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An Equivariant Bivariant Chern Character The Pennsylvania State University The Graduate School Department of Mathematics AN EQUIVARIANT BIVARIANT CHERN CHARACTER A Thesis in Mathematics by Jeff Raven c 2004 Jeff Raven Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2004 We approve the thesis of Jeff Raven. Date of Signature Paul Baum Thesis Adviser Evan Pugh Professor of Mathematics Chair of Committee Nigel Higson Distinguished Professor of Mathematics Head, Department of Mathematics Victor Nistor Professor of Mathematics Pablo Laguna Professor of Physics, Astronomy & Astrophysics Abstract Using notions from homological algebra and sheaf theory Baum and Schneider defined a bivariant equivariant cohomology theory which shares many of the properties of equivariant KK -theory; indeed, these two theories have so much in common that when the group under consideration is profinite they are rationally isomorphic. This, combined with other similar results, led Baum and Schneider to conjecture that the same should be true for any totally disconnected group. We verify the conjecture for a large class of such groups, namely the countable discrete groups. iii Contents Acknowledgments ........................................ vi Chapter 1. Introduction ................................... 1 I Equivariant KK -Theory 5 Chapter 2. Proper Actions and Equivariant K-Theory .................. 6 2.1 ProperActions ................................... 6 2.2 EquivariantVectorBundles . .... 10 2.3 Equivariant K-Theory ............................... 13 2.4 Equivariant Spinc Structures ........................... 14 2.5 EquivariantBottElements. .... 18 Chapter 3. Kasparov’s Equivariant KK -Theory ...................... 21 3.1 C∗-Algebras..................................... 21 3.2 Hilbertmodules.................................. 22 3.3 Kasparov’s KK -Theory .............................. 25 3.4 Vector Bundles and KK G(X,X) ......................... 28 G 3.5 Dirac Operators and KK ∗ (M, pt)......................... 32 3.6 Principal Induction and the Thom Isomorphism . ........ 35 Chapter 4. Topological Equivariant KK -Theory ...................... 39 4.1 TopologicalCyclesandBordism. ..... 39 4.2 VectorBundleModification . ... 45 4.3 Topological KK -Theory .............................. 46 4.4 ATechnicalLemma ................................ 47 4.5 NormalBordism .................................. 50 4.6 TheEilenberg-SteenrodAxioms. ..... 53 4.7 InductionandtheDimensionAxiom . .... 56 II Equivariant Bivariant Homology 60 Chapter 5. Homological Algebra ............................... 61 5.1 AbelianCategories ............................... .. 61 5.2 ComplexesandHomotopy . 63 5.3 TriangulatedCategories . .... 66 5.4 LocalizationofCategories . ..... 70 iv 5.5 TheDerivedCategory .............................. 74 5.6 DerivedFunctors ................................. 75 5.7 Example:Modules................................. 77 Chapter 6. Equivariant Sheaf Theory ............................ 80 6.1 Sheaves ....................................... 80 6.2 DirectandInverseImages . ... 84 6.3 BorelCohomology ................................. 87 6.4 ProperSupports .................................. 90 6.5 EquivariantBivariantCohomology . ...... 94 III An Equivariant Bivariant Chern Character 99 Chapter 7. An Equivariant Bivariant Chern Character .................. 100 7.1 The K-theoryChernCharacter . 100 7.2 The Chern Character of a G-Spinc Manifold................... 103 7.3 TheBivariantChernCharacter . .... 108 7.4 ConcludingRemarks ............................... 112 Appendices 114 Appendix A. Straightening the Angle ............................. 115 Appendix B. Smoothing Spinc Structures ........................... 117 Bibliography ........................................... 120 v Acknowledgments Many thanks to my adviser Paul Baum for our numerous discussions and his enthusiastic support. My thanks also go out to the other members of my committee for their feedback on my thesis. Finally I would also like to thank my wife, Marjorie Raven, for her assistance in proofreading the final draft. vi Chapter 1 Introduction K-theory first appeared on the mathematical landscape in the early sixties, when it played a central role in the proofs of Riemann-Roch and the Atiyah-Singer index theorem. As the natural home for the symbols of elliptic operators on a manifold M, the use of the K-theory group K0(T ∗M) allowed for a much more elegant proof of the index theorem than earlier bordism-based arguments. At the time, it was realized that this process of associating classes in K0(T ∗M) to elliptic operators on M should be a manifestation of Poincar´eduality, so that elliptic operators on M should themselves provide elements in the K-homology of M; unfortunately at the time the only means available for defining K-homology was via the Bott spectrum, a viewpoint unsuited to analytic objects. By generalizing the notion of elliptic operator, Atiyah [Ati70] was ultimately able to provide a notion of analytic K-cycles, but he was unfortunately unable to find a suitable equivalence relation which would recover K-homology. It wasn’t until a decade later, with the work of Brown, Douglas, and Fillmore [BDF73] on extensions of C∗-algebras, that an analytic picture of K-homology emerged. This was quickly followed by a more general theory of extensions due to Kasparov [Kas80b], which provided a natural framework for working with not only K-theory and K-homology, but maps between them as well. The Kasparov groups KK ∗(A, B) form a bivariant theory on C∗-algebras, contravariant in the first variable and covariant in the second. Of particular importance is the existence of a composition product KK i(A, B) KK j (B, C) KK i+j (A, C) ⊗ → which serves to generalize the cup and cap products. One can obtain a theory for spaces simply by letting ∗ KK ∗(X, Y )= KK (C0(X), C0(Y )), where C0(X) denotes the continuous functions on X vanishing at infinity. The fact that C0 is contravariant means that KK ∗(X, Y ) is a homology theory in the first variable and a cohomology in the second; a bit of work shows that setting one of the spaces to be a point allows one to recover K-theory and K-homology. The corresponding equivariant groups have played an important part throughout the develop- ment of these theories. The original K-theoretic proof of the index theorem relied in part on equiv- ariant K-theory, while equivariant K-homology lies at the heart of the Baum-Connes and Novikov ∗ conjectures. As in the non-equivariant case, the equivariant Kasparov groups KK G(A, B) serve to unify these many constructions. Baum and Connes [BC98] were among the first to consider the problem of defining a Chern 1 character in equivariant K-theory, focusing on the case when G is discrete; using a markedly different approach, L¨uck and Oliver [LO01] later extended these results to a much larger class of spaces. Smooth actions of Lie groups were the next to be analyzed, first by Baum, Brylinksi and MacPherson [BBM85], who looked at smooth S1-actions, but later by many others — including work by Block and Getzler [BG94] which approached the problem using equivariant cyclic ho- mology. Meanwhile on the homological side, for discrete groups L¨uck [L¨uc02] produced a Chern character isomorphism from an equivariant homology group to rational equivariant K-homology. However, until recently very little consideration had been given to the problem of defining a Chern character isomorphism ∗∗ ch : KK G(X, Y ) C HH (X, Y ; C), G ∗ ⊗ → G where HHG denotes an appropriate bivariant equivariantd theory. Combining earlier work of Nistor [Nis91] on bivariant Chern characters with more recent work by Voigt [Voi03] on bivariant d equivariant periodic cyclic homology provides one means of approaching the problem (at least for certain spaces), but one might hope that a more general Chern character could be obtained using topological techniques. Matters are substantially simplified by working with a finite group over the complex numbers. Classical representation theory then tells us that R(G) C = C, ⊗ ∼ [γ]∈MG//G so that every R(G) C-module decomposes as a sum of local terms. Applying this to the case of ⊗ G C KK ∗ (X, Y ) , Baum and Schneider [BS02] have shown that the summand corresponding to the ⊗ ∗∗ γ γ conjugacy class [g] can be identified with the bivariant equivariant cohomology HH Z(γ)(X , Y ), and thus KK G(X, Y ) C = HH ∗∗ (Xγ, Y γ). ∗ ⊗ ∼ Z(γ) [γ]∈MG//G A limiting argument then allowed them to extend this result to profinite groups, though in this case the target is more complicated. This, combined with some of the other results mentioned above, led them to conjecture that a similar result should hold for all totally disconnected groups. Theorem. For any countable discrete group G, finite proper G-CW complex X and G-space Y there is a natural Chern character isomorphism ∗∗ KK G(X, Y ) C HH (X, Y ). ∗ ⊗ → G In the end, the only real obstacle to proving suchd a result lies in defining a sufficiently natural Chern character; once this is done, the fact that it is an isomorphism follows more or less by G abstract nonsense. However, to do this one needs a more concrete model for KK ∗ (X, Y ) than is provided by Kasparov’s definition; it is at this point that an approach first used by Baum and Douglas [BD82] to describe K∗(X)= KK ∗(X, pt) becomes quite useful. In this framework, one considers triples (M,ξ,f) consisting of a Spinc manifold M, a class ξ K(M), and a continuous map f : M X. There is a natural notion of bordism on such ∈ → 2 cycles, but it alone
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