Equivariant geometric K-homology with coefficients. Michael Walter

Transcription

1 Equivariant geometric K-homology with coefficients Michael Walter

2

3 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

4

5 Equivariant geometric K-homology with coefficients Michael Walter v 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 Spin c - manifolds with boundary which map to the target, i.e. E (M, BM) f (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 Kasparov 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.

6 vi Acknowledgements. It is a pleasure to acknowledge Prof. Thomas Schick for inspiring discussions and helpful advice, and for sharing his enthusiasm for doing mathematics.

7 Contents Abstract v Chapter 1. Analytic K-theory and K-homology C -algebras and topological spaces with group actions Hilbert G-C -modules Equivariant KK-theory Analytic K-theory Topological K-theory Analytic K-homology Products Elliptic operators and analytic K-homology Dirac operators Spin c -structure and spinor bundles Fundamental classes and Poincaré duality The double of a manifold Equivariant Spin c -structure of the spheres Bott periodicity and Thom isomorphism Gysin maps 70 Chapter 2. Geometric K-homology Cycles and equivalence relations Geometric K-homology groups Isomorphism between geometric and analytic K-homology 82 Outlook 87 Bibliography 89 Notation Index 91 Subject Index 95 vii

8

9 CHAPTER 1 Analytic K-theory and K-homology In his landmark article [Kas81], Kasparov introduced a bivariant theory, KK-theory, which unified 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 ). 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 C 0 (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 Spin c -structure, the K-theory equivalent of an orientation (Sections ). 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 Spin c -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 trouble 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 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 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

10 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 p b B of graded G-C -algebras A and B can be formed in different ways by completing the skew-commutative algebraic tensor product A p b B. We will always use the minimal/spatial tensor product from [Kas81, Section 2]. There exist canonical braiding isomorphisms A p b B B p b A sending a tensor a b b to ( 1) BaBb b 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, Appendix 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: PROPOSITION ([Bla06, Theorem ]). The following classes of C -algebras are nuclear: Commutative 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 PROPOSITION ([WO93, Theorem T.6.26]). Let 0 A 1 A A 2 0 be a short exact sequence of C -algebras and B another C -algebra. If at least one of the C -algebras A 2 or B is nuclear then the tensored sequence remains exact. 0 A 1 b B A b B A 2 b B EXAMPLE ([WO93, Corollary T.6.17, Theorem T.6.20]). In particular, the C -algebra C 0 (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 C 0 (X) p b A Ñ C0 (X, A), C 0 (X) b C 0 (Y) Ñ C 0 (X Y), f b a ÞÑ (x ÞÑ f (x)a), f b g ÞÑ ((x, y) ÞÑ f (x)g(y)) for all graded C -algebras A and spaces Y. Here and in the following, the space C 0 (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 DEFINITION (G-SPACE, G-PAIR). A G-space is a (locally compact, second-countable, Hausdorff) 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 (X, Y) Ñ (X 1, Y 1 ) is a proper G-map X Ñ X 1 sending Y to Y 1. We define the Cartesian product of G-pairs (X, Y) and (X 1, Y 1 ) to be the pair (X X 1, X Y 1 Y Y X 1 ). Similarly, we define their disjoint union to be the pair (X > X 1, Y > Y 1 ).

11 1.1. C -ALGEBRAS AND TOPOLOGICAL SPACES WITH GROUP ACTIONS LEMMA. The group action of a non-compact G-space X extends naturally to its one-point compactification X + = X Y t8u (with the trivial action on 8). PROOF. It suffices to verify continuity of G X + Ñ X + in the point (1 G, 8). Let X + zk with K X compact be a neighborhood of 8 P X +. Then GK, the orbit of K, is a compact set in X and G (X + zgk) is a neighborhood of (1 G, 8) that maps into X + zk. 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 category 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 PROPOSITION. The assignment C 0 : # (X, Y; A) Ñ C0 (XzY, A) C 0 (XzY) p b A (ϕ; Φ) ÞÑ ( f ÞÑ Φ f ϕ XzY ) defines a bifunctor which is contravariant in the G-pair and covariant in the algebra variable. Here, we extend functions by zero so that the second composition makes sense, and G acts on C 0 (XzY, 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. Let (X, Y) be a G-pair and A a graded G-C -algebra. It is well-known that C 0 (XzY, A) is a C - algebra and that C 0 (XzY, A) C 0 (XzY) p b 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 to show that C 0 (XzY) is in fact a G-C -algebra, i.e. that every element is G-continuous. Clearly we may assume that Y = H. Then the orbit map of any function f P C 0 (X) C(X + ) can be written as the composition G g 1 C(X +, X + ) f C(X + ) which in fact maps into C 0 (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 C 0 is well-defined on morphisms. We only need to show that the preimage of a compact set K X 1 zy 1 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 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]) REMARK. The functor C 0 maps disjoint unions to direct sums and Cartesian products to tensor products. Indeed, C 0 ((X > X 1 )z(y > Y 1 )) = C 0 ((XzY) > (X 1 zy 1 )) = C 0 (XzY) ` C 0 (X 1 zy 1 ), C 0 ((X X 1 )z(x Y 1 Y Y X 1 )) = C 0 ((XzY) (X 1 zy 1 )) = C 0 (XzY) b C 0 (X 1 zy 1 ).

12 4 1. ANALYTIC K-THEORY AND K-HOMOLOGY Let us write ϕ := C 0 (ϕ; id A ) and Φ := C 0 (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) COROLLARY. We have natural short exact sequences of G-C -algebras (1.1.11) 0 C 0 (XzY, A) rel C 0 (X, A) incl C 0 (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 C 0 (Y) is nuclear it is also semisplit [Bla98, Theorem ]. 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). For naturality, observe that every morphism (ϕ; Φ) : (X, Y; A) Ñ (X 1, Y 1 ; B) gives rise to a commutative diagram (Y, H; A) (X, H; A) (X, Y; A) (ϕ Y ;Φ) (ϕ;φ) (ϕ;φ) (Y 1, H; B) (X 1, H; B) (X 1, Y 1 ; B) in the product category. The claim thus follows from applying the functor C 0. 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 EXAMPLE (ENDOMORPHISMS OF THE EXTERIOR ALGEBRA). Let W be a finite-dimensional unitary representation 1 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 in the next section EXAMPLE (CLIFFORD ALGEBRA). Let W as in the previous example. The complex Clifford algebra C W is defined as quotient of the tensor algebra of W with respect to the relation x 2 = xx, xy. It inherits the G-action and Z/2-grading from the tensor algebra. We obtain an antilinear involution on C W by setting (x 1 x k ) := x k x 1 for x 1,..., x k P W. The Clifford algebra C W 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 satisfies Φ(x) 2 = xx, xy extends uniquely to a G-equivariant -algebra homomorphism C W Ñ A. In particular, every G-equivariant isometric -endomorphism of W extends to a G-equivariant -algebra automorphism of C W. We can define a canonical inner product on C W by taking any orthonormal basis (e i ) of W and requiring that the monomials (e i1 e ik ) for i 1... i k and 0 k dim W form an orthonormal basis. Then we have a canonical isometric isomorphism of unitary G-representations (1.1.14) W Ñ CW, e i1 ^... ^ e ik ÞÑ e i1 e ik, where (e i ) is any orthonormal basis of W. 1 We will often identify a representation with its corresponding representation space.

13 1.2. HILBERT G-C -MODULES 5 However, we will usually consider C W to be equipped with the C -norm constructed as follows: Denote by λ x the operator of left multiplication by x P W on the exterior algebra. By the universal property, the G-equivariant -homomorphism (1.1.15) W ` ( W) Ñ L( W), x ` y ÞÑ λx+y + λ x y extends to an equivariant homomorphism C W`( W) Ñ L( W) of -algebras which for reasons of dimension has to be an isomorphism. We can use it to equip C W with a C -norm via the canonical embedding C W ãñ C W`( W) extending the inclusion W W ` ( W). Thus C W is canonically a graded G-C -algebra EXAMPLE (STANDARD CLIFFORD ALGEBRA). Consider W = C p+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 C p,q for the Clifford algebra of W. This is the complex Clifford algebra with respect to the quadratic form (z k ) ÞÑ z z2 p z 2 p+1... z2 p+q. We will refer to the algebras C p,q as the standard Clifford algebras. All these algebras are canonically isomorphic for fixed sum p + q = n. Indeed, an isomorphism C p,q+1 Ñ C p+1,q is induced by the universal property from mapping the coordinate basis vector e p+1 to ie p+1 and fixing all other coordinate basis vectors. We also have isomorphisms C p,q b p Cp 1,q 1 Ñ C p+p 1,q+q1 extending the assignment $ '& '% e i b 1 ÞÑ e i (i p) e i b 1 ÞÑ e p 1 +i (i p) 1 b e i ÞÑ ( 1) q e p+i (i p 1 ) 1 b e i ÞÑ e p+q+i (i p 1 ) In the following we will write C p := C p,0 = C C p and C p := C C p where C p = C p+0 as above. We shall always identify C 0,n C n using the isomorphism induced by sending e i to ( 1) i 1 e i REMARK. Our notation and conventions are as in Kasparov s article [Kas81, ]. 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] 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 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.

14 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 := 1 B(V) = id V the identity operator of V. If E and F are two Hilbert G-A-modules then we write B A (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 B A (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 K A (E, F) of compact operators is defined to be the closed ideal generated by the rank one operators T v,w := xv, yw where v P E and w P F. The space B A (E) := B A (E, E) is always a C -algebra (with the operator norm), but the set of G- continuous elements is usually a proper subset of B A (E). However, every G-equivariant operator in B A (E) is of course G-continuous (its orbit map is constant). In particular, the closed subset of G-equivariant operators in B A (E) is a G-C -algebra. We also remark that the ideal of compact operators K A (E) := K A (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 B A (E) can of course be uniquely decomposed into a grading-preserving and a grading-reversing part. This way we can consider B A (E) and K A (E) as graded C -algebras EXAMPLE. Every graded G-C -algebra A and, more generally, every finite direct sum A n is a graded Hilbert G-A-module over A with inner product x(a i ), (b i )y := n i=1 a i b i and diagonal G-action. It is well-known that every A-linear map A n Ñ A m can be written as left multiplication by a matrix with entries in the left multiplier algebra M left (A) of A. It follows that every such map is automatically continuous and has an adjoint. The compact operators on A n 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: L A (A n, A m ) = B A (A n, A m ) = K A (A n, A m ) M m n (A) M m n (C) b A L(C n, C m ) b A. Here, we have used the notation L A (V, W) for the set of A-linear maps from V to W, and we write L(V, W) := L C (V, W) 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-dimensional W all the algebras B(W), K(W) and L(W) are identical and hence graded G-C -algebras DEFINITION (EXTERIOR TENSOR PRODUCT). Let E be a graded Hilbert G-A-module and F a graded Hilbert G-B-module. The skew-commutative (exterior) tensor product E p b F is defined by completing the algebraic skew-commutative tensor product with respect to the A p b B-valued inner product xx b y, x 1 b y 1 y := xx, x 1 y b xy, y 1 y. It is a Hilbert G-A p b B-module with diagonal G-action, right A p b B-action given by (x b y)(a b b) := ( 1) ByBa (xa b yb) and grading B(x b y) = Bx + By. Moreover, we have a canonical embedding B A (E) p b BB (F) Ñ B A p b B (E p b F), T p b S ÞÑ (x b y ÞÑ ( 1) BSBx T(x) b S(y)) which restricts to an isomorphism K A (E) p b KB (F) Ñ K A p b B (E p b F). We will usually write b instead of p b for tensor products of trivially graded Hilbert modules.

15 1.2. HILBERT G-C -MODULES EXAMPLE (FREE HILBERT G-A-MODULE). Let A be a graded G-C -algebra and W a finitedimensional unitary representation of G on a graded Hilbert space. Then, combining the preceding examples, we find that W p b A is a graded Hilbert G-A-module which we will call a finitedimensional free Hilbert G-A-module (we will see in Proposition that this is indeed the correct notion of a free Hilbert module). Note that we recover A n if W is the trivial representation on C n. Now suppose that A and W are trivially graded and that A is unital. If W 1 is another triviallygraded finite-dimensional unitary representation of G then, generalizing Example 1.2.2, we find that L A (W b A, W 1 b A) = B A (W b A, W 1 b A) L(W, W 1 ) b A 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 p b B is not ambiguous DEFINITION (INTERIOR TENSOR PRODUCT). 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 B B (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 xx b y, x 1 b y 1 y := xy, Φ(xx, x 1 y)y 1 y (we may have to quotient out vectors of norm zero first) is a graded Hilbert G-B-module with the diagonal G-action and B-action on the second factor, called the (interior) tensor product E p b Φ F along Φ. It is graded in the same way as the exterior tensor product. Note that for every other Hilbert G-A-module E 1 we have a canonical map B A (E, E 1 ) Ñ B B (E p b Φ F, E 1 p b Φ F), T ÞÑ T b 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: DEFINITION (PUSHFORWARD). The pushforward of a graded Hilbert G-A-module E along a morphism Φ : A Ñ B of graded G-C -algebras is defined to be Φ (E) := E b 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 LEMMA ([Bla98, Example , (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-Bmodules. 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 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 morphism Φ : A Ñ B B (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- module action of F and write E p b B F for the result.

16 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 of graded G-C -algebras by precomposing the morphism B Ñ B C (E) with Φ. In other words, the left action is given by the formula ax := Φ(a)x EXAMPLE. Every morphism Φ : A Ñ B of graded G-C -algebras determines a graded Hilbert G-A-B-bimodule B Φ in the following way. As a Hilbert G-B-module, B Φ 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 K B (B) B (Example 1.2.2). In fact, it is easy to see that the interior tensor product with the bimodule B Φ implements the pullback and pushforward of Hilbert bimodules along Φ. The following lemma is somewhat of an analogon of Lemma for Hilbert bimodules: 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 clearly intertwines the left A-action. Recall that in Example we have defined both an inner product norm and a C -norm on Clifford algebras C W. Their interplay can be understood as follows: EXAMPLE. Let C W be a complex Clifford algebra. Then the Hilbert space C W is a Hilbert G-C W -C-bimodule with respect to each of the following actions of C W : 1. left multiplication, or 2. right multiplication by the transpose (this is the linear involution ( ) t defined by reversing the order of products). PROOF. The nontrivial part of the assertion is that the respective morphisms C W Ñ B(C W ) are -preserving. In order to see this, we will identify the Hilbert space C W with the exterior algebra W (Equation (1.1.14)). For the first action, let x P W, choose a unit vector e 1 = sx in the span of x and extend it to an orthonormal basis (e i ) of W. Then for every v = e i1 e ik e i1 ^... ^ e ik with i 1... i k we have # sei2 e ik (λ x + λ x xv = )v, if i 1 = 1 se 1 e i1 e ik (λ x + λ x )v, if i 1 1 This shows that the left multiplication action corresponds to the canonical left action of C M on the exterior algebra as defined in Equation (1.1.15). For the second action, consider v = e i1 e ik e i1 ^... ^ e ik with i 1... i k. Then vx = # se i1 e ik 1 ( 1) k (λ x λ x )v, if i k = 1 ( 1) k se 1 e i1 e ik ( 1) k (λ x λ x )v, if i k 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 DEFINITION (G-IMPRIMITIVITY BIMODULE, 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 K B (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

17 1.3. EQUIVARIANT KK-THEORY 9 bimodule E := K B (E, B) is a G-imprimitivity bimodule between B and A (with left multiplication on the result and right action of A K B (E) by precomposition; see e.g. [Bla98, Exercise ]). In fact, E b B E A and E b 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 EXAMPLE (IRREDUCIBLE CLIFFORD MODULES). Recall the definition of the standard Clifford algebras from Example We can consider the exterior algebra C p as a Hilbert C p,p -C-bimodule using the left action defined by the composition (1.2.16) C p,p ÝÑ Cp,0 p b C0,p ÝÑ Cp p b C p = C C p p b C C p ÝÑ C Cp`( C p ) ÝÑ L( C p ) All isomorphisms are induced by the canonical ones from Examples and Because their composition is again an isomorphism and every Hilbert C-module is full, we conclude that W p := C p is an imprimitivity bimodule between C p,p and C. Note that in this case the dual imprimitivity bimodule is just the dual space with its canonical right C p,p -action by left multiplication in the argument. We will later need the following fact LEMMA. We have W p+q W p p b Wq as Hilbert C p+q,p+q -C-bimodules. Here, we use the canonical isomorphism C p,p p b Cq,q C p+q,p+q from Example to consider the exterior tensor product as a left C p+q,p+q -module. PROOF. Clearly, it suffices to show that both representations of C p+q,p+q are isomorphic. It is well-known from the representation theory of complex Clifford algebras that all irreducible graded representations of C p,p have dimension 2 2p and that they are classified by the action of the orientation element ω p := e 1 e 2p (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. Both representations of C p+q,p+q we want to compare have dimension 2 2(p+q), 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 ω p+q acts by the identity on 1 P W p+q, hence on any even vector in W p+q. On the other hand, the canonical isomorphism C p+q,p+q C p,p b p Cq,q sends ω p+q to (ω p b 1)(1 b ω q ), hence the volume element also acts by the identity on even vectors in W p b p Wq. We conclude that both representations are isomorphic and the claim is proved 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 definitions 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 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.

18 10 1. ANALYTIC K-THEORY AND K-HOMOLOGY C. There exists an associative bilinear product pb D : KK G p (A 1, B 1 p b D) b KK G q (D p b A2, B 2 ) Ñ KK G p+q(a 1 p b A2, B 1 p b B2 ) called the Kasparov product which is contravariant in A 1 and A 2, covariant in B 1 and B 2 and functorial in D in the sense that Φ (x) p b D2 y = x p b D1 Φ (x) for every morphism Φ : D 1 Ñ D 2. The most important special cases are the composition Kasparov product where A 2 = B 1 = C and the exterior Kasparov product where D = C. D. The exterior Kasparov product is gradedly commutative, i.e. x p b y = ( 1) BxBy y p b 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. E. Every morphism of graded G-C -algebras Φ : A Ñ B defines an element [Φ] P KK0 G (A, B). This assignment is functorial with respect to the composition Kasparov product, and the tensor product Φ p b Ψ of morphisms is sent to the exterior Kasparov product [Φ] p b [Ψ]. We can regard KK0 G as a category whose objects are graded G-C -algebras, where morphisms from A to B are given by the elements of KK0 G (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 KK G. 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). In particular, KK G (A, A) is a graded ring with unit 1 A := [id A ], and all the groups KK G (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 1 A as tensoring with A. G. We have the identity x p b D y = (x p b 1D2 ) p b D1 p b D p b D2 (1 D1 p b y) for all x P KK G (A 1, B 1 p b D1 p b D) and y P KK G (D p b D2 p b A2, B 2 ). It follows using properties C and D that the Kasparov product is gradedly distributive in the sense that (x p b D y) p b (x 1 p b D 1y 1 ) = ( 1) Bx1 By (x p b x 1 ) p b D p b D 1(y p b y 1 ). In particular, pullback and pushforward of an external Kasparov product x p b y along a tensor product of morphisms Φ p b Ψ is the external tensor product of the pullbacks or pushforwards of the individual factors, and tensoring is distributive in the sense that 1 E p b (x p b D y) = (x p b D y) p b 1E = (1 E p b x) p b E p b D (1 E p b y) = (x p b 1E ) p b D p b E (y p b 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 B 1 = A 2 = 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 KK G (A, B) KK G (A p b K, B) KK G (A, B p b K) KK G (A p b K, B p 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.

19 1.3. EQUIVARIANT KK-THEORY 11 J. Every Hilbert G-A-B-bimodule E with compact A-action determines an element [E] P KK0 G (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 B Φ from Example , 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. [E b p F] = [E] b p [F] and [E b p B F] = [E] b p B [F] in the situation of Definition , and every G-C -algebra A considered as a bimodule over itself determines the unit element of KK0 G (A, A). In particular, every G-imprimitivity bimodule determines a KK-equivalence (this follows from the discussion after Definition ). 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 product, note that we now have a Hilbert G-C-C-bimodule and send it to the element [W] P R 0 (G). Furthermore, R 1 (G) is always zero. L. Every short exact sequence 0 J j A q 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 element B P KK G 1 (Q, J). Moreover, for every G-C -algebra D we get long exact sequences... KK G (D, J) j KK G (D, A) q KK G (D, Q) B KK G 1 (D, J) KK G (Q, D) q KK G (A, D) j KK G (J, D) B KK G 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 1 D. 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 KK H (A, B) Ñ KK G (C 0 (X, A) H, C 0 (X, B) H ) compatible with Kasparov product and unit elements (for A = B). Here, ( ) H denotes the subalgebra of H-invariant elements. Note that if A is a finite-dimensional H-C -algebra then C 0 (X, A) H 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. N. We have natural isomorphisms KK G KK +2 G Kasparov product and induction. (Clifford periodicity) which commute with the 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.

20 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 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 B B (E) such that [T, a], a(t 2 1), a(t T ), g(t) T P K B (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-Bbimodules 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 DEFINITION (HOMOTOPY). A homotopy for (A, B) is a cycle for (A, C([0, 1], B)), and we say that the pushforwards along the evaluation maps eval k : C([0, 1], B) Ñ B, f ÞÑ f (k), k = 0, 1 are homotopic. We can now define the equivariant KK-groups DEFINITION (EQUIVARIANT KK-GROUPS). The equivariant KK-group KK0 G (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 KK G n (A, B) := KK G 0 (A p b Cn,0, B) KK G n(a, B) := KK G 0 (A p b C0,n, B) for n P N 0, and this way we can consider KK G (A, B) as a Z-graded Abelian group REMARK. A cycle (E, T 1 ) is a compact perturbation of another cycle (E, T) for (A, B) if a(t 1 T) P K B (E) (@a P A). Such cycles are homotopic via the linear path from T to T 1 (in the obvious sense) and hence determine the same element in KK0 G (A, B). The next lemma gives another useful means of identifying cycles LEMMA ([Bla98, Proposition ]). Two cycles (E, T), (E, T 1 ) for (A, B) with a[t, T 1 ]a 0 (mod K B (E)) for all a P A determine the same element in KK0 G (A, B) 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.

21 1.3. EQUIVARIANT KK-THEORY 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 ]). 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 SKETCH OF PROPERTIES A AND B. Pullback and pushforward are implemented in terms of the same operations on Hilbert bimodules (Definition ). Homotopy invariance follows from the homotopy equivalence relation we have quotiented out in the definition of the KK-groups. Indeed, if Φ : A Ñ C([0, 1], B) is a homotopy of morphisms Φ 0 and Φ 1 then the pushforward of an arbitrary cycle along Φ is a homotopy of the pushforwards of that cycle along Φ 0 and Φ 1 (similarly for pullbacks, see [Bla98, Proposition ]) SKETCH OF PROPERTY C. One first defines the composition Kasparov product pb D : KK0 G(A, D) b KKG 0 (D, B) Ñ KKG 0 (A, B). On the level of bimodules, the product is simply given by the interior tensor product over D, but because there is no canonical embedding B B (F) Ñ B B (E b p D F) (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 p b D, T p b 1), considering D as a Hilbert G-bimodule over itself, and establishes that this assignment induces a homomorphism σ D : KK G 0 (A, B) Ñ KKG 0 (A p b D, A p b D). This map is then used to define a more general Kasparov product by the formula # KK G 0 (A 1, B 1 p b D) b KK G 0 (D p b A2, B 2 ) Ñ KK G 0 (A 1 p b A2, B 1 p b B2 ) x b y ÞÑ σ A2 (x) p b B1 p b D p b A2 σ B1 (y) 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] SKETCH OF 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 [E] := [E, 0], [Φ] := [B Φ ] = [B Φ, 0] P KK0 G (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 ]) and refer to [Bla98, Examples , (a) and (b)] for a proof of the second fact. One similarly verifies that the

22 14 1. ANALYTIC K-THEORY AND K-HOMOLOGY exterior tensor product with 1 D agrees with the tensoring homomorphism σ D 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 ], [Kas88, Remark after Definition 2.15] and [Bla98, Section 19.5 and Exercise ] together with [BS89, Theorème 7.2], respectively SKETCH OF PROPERTY M. The induction homomorphism is defined as follows KK H (A, B) Ñ KK G (C 0 (X, A) H, C 0 (X, B) H ), [E, T] ÞÑ [C 0 (X, E) H, T ] for cycles with H-equivariant operator T (cf. Remark 1.3.7). Here, C 0 (X, E) H 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]) SKETCH OF PROPERTY N. One first proves that the canonical isomorphisms (1.3.14) C p,q p b Cp 1,q 1 Ñ Cp+p 1,q+q 1, C p,q+1 Ñ Cp+1,q from Example are compatible with each other in the sense that any combination of them yields identical pushforward and pullback homomorphisms in KK-theory ([Kas81, Theorem 5.3]). In particular these isomorphisms allow us to identify C n+2,0 C n,0 p b C1,1 and C 0,n+2 C 0,n p b C1,1 for all n P N 0. Consequently, it suffices to fix a single KK-equivalence in KK0 G(C 1,1, C) since it follows from the above considerations and graded commutativity that exterior Kasparov multiplication by any such element implements natural isomorphisms KK G KK +2 G compatible with the Kasparov product. These are also compatible with induction since the latter commutes with exterior Kasparov 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 KK0 G(A 1 b p Cp,q, B 1 b p D) b KK G 0 (D b p A2 b p Cp 1,q 1, B b 2) p ÝÑ D KK G 0 (A 1 b p A2 b p Cp,q b p Cp 1,q 1, B 1 b p B2 ) ÝÑ KK G 0 (A 1 p b A2 p b Cp+p 1,q+q 1, B 1 p b B2 ). By repeatedly applying the inverse of the KK-equivalence we can reduce C p+p 1,q+q1 to either of the forms C p+p 1 q q 1,0 or C 0,q+q 1 p p1. Observe that this specializes to a product on all KKgroups. The KK-equivalence we shall fix is the one induced by the imprimitivity bimodule coming from the canonical representation W 1 of C 1,1 on the exterior algebra of C (Example ). Note that it follows from the isomorphism of Hilbert bimodules W p+q W p b p Wq (Lemma ) that the induced higher periodicity isomorphisms KK G KK +2p G can also be written as Kasparov multiplication by the element [W p ] P KK0 G(C p,p, C) REMARK. An equivalent picture of the higher KK-groups is given by the canonical isomorphisms KK G p (A, B) = KK G 0 (A p b Cp,0, B) ÝÑ KK G 0 (A, B p b C0,p ) KK G p(a, B) = KK G 0 (A p b C0,p, B) ÝÑ KK G 0 (A, B p b Cp,0 )