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 (ab- stract) 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 geome- try. 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 con- structions 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 geo- metric 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 in- volved 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 gen- eralize 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.