general index theorem from a similar approach (see [BDT89, Section 6]). It is well-known that in various special cases there exist cohomological formulae for the left-hand side (see e.g. [LM89, Theorem D.15], [MF79, Theorem 3.4], [Sch05, Corollary 6.14]).

G Similarly, if A = C then the right-hand side is simply index DE, the equivariant index of the twisted Dirac operator (Corollary 1.10.38). For more general coefficients, the index is harder to define since the kernel will in general not be a finitely generated, projective A-module, and thus additional care is necessary (see [Sch05, Section 7] for further discussion and the case where A is a von Neumann algebra). 82 2. GEOMETRIC K-HOMOLOGY

2.2.12 REMARK (GEOMETRIC OPERATIONS). Many constructions and operations of analytic K- homology with geometrical content have natural analogues in the geometric picture. We have already seen this for pullbacks, pushforwards and boundary maps. For cap products, it follows readily from the projection formula that the correct definition in geometric K-homology is given by ¦ G [E] X [M, F, f ] := [M, f E b F, f ] P K¦ (X, Y; A b B) P 0 P G for every compact G-pair (X, Y), [E] KG(X; A) and [M, F, f ] K¦ (X, Y; B). As the fundamen- c tal class of a G-Spin -manifold is represented by the geometric cycle [M, M ¢ C, idM] this gives in particular an elegant representation of twisted fundamental classes — but this is of course precisely the idea behind geometric K-homology. The cap product can also be used to define Kasparov’s assembly map in the geometric picture. For every (torsion-free) group π it is given by ¦ ¦ ¦ K¦(Bπ) Ñ K¦(¦; C π) = K¦(C π), [M, E, f ] ÞÑ [M, f L b E, collapse] ¦ 0 ¦ where L = [Eπ ¢π C π] P K (Bπ; C π) is the Mišˇcenkoline bundle (cf. [Hig08]). It should also be possible to develop some sort of geometric induction mechanism which com- plements the one in KK-theory. Conceivably, such an approach might be used to generalize from compact Lie group actions to more general non-compact Lie group actions. We will briefly discuss this issue in the outlook.

2.3. Isomorphism between geometric and analytic K-homology

We will now establish that the natural transformation α is an isomorphism for a certain category of compact topological pairs, using the method of [BOOSW10].

C c 2.3.1 DEFINITION (COMPACT G-SPIN -HOMOTOPYRETRACTS). The category of compact G-Spin - homotopy retracts is the full subcategory of the category of G-pairs consisting of those compact G-pairs (X, Y) for which there exists a compact G-Spinc-manifold N with boundary and maps

j r (X, Y) / (N, BN) / (X, Y) such that the composition r ¥ j is G-homotopic to the identity (by a homotopy preserving Y). Of course, the deﬁnition of this category is somewhat ad-hoc. It is merely the category where the technique we have developed so far can be used to prove that α is an isomorphism. However, the following lemma shows that a rather large class of G-pairs are compact G-Spinc-homotopy retracts. We will later see how to loosen the compactness assumption.

2.3.2 LEMMA ([BOOSW10, Lemma 2.1]). Every compact G-Euclidean neighborhood retract, in particular every compact smooth G-manifold with boundary (absolute or relative to its boundary), is in this category.

The following then is the main theorem of this thesis.

2.3.3 ISOMORPHISM THEOREM. Equivariant geometric K-homology with coefﬁcients in a triv- ¦ ially graded unital separable C -algebra is a Z/2-graded generalized homology theory on the category of compact G-Spinc-manifold homotopy retractions. It is naturally isomorphic to equivariant analytic K-homology with coefﬁcients via α.

PROOF. Let us try to define a candidate for the inverse of α: For any G-pair (X, Y) and p P Z/2 let G,geom ¡ G Ñ ÞÑ 1 ¦ β : Kp (X, Y; A) Kp (X, Y; A), x [N, PDN j x, r] 2.3. ISOMORPHISM BETWEEN GEOMETRIC AND ANALYTIC K-HOMOLOGY 83 where (N, j, r) are as in Definition 2.3.1 and the dimension of N modulo 2 is p (this requirement can always be satisfied by replacing N with N ¢ T). Reduction. If we can prove that β is well-defined and that α and β are inverse to each other then the main theorem follows from naturality of the transformation (Lemma 2.2.10) and the corresponding properties of analytic K-homology (Theorem 1.6.3). Now, β is a right inverse for any choice of data (N, j, r) since for any x P K¦G(X; Y; A) we have ¡ ¡ 1 ¦ ¦ 1 ¦ αβ(x) = α[N, PDN , j x, r] = r PDNPDN j x = x. Thus we only have to show that α admits any left inverse since then it follows that α is invertible and, by uniqueness of inverses, that β does not in fact depend on these choices. Assume that we have proved this for every G-Spinc-manifolds (N, BN) with boundary. We will show that the general result then follows automatically. To see this, fix a compact G-Spinc- P ¥ homotopy retract (X, Y), p Z/2 and data (N, j, r) as above. Because r j id(X,Y), we can G,geom G,geom B regard Kp (X, Y; A) as a subgroup of K (N, N; A) via j¦ (it is in fact a direct summand). Consider the diagram

j¦ G,geom / G,geom B Kp (X, Y; A) Kp (N, N; A)

βα id=βα G,geom G,geom K (X, Y; A) / K (N, BN; A) p j¦ p where we use the data (N, j, r) for (X, Y). This diagram is commutative. Indeed, we have the identity ¡ ¦ 1 ¦ ¦ ¥ αj βα[M, E, f ] = α[N, PDN j f PDM[E], j r] ¡ ¦ ¦ 1 ¦ ¦ = j r PDNPDN j f PDM[E] = j¦ f¦PDM[E] = αj¦[M, E, f ] and α is by assumption an isomorphism for (N, BN). We conclude that βα for (X, Y) is the re- G,geom striction of an identity map, hence the identity map on Kp (X, Y; A). Proof for G-Spinc-manifolds. We will now establish that β is a left inverse of α in the case of a compact G-Spinc-manifold (N, BN) with boundary and trivial data (N, id, id). More precisely, for every cycle [M, E, f ] we will show the last identity in

¡ ! 1 ¦ βα[M, E, f ] = [N, PDN f PDM[E], id] = [N, f![E], id] = [M, E, f ]. By the Mostow-Palais Embedding Theorem ([Mos57], [Pal57]), there exists a G-equivariant embedding e : M ãÑ W of M into a unitary G-representation W. We can embed W further into its one-point compactiﬁcation W+ which is a smooth compact even-dimensional G-Spinc-manifold in a compatible way (Example 1.13.2). We thus get the following diagram of G-equivariant embeddings ( f ,e) ( f ,0) (idN,0) M / N ¢ W+ o N ¥ ¥ which satisfy the identities projN ( f , e) = f and (idN, 0) f = ( f , 0). Using the following lemma twice as well as homotopy invariance of the Gysin maps we ﬁnd that

2.3.4 ¢ + ¢ + 2.3.4 [M, E, f ] = [N W , ( f , e)![E], projN] = [N W , ( f , 0)![E], projN] = [N, f![E], id]. 84 2. GEOMETRIC K-HOMOLOGY

c 2.3.4 LEMMA. Let g : (M, BM) ãÑ (N, BN) be an embedding of compact G-Spin -manifolds with boundary, E a complex G-A-vector bundle over M and f : (N, BN) Ñ (X, Y) a continuous map. Assume the submanifold has even codimension and denote its normal bundle by νM. Then the vector bundle modiﬁca- ¥ 2 ¢ ` tion of (M, E, f g) along (R M) νM is bordant to the vector bundle modiﬁcation of (N, g![E], f ) along N ¢ R2 (where G acts trivially on R2 ¢ M and R2 ¢ N). In particular, ¥ P G,geom [M, E, f g] = [N, g![E], f ] K¦ (X, Y; A).

PROOF. Observe that we have written both trivial vector bundles in the form R2 ¢ M. This is because then the canonical G-Spinc-structure of the total space as defined in Proposition 1.14.11 matches the Cartesian product G-Spinc-structure. In the following we will also do this for trivial unit sphere bundles where the analogous observation holds (see Example 1.15.8). ¥ ¥ Ñ The first modification is bordant to the cycle (Mˆ , s![E], f g π) where π : Mˆ M is the unit 3 sphere bundle of (R ¢ M) ` νM and s : M Ñ Mˆ is the north pole section. The second modifi- 2 ¢ ¥ Ñ 2 ¢ cation is bordant to the cycle (S N, t!g![E], f proj2) where t : N S N is the north pole section sending a point n P N to ((1, 0, 0), n). We will now construct an explicit bordism between these two cycles.

Recall that we can consider νM as an open tubular neighborhood of M in N. Thus we can also 3 3 3 embed the e-unit disk bundle of (R ¢ M) ` νM into the unit disk bundle D ¢ N of R ¢ N for any e P (0, 1). Consider the compact G-space W we get by removing from D3 ¢ N the open e-disk bundle. It is a manifold with corners whose boundary contains both S2 ¢ N and a scaled version of the opposite of Mˆ c (see Figure 2 for an illustration). Note that the G-Spin -structure can be Figure 2: Illustration of verified locally using Lemma 1.10.24. Using a similar procedure as in the the bordism W for M = N. proof of Lemma 2.1.14, we can G-equivariantly smooth its corners in such a way that we get a (G-equivariantly homeomorphic) G-Spinc-manifold (which we also denote by W) with boundary ¡ as well as a smooth G-invariant map h : BW Ñ R with regular values ¨1 such that h 1([1, 8)) ¡ ¡ S2 ¢ N, h 1((¡8, ¡1]) ¡Mˆ and h 1([¡1, 1]) corresponds to a subset of (D3 ¢ BN) X W. ¥ B Ñ This last property ensures that the canonical extension f proj2 of f : (N, N) (X, Y) to W ¡ sends h 1([¡1, 1]) to Y so that we get a map (W, BW) Ñ (X, Y). Its restriction to S2 ¢ N is ¥ ˆ ¥ ¥ already the correct map f proj2, but its restriction to M is only homotopic to f g π. However, ¥ 1 an easy modification of f proj2 will produce a correct map f (one has to perform some scaling in the S2-factor; this is of course possible because the group acts trivially there). We have constructed the desired bordism except for the equivariant K-theory datum. For this, 3 consider the canonical embedding e : M ¢ [e, 1] ãÑ D((R ¢ M) ` νM) W which sends the scalar to the first R-coordinate (the one corresponding to the north pole direction) and define ¦ F := e! proj1 [E]. We claim that F restricts to the correct K-theory data. Indeed, from Lemma 1.15.11 applied to the following pullback diagrams of G-equivariant embeddings

id ¢e id ¢1 M / M ¢ [e, 1] M / M ¢ [e, 1]

s e t¥g e / 2 Mˆ W S ¢ N / W we get the desired identities: = ¦ ¦[ ] 1.15.11= ( ¢ )¦ ¦[ ] = [ ] F Mˆ inclMˆ W e! proj1 E s! id e proj1 E s! E , = ¦ ¦[ ] 1.15.11= ( ¥ ) ( ¢ )¦ ¦[ ] = [ ] F N¢S2 inclS2¢NW e! proj1 E t g ! id 1 proj1 E t!g! E . 2.3. ISOMORPHISM BETWEEN GEOMETRIC AND ANALYTIC K-HOMOLOGY 85

2.3.5 REMARK (NON-COMPACT SPACES). Although we have proved the isomorphism only for compact G-Spinc-homotopy retracts (X, Y), it is clear from the excision properties of analytical G,geom K-homology that we can take K¦ (X, Y; A) as the deﬁnition of the geometric K-homology of the G-space XzY which in general is not compact. This way we get a geometric description of equivariant K-homology for a much more general class of spaces (cf. Lemma 2.3.2). Observe that this is similar in spirit to the topological description of K-theory of non-compact spaces in terms of certain vector bundles over the one-point compactiﬁcation (see Remark 1.5.17).

2.3.6 REMARK (BAUM-CONNESCONJECTURE). Our result that α is an isomorphism can also be seen as a proof of the topological Baum-Connes conjecture3 for compact Lie group actions and coef- ﬁcients (see [BC00, Section 10] and also [BHS08, Section 9]).

3The analytical Baum-Connes conjecture is of course trivial for compact groups.

Outlook

In this thesis, we have described a powerful analytic framework, developed its index theoretic relationship to geometry, and used it to construct a geometric model of equivariant K-homology with coefﬁcients for actions of a compact Lie group.

Already in his original article on equivariant KK-theory, Kasparov allowed for actions of general locally compact (second countable) topological groups1 and consequently our deﬁnition of analytic K-homology makes good sense in this generality. Similarly, geometric cycles and their equivalence relations make almost literally sense in this situation so that we can also talk about equivariant geometric K-homology. It is thus natural to ask whether our approach generalizes to certain classes of non-compact group actions.

Let us now sketch some of the difficulties. On the analytical side, it is known that equivariant KK- theory has good properties for arbitrary locally compact group actions, including induction and Kasparov’s Poincaré duality. At least for equivariantly semisplit extensions natural long exact sequences also exist by a result of Baaj and Julg ([BS89, Theorème 7.2]; note that this is automatic for compact groups). The first main difficulty will thus be to develop a reasonable theory of fundamental classes (or more conceptually, Poincaré duality) and Gysin maps. As we had to use averaging on several occasions, this will certainly require some sort of finiteness assumption on the manifolds used in geometric cycles (for example, in the case of discrete group actions, a good assumption seems to be to require that all manifolds be G-compact proper G-spaces, see [BHS08]). Compactness of the group was also implicit in several of our KK-arguments. Once these analytical–geometrical obstacles have been overcome, the natural transformation from the geometric to analytic theory can be defined. For surjectivity, we need to establish that there exist enough equivariant vector bundles to generate the equivariant (representable) K-theory of the G- manifolds used in geometric cycles (cf. Remark 1.5.17). For injectivity, our main ingredient was the Mostow-Palais Embedding Theorem which generalizes to proper actions of matrix groups of finite orbit type (see discussion in [GGK02, Remark B.51]).

In a certain sense the above program (and a lot more) has been carried out in great generality by Emerson and Meyer in their work on equivariant correspondences for locally compact groupoid actions (for trivial coefficients and the absolute case). In a series of articles they have developed abstract duality results ([EM09b]), representable K-theory ([EM09d]) and generalizations of the Mostow-Palais Embedding Theorem to smooth actions of proper groupoids and of functorial Gysin elements for K-oriented normally non-singular maps (these are maps which admit a certain factorization condition which is automatic in the setting of this thesis, see [EM09c]). These results are then combined in [EM09a] to construct an equivariant theory of correspondences for very general groupoid actions which agrees with Kasparov’s theory under certain assumptions (which are satisfied in the situation considered in this thesis). Their cycles are somewhat less concrete but rather flexible, even in the smooth case, so that for example their version of vector bundle modification can simply use the total space instead of the unit sphere bundle. We refer to the above articles and the survey [Eme09] for details.

1There is even a generalization of equivariant KK-theory to locally compact groupoid actions due to Le Gall (see [LG99]).

87 88 OUTLOOK

A different possible line of attack has been suggested by Oyono-Oyono and Schick. Instead of proving explicitly that the geometric and analytic theory are isomorphic in more general situations, they suggest to develop a induction machinery which plays the geometrical counterpart to Kasparov’s induction and use it to reduce the case of proper actions of non-compact Lie groups G to their maximal compact subgroup K (cf. [BHS08, Lemma 8.2]). The obvious approach would c be to send a K-Spin -manifold M to G ¢K M, however it is not clear how to equip this space with an equivariant Spinc-structures in a generic fashion (already in the case of homogeneous spaces G/K this will not in general be possible). In the end, one might need to replace Spinc-structures by Dirac bundles, but the details of such a theory have yet to be worked out. Bibliography

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¡ ¡ [ , ] graded commutator, 12 ¦ ¦ T M exterior algebra bundle, 35 ¦ W exterior algebra, 4 ¦ ¦ C T M complexified exterior algebra bundle, 35 X cap product, 26 Y cup product, 25 bp exterior Kasparov product, 10 p b D Kasparov product, 10 x¡, ¡y inner product of a Hilbert G-A-module, 5 x¡, ¡y inner product of a Hilbert G-A-vector bundle, 19 [Φ] KK-element determined by a morphism, 10 ∇ connection, 38 ∇LC Levi-Civita connection, 39 ∇X covariant derivative, 38 ¡M manifold with opposite orientation and G-Spinc-structure, 46 ¡W representation space with negated involution, 4 [D] K-homology class determined by a differential operator, 32 [D]A K-homology class determined by a differential operator w.r.t. an admissible algebra bundle, 31 [E] K-theory class determined by a G-A-vector bundle, 23 [E] KK-element determined by a Hilbert G-A-B-bimodule, 11 [M, BM; W] twisted fundamental class, 54 [M, BM] fundamental class, 51 [W/M] fiberwise Dirac element, 67 [S] K-homology class determined by a twisted spinor bundle, 49 [/S ˚ ] reduced fundamental class, 54 M ¦ [[d + d ]] Dirac element, 36 1 identity operator, 6 1 trivial group, 15 1, 1A unit element, 3 G 1A identity element of KK¦(A, A), 10 ¦ A bp B algebraic tensor product of graded G-C -algebras, 2 ¦ A bp B tensor product of graded G-C -algebras, 2 ¦ A b B tensor product of trivially graded G-C -algebras, 2 a b b elementary tensor, 2 AG subalgebra of invariant elements, 11 Ad adjoint action, 40 Ad twisted adjoint action, 40 α natural isomorphism from geometric K-homology to analytic K-homology, 80 B(V, W) bounded operators, 6 b2k reduced Bott element, 65 BA(E, F) adjointable operators, 6 BΦ Hilbert G-A-B-bimodule determined by a morphism, 8 β natural isomorphism from analytic K-homology to geometric K-homology, 82 βn Bott element, 67 βW Thom element, 67 Cp+q complex vector space with standard involution, 5 C¡E Clifford algebra bundle constructed from C¡n, 34 C¡M Clifford algebra bundle constructed from C¡n, 34 p C¡p Clifford algebra of ¡C , 5 ¦ z C0 bifunctor sending a G-pair (X, Y) and G-C -algebra A to C0(X Y, A), 3 C0(X) continuous complex-valued functions vanishing at infinity, 3 C0(X, A) continuous A-valued functions vanishing at infinity, 3

91 92 NOTATION INDEX

Cc(X) compactly supported continuous functions, 31 CE Clifford algebra bundle, 34 CM Clifford algebra bundle, 34 p Cp Clifford algebra of C , 5 p+q Cp,q Clifford algebra of C , 5 CW Clifford algebra, 4 collapse collapse map sending everything to a point, 34 1 D ¢ D product operator, 33 B boundary element in KK-theory, 11 B boundary map in KK-theory, 11 B boundary map in geometric K-homology, 80 ¦ d + d de Rham operator, 35 Dn+1 closed unit ball, 62 Ba degree of a homogeneous element, 2 DM double of a manifold, 60 BM boundary of a manifold, 45 E bp F exterior tensor product of graded Hilbert G-A-modules, 6 E bp F tensor product of graded G-A-vector bundles, 24 p E b BF interior tensor product of graded Hilbert G-A-B-bimodules, 7 bp E ΦF interior tensor product of graded Hilbert G-A-modules, 7 E b F exterior tensor product of trivially graded Hilbert G-A-modules, 6 E b F tensor product of G-A-vector bundles, 24 b E Φ F interior tensor product of trivially graded Hilbert G-A-modules, 7 ¦ E dual G-imprimitivity bimodule, 9 E G subspace of invariant elements, 14 eval evaluation map, 20 Φ¦ pushforward of functions, 4 ¦ Φ pullback in KK-theory, 9 ¦ ϕ pullback of G-pairs in analytic K-theory, 15 ¦ ϕ pullback of functions, 4 ¦ ¦ ϕ pushforward along ϕ in KK-theory, 15 ¦ Φ (E) pullback of a Hilbert G-A-B-bimodule, 8 Φ¦ pushforward in KK-theory, 9 Φ¦ pushforward of coefficients in analytic K-homology, 25 Φ¦ pushforward of coefficients in analytic K-theory, 15 Φ¦ pushforward of coefficients in geometric K-homology, 80 ϕ¦ pushforward of G-pairs in analytic K-homology, 25 ϕ¦ pushforward of G-pairs in geometric K-homology, 80 Φ¦(E) pushforward of a Hilbert G-A-module, 7 Φ¦(E) pushforward of a G-A-vector bundle, 22 ¦ ϕ (F) pullback of a G-A-vector bundle, 22 Φ¦(P) pushforward of finitely generated, projective G-A-module, 17 F2k Bott bundle, 65 8 F2k pullback of the south pole fiber of the Bott element, 65 f! Gysin map, 70 FW Thom bundle, 68 8 FW pullback of the south pole restriction of the Thom bundle, 68 Γ section functor, 19 Γ0(E) space of sections vanishing at infinity, 19 Γc(E) compactly supported continuous sections, 32 g! Gysin element determined by a K-oriented map, 76 + GLn orientation-preserving linear group, 43 + GLW oriented frame bundle, 43 id identity map, 4 incl inclusion of a subspace, 4 incl! pullback in KK-theory induced by extension of functions by zero, 25 inclk inclusion of the k-th summand, 16 incl! extension of functions by zero, 16 incl! pushforward in KK-theory induced by extension of functions by zero, 16 indexG equivariant index, 15 G ¦ K¦(A) analytic K-theory of graded G-C -algebras, 15 G K¦(X, Y; A) analytic K-homology of G-pairs with coefficients, 25 G,geom K¦ (X, Y; A) geometric K-homology of G-pairs with coefficients, 79 NOTATION INDEX 93

KA(E, F) compact operators, 6 KG equivariant topological K-bifunctor, 22 KG algebraic K-functor, 17 ¦ ¦ K (A) analytic K-homology of graded G-C -algebras, 25 G¦ KG (X, Y; A) analytic K-theory of G-pairs with coefficients, 15 G KK¦(A, B) KK-groups, 9 G KK[¦](A, B) Z/2-graded KK-groups, 75 L(W) linear operators, 4 L2(E) square-integrable sections, 29 LA(V, W) A-linear operators, 6 2 λ double covering of U1, 41 ΛM determinant line bundle, 42 ΛW determinant line bundle, 42 λx left exterior multiplication, 5 λξ left exterior multiplication, 35 Mm¢n(A) matrix algebra, 6 Mleft(A) left multiplier algebra, 6 ν normal bundle, 45 ω connection form, 38 ¦ Ω (M) differential forms, 35 ¦ ΩC (M) differential forms, 35 OE orthonormal frame bundle, 34 On orthogonal group, 34 P b Q algebraic exterior tensor product of finitely generated, projective G-A-modules, 17 b P Φ Q algebraic inner tensor product of finitely generated, projective G-A-modules, 17 P b Q exterior tensor product of finitely generated, projective G-A-modules, 17 P ¢H W associated vector bundle, 38 PDM Poincaré duality isomorphism, 57 Pinn real Pin group, 40 c c Pinn complex Pin group, 40 projk projection onto the k-th factor, 54 R(G) representation ring, 11 n Rn real Clifford algebra of R , 40 G RKK¦ representable KK-theory, 23 s north pole section, 73 Sn unit sphere, 62 n S¨ northern and southern hemisphere, 62 σD symbol of the differential operator, 29 SM full spinor bundle, 47 /SM reduced spinor bundle, 47 S/ M conjugate of the reduced spinor bundle, 47 + S/ M conjugate of the even part of the reduced spinor bundle, 64 taut SM tautological spinor bundle, 47 c SW/M full spinor bundle of a G-Spin -vector bundle, 69 taut c SW/M tautological spinor bundle of a G-Spin -vector bundle, 67 SOE oriented orthonormal frame bundle, 34 son Lie algebra of SOn, 39 SOn special orthogonal group, 34 c c spinn Lie algebra of Spinn, 41 Spinn real Spin group, 40 c c SpinM principal Spinn-bundle, 41 c c Spinn complex Spin group, 40 Spinc principal Spinc -bundle, 41 ¦ W n T adjoint of an operator, 6 ¦ T M cotangent bundle, 34 TM tangent bundle, 41 u1 Lie algebra of U1, 41 Un unitary group, 44 Wp standard imprimitivity bimodule between Cp,p and C, 9 X+ one-point compactification, 3 Xˆ unit sphere bundle, 68 r X universal covering, 43

Subject Index

adjoint action that preserves the G-Spinc-structure, 43 of Spinn, 40 difference bundle, 23 c of Spinn, 41 differential operator, 29 admissible algebra bundle, 31 Dirac bundle, 35 algebraic K-functor, 17 Dirac element, 36 aligned cycle, 31 Dirac operator, 34, 35 analytic K-homology, 25 Dolbeault operator, 34 analytic K-theory, 15 double of a manifold, 60 dual Dirac element, 70 bordism, 77 Bott bundle, 65 elliptic differential operator, 29 Bott element, 66 embedding, 29 Bott periodicity of G-Spinc-manifolds, 71 analytical, 67 equivariant correspondences, 87 topological, 66 equivariant index, 15 boundary construction, 45 essential morphism, 7 boundary elements in KK-theory, 11 essentially self-adjoint differential operator, 29 boundary map in KK-theory, 11 exterior algebra, 4 boundary map in geometric K-homology, 80 exterior Kasparov product, 10 braiding isomorphisms, 2 exterior tensor product

¦ ¦ of KK-cycles, 12 C -algebra, see also G-C -algebra of finitely generated, projective G-A-modules, 17 G-continuous element, 1 of graded Hilbert G-A-B-bimodules, 7 cap product, 26 of graded Hilbert G-A-modules, 6 Clifford algebra, 4 standard Clifford algebra, 5 fiberwise Dirac element, 67 universal property, 4 finitely generated, projective G-A-module, 16 Clifford algebra bundle, 34 finitely generated, projective G-A-module bundle, see Clifford multiplication, 35, 35 also G-A-vector bundle Clifford periodicity, 11 free Hilbert G-A-module, 7 coefficient algebra, 17 free trivial Hilbert G-A-vector bundle, 20 coefficient category, 17 full Hilbert G-A-module, 8 compact G-Spinc-homotopy retract, 82 full spinor bundle, 47, 68 compact perturbation of a KK-cycle, 12 fundamental class, 51 composition Kasparov product, 10 fundamental vector field, 38 compression of a KK-cycle, 13 G-A-vector bundle, 18 connection ¦ in KK-theory, 50 G-C -algebra, 2 on a principal bundle, 37 graded, 2 on a vector bundle, 38 nuclear, 2 connection form, 38 G-Spin-structure, 43 c correspondences, 76 G-Spin -manifold, 41 ¦ c covariant system, see also G-C -algebra G-Spin -structure, 41 c cup product, 24, 25 G-Spin -structure-preserving diffeomorphism, 43 c cycle for KK-theory, 12 G-Spin -structure-preserving isomorphism, 43 G-Spinc-vector bundle, 41 ¦ de Rham operator, 35 G-algebra, see also G-C -algebra degenerate cycle, 12 G-continuous element, 1 degree, 2 G-equivariant connection, 38 determinant line bundle, 42 G-imprimitivity bimodule, 8 diffeomorphism, 29 dual, 8

95 96 SUBJECT INDEX

G-index, 15, 33 orientation element, 9 G-manifold, 29 with boundary, 29 projection formula, 27 G-pair, 2 pullback Cartesian product, 2 of KK-cycles, 12 disjoint union, 2 of a G-A-vector bundle, 22 G-space, 2 of graded Hilbert G-A-B-bimodules, 8 Gelfand-Naimark theorem, 3 pushforward geometric K-homology, 79 of KK-cycles, 12 geometric cycle, 75 of a G-A-vector bundle, 22 graded G-A-vector bundle, 24 of a ﬁnitely generated, projective G-A-module, 17 graded commutativity of the Kasparov product, 10 of graded Hilbert G-A-B-bimodules, 7 graded distributivity of the Kasparov product, 10 of graded Hilbert G-A-modules, 7 Gysin element, 70 reduced Bott element, 65 Gysin map, 70 reduced fundamental class, 54 reduced spinor bundle, 47 Hilbert G-A-B-bimodule, 7 reduced Thom element, 68 Hilbert G-A-module, 5 representable K-theory, 23 adjointable operators, 6 representation ring, 11 compact operators, 6 free, 7 Spin-structure, 40, see also G-Spin-structure full, 8 Spinc-structure, see also G-Spinc-structure Hilbert G-A-modules semisplit short exact sequence, 4 graded, 6 smooth structure on a Hilbert G-A-vector bundle, 19 Hilbert G-A-vector bundle, 19 spinor action on Rn, 43 free trivial, 20 spinor bundle, 46 Hodge Laplacian, 36 square-integrable sections, 29 homotopy of KK-cycles, 12 symbol of a differential operator, 29 homotopy-invariant, 9 symmetric differential operator, 29 symmetric monoidal category, 2 induction homomorphism, 11 interior tensor product tautological spinor bundle, 47, 67 of graded Hilbert G-A-B-bimodules, 7 tensor product of graded Hilbert G-A-modules, 7 in KK-theory, 10 c ¦ isometry of G-Spin -manifolds, 41 of graded G-C -algebras, 2 of graded Hilbert G-A-modules, 6 K-homology of Hilbert G-A-B-bimodules, 7 analytic, 25 tensor product of G-A-vector bundles, 24 geometric, 79 Thom bundle, 68 K-orientation, 41 Thom element, 67 K-oriented maps, 70, 76 Thom isomorphism K-theory analytical, 67 analytic, 15 topological, 68 compactly supported, 23 topological K-bifunctor, 22 representable, 23 topological K-theory, 18 topological, 18 topological Baum-Connes conjecture, 85 KK-cycle, 12 twisted adjoint action of Pinn, 40 degenerate, 12 twisted fundamental class, 54 KK-equivalence, 10 twisted spinor bundle, 47 KK-theory, 9 twisting of a Dirac bundle, 35 Kasparov product, 10 composition, 10 unit disk bundle, 78 exterior, 10 unit sphere bundle, 68

Levi-Civita connection, 39 vanishing submodule, 20 vector bundle modiﬁcation, 78 manifold, see also G-manifold Morita equivalent, 8 normalizing function, 30 north pole section, 73 ¦ nuclear C -algebra, 2 opposite G-Spinc-structure, 46 opposite KK-cycle, 12 opposite of a geometric cycle, 76