An Equivariant Bivariant Chern Character

Home , Equivariant topology

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 Poincaréduality, 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 development of these theories. The original K-theoretic proof of the index theorem relied in part on equivariant 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, Lück 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 homology. Meanwhile on the homological side, for discrete groups Lück [Lüc02] 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 is not sufficient to produce K∗(X). One must also make use of ‘vector bundle modification’ — given a smooth Spinc bundle E M one can construct a new cycle → (M,ξ,f)E := (S(E 1),β π∗ξ,f π). ⊕ E ⊗ ◦ Two triples are then equivalent if they can be connected by a series of bordisms and vector bundle modifications. That the resulting group is isomorphic to K∗(X) would quickly follow from the fact that it forms a homology theory, but it is not entirely obvious that this is the case. In particular, it is unclear whether one has a six-term exact sequence associated to a CW-pair (X, A) — the essential difficulty being that the equivalence relation does not provide a concise condition for when a triple is equivalent to zero. There are a variety of arguments that establish the existence of the six-term exact sequence, yet they inevitably involve the introduction of an auxiliary group involving framed bordism. However, a more direct (though similar) approach is available. One can define a relation of

‘normal bordism’ on triples, in which two triples (Mi, ξi,fi)i=0,1 are equivalent if and only if

c νi there exist Spin normal bundles νi for each Mi such that the triples (Mi, ξi,fi) are bordant. It turns out that this produces the same equivalence relation as that of Baum and Douglas, but is explicit enough that the usual arguments for the long exact sequence in bordism can be carried out with only slight modiﬁcation.

These ideas extend naturally to the study of KK ∗(X, Y ); in fact, so long as G is compact G they even apply to KK ∗ (X, Y ). However, while the same result is ultimately true when G is a countable discrete group and X is a ﬁnite proper G-CW complex, the argument in this case is more subtle. The problem is that normal bordism relies heavily on the fact that every Spinc manifold has a Spinc normal bundle. In the equivariant case one must instead work with G-Spinc normal bundles, and unfortunately these need not exist when G is non-compact.

To deal with this, one must shift to the category G-TopZ of G-spaces over a proper G-space Z. In this context one defines a normal bundle for a G-Spinc manifold M Z to be a G-Spinc → bundle ν such that TM ν is isomorphic to the pullback of a G-Spinc bundle on Z. When ⊕ G is compact and Z is a point this reduces to the usual notion of normal bundle, but when G is non-compact the extra G-Spinc bundles provided by Z can help one sidestep the difficulties mentioned above. In particular, a theorem of Lück and Oliver allows one to conclude that when G is discrete and Z is a finite proper G-CW complex every G-Spinc manifold M Z will have → a normal bundle. This in turn allows normal bordism to be used to define a homology theory G|Z tKK ∗ (X, Y ) on G-TopZ . A little more work shows that this group is actually independent of G the choice of Z and, in fact, is isomorphic to KK ∗ (X, Y ) whenever X is a finite proper G-CW complex. Before finally defining the Chern character there is still one last matter to attend to — the ∗∗ target of the map, the group denoted HH G (X, Y ). As when G was finite, the target arises as a sum of local contributions, one for each finite order conjugacy class of G. Specifically, d ∗ ∗ γ γ HH G(X, Y ) := HH Z(γ)(X , Y ), [γ]∈GMtor//G d 3 where G G denotes the elements of finite order in G, G //G its conjugacy classes and tor ⊆ tor ∗ γ γ HH Z(γ)(X , Y ) is the bivariant Z(γ)-equivariant cohomology of the γ-fixed sets. ∗ C G Thus defined, HHG(X, Y ; ) naturally enjoys many of the same properties as KK ∗ (X, Y ); in particular, it has long exact sequences in each variable, and there is an associative product d ∗ ∗ ∗ HH (X, Y ) HH (Y,Z) HH (X,Z). G ⊗ G → G Moreover any proper G-equivariantd mapdf : Y X dnaturally yields a corresponding class 0 → [f] HH (X, Y ). ∈ G With all of that said, defining the Chern character d ∗∗ ch : KK G(X, Y ) C HH (X, Y ) G ∗ ⊗ → G for a discrete group G now simplyb comes down to describingd for each triple (M,ξ,f) corresponding classes

∗∗ ch (M) HH (M, pt) G ∈ G ∗∗ ch (ξ) HH (M Y,M Y ) b G ∈ d G × × and using the composition productb to constructd

∗∗ [f] [π ] ch (ξ) (ch (M) 1 ) HH (X, Y ). ◦ M ◦ G ◦ G × Y ∈ G The content of this thesis fallsb quite naturallyb into three pdarts. In the first part we begin by reviewing the necessary material regarding proper actions of discrete groups and their equivariant K-theory; we then move on to review the properties of Kasparov’s equivariant bivariant theory. We conclude by using the notion of topological KK G-cycles described above to define the tKK G- groups, and prove that they are isomorphic to Kasparov’s analytic groups in those cases of interest. The next part of the thesis sees the development of the homological algebra and equivariant sheaf theory needed to define the bivariant equivariant cohomology groups and establish its properties. Finally, in the third part we define the bivariant equivariant cohomology classes needed to construct the equivariant bivariant Chern character and show that it gives a well-defined isomorphism. We conclude by discussing some extensions of this result, as well as describing an alternate approach to the problem which might hold promise for (non-discrete) totally disconnected groups.

Conventions

Unless speciﬁcally noted, all spaces are assumed to be compactly generated, paracompact and Hausdorﬀ. Note that in particular this implies that all spaces are assumed to be normal.

4 Part I

Equivariant KK -Theory

5 Chapter 2

Proper Actions and Equivariant K-Theory

2.1 Proper Actions

Let G be a countable discrete group. Definition 2.1.1. A G-space consists of a topological space X along with a continuous action of G on X. Given a closed G-invariant subspace A X we will refer to (X, A) as a G-pair. If ⊆ X and Y are two G-spaces, a G-map from X to Y is a continuous G-equivariant map X Y ; → a G-map between G-pairs (X, A) and (Y,B) is a G-map X Y that carries A into B. → In general the action on a G-space X can be rather pathological, in that there is nothing that ensures a reasonable quotient topology on X/G. The prototypical example of this is the action of Z on S1 = [0, 1]/ 0, 1 generated by an irrational translation T (x)= x + α. Indeed, while the { } quotient S1/Z is quite large, consisting of uncountably many orbits, the topology induced by the quotient map contains only the trivial open sets. Ultimately the problem with this example is that although the action is free, it is not proper — there are no Z-invariant neighborhoods of the form U = Z V . × Definition 2.1.2 ([BCH94]). A G-space X is proper if for every point p X there exists ∈ a G-invariant open neighborhood U, • a finite subgroup H G, and • ≤ a G-map φ : U G/H. • → There are various other notions of proper action in the literature, but most are equivalent given some minor assumptions about the spaces involved. The advantage of the definition given here is two-fold. First it allows for very concise proofs of the inheritance properties for proper actions, and second it emphasizes a view of proper spaces which is central to many arguments — namely, that every proper G-space is locally obtained from a finite group action. Indeed, given (U,H,φ) as in the definition above and letting V = φ−1([H]), one finds that U is G- homeomorphic to the induced space G V . This phenomenon will often allow us to reduce ×H global questions about proper G-actions to local ones regarding finite group actions, for which many more techniques are available. Lemma 2.1.3. Let X and Y be G-spaces, f : X Y a G-map. Then if Y is a proper G-space → so is X. In particular, any subspace of a proper G-space is a proper G-space. Proof. Choose a point x X and consider the image f(x) Y . Since Y is a proper G-space, ∈ ∈ we can find a triple (U,H,φ) for f(x); one can then easily check that (f −1(U),H,φ f) gives a ◦ triple for x X, and it follows that X is a proper G-space. ∈ 6 Lemma 2.1.4. Let X be a proper G-space and G0 G a subgroup. Then X is a proper G0-space ≤ under the restricted action.

Proof. Consider any point x X and let (U,H,φ) be the triple guaranteed by the properness ∈ of the G-action. Let V denote the component of U containing x; without loss of generality we may assume that H ﬁxes V . Since (G0 H)/H = G0/(G0 H) the restriction of φ to G0 V yields · ∼ ∩ · a G0-map φ0 : G0 V G0/(G0 H). Together this gives a triple (G0 V, G H0, φ0) for the · → ∩ · ∩ G0-action around x, and thus X is a proper G0-space.

One of the more signiﬁcant consequences of properness is that the action eventually separates various sets. To allow for a more concise discussion of this phenomena we will need the following extra bit of notation.

Definition 2.1.5. Let X be a G-space with subsets A and B. Define (A, B) to be the subset G of G given by g G : A (g B) = ∅ . { ∈ ∩ · 6 } Proposition 2.1.6. A G-space X is proper if and only if for every pair of points x, y X there ∈ exist neighborhoods W and W of x and y respectively such that (W , W ) is finite. x y G x y Proof. First suppose that X is proper and we wish to prove the finiteness claim. Choose two points x, y X and let (U,H,φ) be the triple for x guaranteed by the properness of the G-action. ∈ Let V denote the component of U containing x; without loss of generality we may assume that H fixes V and U = G V . ×H There are essentially two cases to consider. The simplest is when y = γ x. In this case we · simply let W = V and W = γ V — it is then easy to check that # (W , W ) = #H. So x y · G x y let us instead assume that y / G x. Note that the properness of the action implies that orbits ∈ · are closed in X, and thus U G y is open. Furthermore since X is normal we can find an open \ · G-invariant neighborhood W G x with W U G y. Let W = X W ; by construction W x ⊇ · x ⊆ \ · y \ x y is a G-invariant open set containing y such that (W , W )= ∅. G x y Now for the converse. For this it suffices to consider the case of x = y; the finiteness condition then guarantees an open neighborhood W with (W , W ) finite. In particular this implies that x G x x the stabilizer G (W , W ) is finite, and thus to prove properness we need only find a G - x ⊆ G x x x invariant neighborhood x V W such that U = G V = G V . ∈ ⊆ x · ∼ ×Gx Since X is normal we can assume that W G x = x . Let Z be the union of the translates x ∩ · { } γ W for γ (W , W ) G . This is a closed set which doesn’t contain the point x, and hence · x ∈ G x x \ x V = W Z is an open neighborhood of x. Note that since G is finite we can also choose V to x\ x be a G -invariant open neighborhood. It is now straightforward to verify that G V = G V , x · ∼ ×Gx and thus the action is proper.

Note that the proof of the proposition has the following corollary.

Corollary 2.1.7. Every proper G-space has Hausdorﬀ quotient.

Proposition 2.1.8. Let X be a G-space. If X is proper, then for every pair of compact subsets K,L X the set (K,L) is finite. The converse holds whenever X is locally compact. ⊆ G 7 Proof. Suppose that X is proper and we are given compact sets K and L. By the previous proposition for each (x, y) K L we can find open neighborhoods W and W such that ∈ × x y (W , W ) is finite. Together the open sets W W form a cover of K L, and hence by G x y x × y × compactness there is a finite subcover W W . But then { x(i) × y(i)} (K,L) (W , W ), G ⊆ G x(i) y(i) i [ and this last set is a finite union of finite sets, hence finite. Now suppose the finiteness condition is satisfied and we wish to prove that the action is proper. Consider two points x, y X. When X is locally compact there exist precompact ∈ neighborhoods Wx and Wy for x and y respectively. The finiteness condition then implies that the set (W , W ) is finite, and thus (W , W ) is finite. The previous proposition then shows G x y G x y that X is proper.

Corollary 2.1.9. Let X be a locally compact and proper G-space. Then averaging over G gives a well-deﬁned -linear map from C (X) to C (X)G. ∗ c b A G-space X is cocompact if the quotient space X/G is compact. Note that when the G-action is proper this implies that X is both locally compact and σ-compact.

Lemma 2.1.10. Let X be a proper cocompact G-space. Then there exists f C (X), f 0, ∈ c ≥ such that γ f =1. γ∈G · Proof. LetPπ : X X/G denote the quotient map. Through the combination of compactness → and properness we can find a finite open cover = U of X/G such that π−1(U ) = G V U { i} i ∼ ×Hi i and each V is precompact. Now let φ be a partition of unity for ; since π(V )= U we can lift i { i} U i i each φ to obtain a function φ˜ supported only on V X. It follows that the function g = φ˜ i i i ⊆ i i is compactly supported and g = γ g is everywhere non-zero. Now set f = g/g. γ∈G · P Given a G-space X, we will spendP a good deal of our time working with various fixed-point subsets. For any subgroup S G let N(S) denote the normalizer of S in G; note that the ≤ centralizer Z(S) sits within N(S) as a normal subgroup. Then the action of G on X restricts to give actions of N(S) and Z(S) on the space XS of S-fixed points.

Proposition 2.1.11. Let X be a proper G-space. Then for any subgroup S G the ﬁxed-point ≤ subspace XS is a proper N(S)-space. Moreover, if X is a cocompact G-space then XS is a cocompact N(S)-space. Similar statements hold with N(S) replaced by Z(S).

Proof. It follows immediately from Lemmas 2.1.3 and 2.1.4 that XS is both a proper N(S)-space and a proper Z(S)-space, so it only remains to prove cocompactness. Note that since every proper cocompact G-space is a finite union of closed induced subspaces it suffices to prove the claim for spaces of the form X = G K, where H G is a finite subgroup and K is a compact H-space. ×H ≤ Moreover, since XS/N(S) is a quotient of XS/Z(S) we need only consider the Z(S)-action. Consider the projection π : X = G K G/H. By assumption the preimage of each point ×H → is compact; thus the same is true for π : XS/Z(S) (G/H)S/Z(S). The desired result would → 8 then follow so long as the set (G/H)S /Z(S) were finite, and this is the content of the following lemma.

Lemma 2.1.12. Let H,S G be two ﬁnite subgroups of G. Then the set (G/H)S/Z(S) is ≤ ﬁnite.

Proof. A simple calculation shows that (G/H)S/Z(S) = Z(S) γ G : γ−1Sγ H /H. ∼ \{ ∈ ⊆ } But this set maps injectively into Hom(S, H)/H, where Hom(S, H) denotes the set of group homomorphisms from S to H. Since S and H are ﬁnite groups this last set is ﬁnite.

The concept of a CW-complex plays a vital rˆole throughout much of algebraic topology, allowing one to prove results by inductively climbing up the skeleta. For the same reason it will be useful to have an equivariant notion of CW-complex.

Deﬁnition 2.1.13. A G-space X is a G-CW complex if it can be constructed as the limit of an ascending chain of closed G-invariant subsets

∅ = X X X X X −1 ⊆ 0 ⊆ 1 ⊆···⊆ n ⊆··· where Xn is constructed from Xn−1 as a pushout of the form

n−1 ∆ S / Xn−1 n ×

∆ Dn / X n × n for some discrete G-space ∆n.

Remark 2.1.14. A G-CW complex is proper if and only if each of the G-spaces ∆n are proper.

The quotient of a G-CW complex is in a natural way a CW-complex. Our primary interest will be with finite G-CW complexes — those G-CW complexes whose quotients are finite CW- complexes. Note that in the world of CW-complexes being finite is equivalent to being compact, and hence a G-CW complex is finite if and only if it is cocompact. Unfortunately there will be occasional moments when we will need to work with spaces with slightly more structure than a G-CW-complex.

Deﬁnition 2.1.15. Let V be a G-set. A G-simplicial complex consists of a family K of subsets of V , called simplices, such that

v K for each v V , •{ } ∈ ∈ if s K and s0 s, then s0 K, and • ∈ ⊆ ∈ if s K and γ G, then γ s K. • ∈ ∈ · ∈ As a extension of simplicial complexes every G-simplicial complex K possesses a geometric realization K ; thanks to the conditions placed on the simplices of K this realization comes with | | a natural G-action. A G-simplicial complex is proper (i.e. has a proper geometric realization) if and only if the G-set V is proper, and is cocompact if and only if the G-set V is coﬁnite.

9 Note that a G-simplicial complex is generally not a G-CW complex — the problem is that an element which preserves a simplex may nonetheless act non-trivially on its vertices. However this is easily remedied by replacing the G-simplicial complex with its barycentric subdivision, which does carry a natural G-CW structure. Though we shall have no use for it, the following proposition shows that the reverse is also true, at least up to G-homotopy; the proof mirrors that of the non-equivariant case (see Theorem 2C.5 of [Hat02]).

Proposition 2.1.16. Every G-CW-complex X is G-homotopy equivalent to a G-simplicial complex, which can be chosen to be of the same dimension, proper if X is proper, and cocompact if X is cocompact.

As noted earlier we will often be working with the ﬁxed-point subspaces of proper G-spaces; when those spaces are G-CW or G-simplicial complexes we have the following elaboration on Proposition 2.1.11.

Proposition 2.1.17. Let X be a proper G-CW complex. Then for any subgroup H G the ≤ ﬁxed-point subspace XH is a proper N(H)-CW complex. Moreover, if X is a cocompact G- CW complex then XH is a cocompact N(H)-CW complex. Similar statements hold with N(H) replaced by Z(H), and for G-simplicial complexes in place of G-CW complexes.

Finally, despite our fondest wishes we will often be forced to work with arbitrary proper cocompact G-spaces. In these cases the following result will often prove to be quite useful.

Lemma 2.1.18. Every proper cocompact G-space has a G-map to a proper cocompact G-simplicial complex, and hence to a ﬁnite proper G-CW complex.

Remark 2.1.19. In fact, analogous to the non-equivariant case every proper cocompact G-space can be realized as the inverse limit of a system of ﬁnite proper G-CW complexes. However, for our purposes a single map to a G-CW complex will suﬃce.

Proof. Let f C (X) be the function guaranteed by Lemma 2.1.10, and let U = f −1(t > 0). ∈ c The properties of f guarantee that the collection = γ U is a locally finite cover of X. Let U { · }γ∈G ( ) denote the nerve of ; it is easily verified that ( ) is a proper cocompact G-simplicial N U U N U complex. Finally, the assignment x (f(γ−1x)) 7→ γ∈G defines a G-map from X to ( ) . |N U |

2.2 Equivariant Vector Bundles

Deﬁnition 2.2.1. A complex (resp. real) G-vector bundle on a G-space X consists of a complex (resp. real) vector bundle E X along with a continuous action of G on E given by bundle → maps which cover the action on X.

The set of all isomorphism classes of complex (resp. real) G-vector bundles over X will be C R denoted by VectG(X) (resp. VectG(X)).

10 Lemma 2.2.2. Let Z be an H-space for some subgroup H G. Then there are natural isomor- ≤ phisms C ∼ C indG : Vect (Z) = Vect (G Z) H H −→ G ×H R ∼ R indG : Vect (Z) = Vect (G Z) H H −→ G ×H given by sending an H-vector bundle E Z to the induced vector bundle indG (E)= G E. → H ×H Unfortunately, without further assumptions this is more or less the limit of what one can say about the G-vector bundles on a G-space X; the same pathologies which can conspire to make X/G so poorly behaved also limit our ability to use local arguments to understand G-vector bundles. However, when the space X is proper and cocompact most non-equivariant results can be made to carry over in one form or another.

Lemma 2.2.3. Let (X, A) be a proper cocompact G-pair, and let E be a G-vector bundle on X. Then any G-invariant section s : A E can be extended to a G-invariant section on X. → |A Proof. Since X is proper and cocompact, by Lemma 2.1.10 there exists a positive f C (X) such ∈ c that γ f = 1. Then s = fs is a compactly supported section of E , and we can apply γ∈G · 0 |A the correspondingP non-equivariant extension theorem to obtain a compactly supported section s on X. Finally, setting s = γ s yields the desired invariant extension, since 0 γ∈G · 0 P b b s A = bγ s0 A = γ fs = s. | · | · γX∈G γX∈G b b Given the previous lemma, arguments identical to the non-equivariant case lead to the following results (see [Ati67]).

Lemma 2.2.4. Let (X, A) be a proper cocompact G-pair, and let E and F be G-vector bundles over X. Then any G-equivariant bundle map φ : E F extends to a G-equivariant bundle |A → |A map φ : E F . Moreover, if φ is an isomorphism (resp. monomorphism) then there exists a G- → invariant open set U containing A such that φ U is also an isomorphism (resp. monomorphism). b | Lemma 2.2.5. Let X and Y be proper cocompactb G-spaces, ft : X [0, 1] Y a G-homotopy × → ∗ ∗ and E a G-vector bundle on Y . Then f0 E ∼= f1 E. Lemma 2.2.6. If f : X Y is a G-homotopy equivalence between proper cocompact G-spaces → then the induced maps C C f ∗ : Vect (Y ) Vect (X) G → G R R f ∗ : Vect (Y ) Vect (X) G → G are bijective.

It will often be essential to have some form of inner product on our G-vector bundles. To this end we deﬁne a Hermitian (resp. Euclidean) G-vector bundle to be a G-vector bundle with a G-invariant Hermitian (resp. Euclidean) structure. The following lemma guarantees that such beasts do in fact exist.

11 Lemma 2.2.7. Let X be a proper G-space, and E be a complex (resp. real) G-vector bundle on X. Then E admits a G-invariant Hermitian (resp. Euclidean) structure.

Proof. We shall only worry about the complex case; the argument in the real case is identical. Through the combination of paracompactness and properness we can find a partition of unity φ for X consisting of G-invariant functions with cocompact support; it therefore suffices to { i} consider only the case when X is cocompact. Since X is paracompact there exist Hermitian inner products on E; let h : E E C be one ⊗ → of them. As X is proper and cocompact, Lemma 2.1.10 tells us that there is a positive f C (X) ∈ c such that γ f =1 > 0. Setting γ∈G · P h(u, v)= f(γ x) h(γ u,γ v) · · · γX∈G b for u, v E then yields a G-invariant Hermitian inner product for E. ∈ x Unfortunately there are some results which have no direct equivariant generalization, the most notable being the properties of trivial bundles. Recall that a vector bundle on X is trivial if and only if it is isomorphic to a pullback over the map X pt; unfortunately, while this → definition has an obvious generalization to G-vector bundles, it is largely useless when G is not compact. For one thing, there need not actually be any interesting G-vector bundles over a point. Indeed, a G-bundle over a point is simply a finite-dimensional representations of G, and there are several well-known examples of discrete groups which have no non-trivial finite-dimensional representations1. Ultimately this is merely a symptom of a more severe problem — for countable groups there is no finite-dimensional analogue of the regular representation. After all, the value of trivial bundles lies not in their mere existence, but in their containment properties — in the fact that every vector bundle can be realized as a summand of a trivial bundle. Without some form of finite-dimensional regular representation there is no hope of extending this result to the equivariant case. Thankfully there is a way around these difficulties. Morally speaking, when G is infinite working with pullbacks from a point is a rather questionable affair — after all, the action of G on a point is far from being proper. Thus a different notion of trivial bundle is required when dealing with infinite discrete groups.

Deﬁnition 2.2.8. Let X and Z be proper cocompact G-spaces, and a : X Z a G-map. A X → complex (resp. real, resp. G-Spinc) G-vector bundle on X is Z-trivial if it is isomorphic to the c pullback over aX of a complex (resp. real, resp. G-Spin ) G-vector bundle on Z.

When G is finite, we have a universal choice for Z in the form of a single point, but when G is infinite there is usually no such option. One might hope to use the universal proper G-space G, but in general it need not admit a cocompact model. Nonetheless, the following results of E Lück and Oliver show that this notion of triviality has the necessary containment property.

1If a finitely-generated group has a non-trivial finite-dimensional representation then a result of Mal’cev [Mal65] implies that it contains a finite index normal subgroup. But there are numerous examples in the literature of infinite finitely-generated simple groups.

12 Proposition 2.2.9 ([LO01]). Let Z be a proper cocompact G-space. Then there exists a complex G-vector bundle E Z such that for each z Z the fiber E is a multiple of the regular → ∈ z representation of Gz. Corollary 2.2.10 ([LO01]). Let a : X Z be a G-map between proper cocompact G-spaces X → and Z. Then for any complex G-vector bundle E over X there exists a complex G-vector bundle F such that E F is Z-trivial. ⊕ Remark 2.2.11. In fact Lück and Oliver only consider the case when X and Z are finite G-CW complexes; however it follows immediately from Lemma 2.1.18 that the proposition also holds whenever Z is proper and cocompact. The results from earlier in this section then show that their proof of the corollary carries over without change.

2.3 Equivariant K-Theory

C Let X be a proper cocompact G-space. The set VectG(X) possesses a natural addition operation in the form of the Whitney sum, and we shall let KG(X) denote the corresponding Grothendieck group. The results of the previous section show that KG(X)is a G-homotopy invariant functor; moreover for any subgroup H G and H-space Y there is a natural isomorphism ≤ ∼ indG : K (Y ) = K (G Y ). H H −→ G ×H

However, far more is true — L¨uck and Oliver have shown that KG(X) extends to a Z2-graded cohomology theory on ﬁnite proper G-CW pairs. In fact, the theory even extends to proper cocompact G-pairs; unfortunately the arguments in [LO01] rely heavily on CW machinery, and thus must be redone for this more general case. On a positive note, given the results from the previous sections their argument for the following lemma carries over unchanged, and ultimately this is all we shall require. Lemma 2.3.1 ([LO01]). Let X be a proper cocompact G-space with closed G-invariant subspaces A and B such that X = A B. Then the sequence ∪ K (X) K (A) K (B) K (A B) G → G ⊕ G → G ∩ is exact in the middle.

Remark 2.3.2. In other words, given elements of KG(A) and KG(B) which agree on the intersection A B we can ‘glue’ them to obtain a (not necessarily unique!) element of K (X). ∩ G In Chapter 4 we shall often be working with spaces of the form M Y , where M is a proper × cocompact G-space and Y is a compact — but generally not proper — G-space. Note that the fact that M is proper implies the same of the product M Y , while the fact that Y is compact × means that M Y is also cocompact. It therefore makes sense to consider the group K (M Y ); × G × the preceding lemma implies that the sequence

K (M Y ) K (A Y ) K (B Y ) K ((A B) Y ) G × → G × ⊕ G × → G ∩ × will be exact in the middle for any pair of closed G-subsets A and B which cover M.

13 2.4 Equivariant Spinc Structures

Deﬁnition 2.4.1. Let X be a proper G-space and G a topological group. A principal (G, G)- bundle over X consists of a principal G-bundle π : P X together with a commuting left action → of G on P such that π is a G-map.

Perhaps the simplest non-trivial example of a principle (G, G)-bundle is the orthonormal frame bundle of a Euclidean G-vector bundle (E, q). Speciﬁcally, let F (E, q) X be the principal O(n)- → bundle whose ﬁbre at a point p X consists of all orthonormal frames in E . The G-invariance ∈ p of the inner product q implies that the G-action on E lifts to a corresponding action on F (E, q); together this provides F (E, q) with the structure of a principal (G, O(n))-bundle. If E happens to possess a G-invariant orientation we can restrict ourselves to only the oriented orthonormal frames, and thus obtain a principal (G, SO(n))-bundle F +(E, q) F (E, q). In fact, ⊂ the existence of a G-invariant orientation is equivalent to the existence of such a sub-bundle; this suggests that one way to impose additional structure on a G-vector bundle is to require the existence of a suitable principal (G, G)-bundle which maps to F (E, q). However, before delving further in this there is one technical point that should be addressed. The frame bundle F (E, q) quite clearly depends on the choice of the G-invariant inner product, and one might reasonably worry that a statement about F (E, q) might not hold for F (E, q0). The following lemma should put any such concerns to rest.

Lemma 2.4.2. Let E be a real G-vector bundle with G-invariant inner products q and q0. Then there is a canonical isometric isomorphism

∼ φ : (E, q0) = (E, q). −→ As a result there is a canonical (G, O(n))-bundle isomorphism

∼ F (φ) : F (E, q0) = F (E, q). −→ Proof. Note that the G-invariant inner product q induces a G-equivariant isomorphismq ˆ : E → E∗ such that q(u, v)= qˆ(u), v . Let ψ = q−1 q0; then h i ◦ q(ψ(u),u)= (q ψ)(u),u = q0(u),u = q0(u,u) > 0. h ◦ b ib h i Thus ψ is a ﬁbrewise positive operatorb and φ = √ψbyields the desired map.

For n 3 let Spin(n) denote the unique non-trivial double-cover of SO(n). The Spin groups ≥ play a central part in the development of real index theory; however our interests will lie with the corresponding complex theory, and for this we shall need their complex analogues.

Deﬁnition 2.4.3. For n 3 deﬁne the nth Spinc group to be ≥ c 1 Spin (n) = Spin(n) Z S , × 2 where Z , Spin(n) is the kernel of the surjection onto SO(n). 2 →

14 Remark 2.4.4. It will often be convenient to view Spinc(n) as an S1-bundle over SO(n). It is 2 Z Z a rather non-trivial exercise in the cohomology of Lie groups to show that H (SO(n); ) ∼= 2 when n 3; hence up to isomorphism Spinc(n) is the unique non-trivial line bundle over SO(n). ≥ Deﬁnition 2.4.5. Let (E, q) be a Euclidean G-vector bundle of (locally constant) rank n over a proper G-space X. A G-Spinc structure for (E, q) consists of a principal (G, Spinc(n))-bundle P on X together with a G-equivariant bundle map σ : P F (E, q). Two G-Spinc structures → (P, σ) and (P 0, σ0) are isomorphic if there exists a (G, Spinc(n))-bundle map τ : P P 0 such → that σ = τ σ0. ◦ Note that thanks to Lemma 2.4.2, a G-Spinc structure for (E, q) induces a G-Spinc structure on (E, q0) — we simply let P 0 be the pullback

P 0 / P

φ F (E, q0) / F (E, q).

Deﬁnition 2.4.6. Let E be a real G-vector bundle on a proper G-space X. A G-Spinc structure for E consists of a choice of G-Spinc structure for some (and hence every) G-invariant inner product on E.

A G-Spinc structure (P, σ) endows the G-vector bundle E with a number of ancillary structures. For instance, every G-Spinc vector bundle is naturally oriented — the quotient P/S1 is a principal (G, SO(n))-bundle, and is identiﬁed under σ with a sub-bundle F +(E, q) F (E, q). ⊂ Similarly, since Spin(n) C Spinc(n) we can form the quotient L = P/Spin(n); as Spinc(n)/Spin(n) is isomorphic to S1 this quotient is naturally a principal (G, S1)-bundle over X.

Deﬁnition 2.4.7. Let X be a proper G-space. The equivariant Picard group PicG(X) is the group of principal (G, S1)-bundles on X.

Remark 2.4.8. It is a well-known result of Hattori and Yoshida [HY76] that for a compact Lie group G and ﬁnite G-CW-complex X one has

2 PicG(X) = H (EG G X; Z). ∼ × A careful examination of the argument in [LMS83] shows that the same result also holds for a discrete group G acting on a ﬁnite proper G-CW-complex.

The aforementioned (G, S1)-bundle L thus associates to any principle (G, Spinc(n))-bundle G c a Chern class c1 (P, σ) = L PicG(X); when the choice of G-Spin structure on E is clear we G ∈ shall simply write c1 (E).

Proposition 2.4.9. Every Hermitian G-vector bundle (E,h) possesses a canonical G-Spinc structure whose Chern class is the determinant line bundle of E.

15 Proof. Let U(˜ n) denote the non-trivial double-cover of U(n). The inclusion U(n) , SO(2n) and → determinant U(n) S1 lift to give maps U(˜ n) , Spin(2n) and U(˜ n) S˜1 = S1 respectively. → → → Together these define a map U(˜ n) , Spin(2n) S1 which descends to give a homomorphism → × U(n) , Spinc(2n); by construction the compositions → U(n) , Spinc(2n) SO(2n) → U(n) , Spinc(2n) S1 → are the inclusion U(n) , SO(2n) and determinant U(n) S1. → → Let F u(E,h) be the unitary frame bundle of E — the principal (G, U(n))-bundle whose fibre at a point p B consists of all unitary frames of E . Setting q = Re(h) defines a ∈ p G-invariant inner product on the underlying real G-vector bundle ER, and one can identify + u u c F (ER, q) with F (E,h) U(n) SO(2n). Setting P = F (E,h) U(n) Spin (2n) then yields a c × × principal (G, Spin (2n))-bundle on X which maps to F (ER, q) via

u c u + P = F (E,h) Spin (2n) F (E,h) SO(2n)= F (ER, q) F (ER, q). ×U(n) → ×U(n) ⊂ The statement about the Chern class now follows from the fact that the composition U(n) , → Spinc(2n) S1 is the determinant. → Proposition 2.4.10 (Two-out-of-three). Let E , E and E = E E be G-vector bundles 1 2 1 ⊕ 2 on a proper G-space X. Then any choice of G-Spinc structures for two of these bundles induces a canonical G-Spinc structure on the third such that

cG(E)= cG(E ) cG(E ). 1 1 1 · 1 2

Proof. Choose G-invariant inner products q1 and q2 for E1 and E2; together these provide a G-invariant inner product q on E. With these choices there is a natural map

F (E , q ) F (E , q ) F (E, q) 1 1 ×X 2 2 → and the frame bundle F (E, q) can be obtained as

[F (E , q ) F (E , q )] SO(n). (??) 1 1 ×X 2 2 ×SO(n1)×SO(n2) c Now suppose that E1 and E2 have G-Spin structures (P1, σ1) and (P2, σ2) respectively, and set c P = [P P ] c c Spin (n). 1 ×X 2 ×Spin (n1)×Spin (n2) The various projections then yield an equivariant map from P to (??), and hence to F (E, q); this deﬁnes the desired G-Spinc structure on E. c On the other hand, suppose E1 and E have G-Spin structures (P1, σ1) and (P, σ). Let Q be the pullback Q / P

P F (E , q ) / F (E, q). 1 ×X 2 2 16 It is easily checked that Q is a principal (G, Spinc(n ) Spinc(n ))-bundle over X. Now set 1 × 2 P = Q/Spinc(n ). Then P is a principal (G, Spinc(n ))-bundle over X, and the map Q 2 1 2 2 → P F (E , q ) descends to the quotient to give 1 ×X 2 2 σ : P F (E , q ). 2 2 → 2 2

Checking the Chern class formula is now simply a matter of chasing through the two constructions.

This proposition allows one to make the following deﬁnition.

Deﬁnition 2.4.11. Let E be a G-Spinc vector bundle on a proper G-space X. The G-Spinc dual of E is the G-Spinc vector bundle E\ such that E E\ = E C as G-Spinc vector bundles, ⊕ ∼ ⊗ where E C is given the G-Spinc structure associated to its complex structure. ⊗

Remarks 2.4.12. (a) It follows from the two-out-of-three result that E\ exists and has c (E\)= c (E). 1 − 1 (b) Let E be a complex G-vector bundle and let E∗ denote its complex dual. Then since E R C = E E∗ as complex G-vector bundles it follows that E∗ is a model for the ⊗ ∼ ⊕ G-Spinc dual of E.

As noted earlier, a G-Spinc structure for E naturally provides it with an orientation. However, it takes only a slight change to the G-Spinc structure to obtain the opposite orientation. To understand this, we ﬁrst need the following lemma, which is an immediate consequence of the construction of the Spin-groups.

Lemma 2.4.13. The adjoint action Ad : O(n) Aut(SO(n)) lifts to a homomorphism Ad : → O(n) Aut(Spin(n)). → c Note that Ad extends in an obvious way to give an action of O(n) on Spinc(n) which preserves the subgroups Spin(n) and S1. c Deﬁnitions 2.4.14. (a) Let T = diag(1,..., 1, 1) O(n) and let α represent the automorphism Ad(T ) n − ∈ n n ∈ Aut(Spinc(n)). Then from any principal (G, Spinc(n))-bundle P one can form a new prin- c cipal (G, Spinc(n)) bundle P op which is isomorphic to P as a G-space but has a right Spinc(n)-action given by pop m = (p α (m))op for any p P , m Spinc(n). · · n ∈ ∈ (b) Let (P, σ) be a G-Spinc structure for E and let α : F (E, q) F (E, q) be the map which E → sends the frame (v , v ,...,v ) to (v , v ,..., v ). Then the opposite G-Spinc structure 1 2 n 1 2 − n is that given by the pair (P op, α σ). E ◦ Note 2.4.15. The opposite G-Spinc structure has the same Chern class as the original G-Spinc structure, but induces the opposite orientation on E.

17 2.5 Equivariant Bott Elements

Let X be a proper cocompact G-space, and let (E, q) be a Euclidean G-vector bundle over X; though X is cocompact the same is obviously not true of E. In classical K-theory this is remedied by working with the one-point compactification E+, but since the induced G-action on E+ has a fixed point this is usually not a proper G-space. We will therefore find it more useful to work with the unit sphere bundle

XE = S(E 1) ⊕ = (v,t) E 1 : q(v, v)+ t2 =1 , { ∈ ⊕ } which has the twin virtues of being both proper and cocompact.

Note 2.5.1. We can also obtain XE by gluing two copies of the unit disc of E; namely, if we set

D = (v,t) S(E 1) : t 0 = D(E) + { ∈ ⊕ ≥ } ∼ and D = (v,t) S(E 1) : t 0 = D(E) − { ∈ ⊕ ≤ } ∼ then XE is simply the identiﬁcation D D . + ∪S(E) − Now suppose that the bundle E is even-dimensional and possesses a G-Spinc structure (P, σ). Recall that the theory of Cliﬀord algebras provides Spinc(n) with a complex ‘Spin’ representation 1 ± ∆, which when n is even splits into two distinct complex ‘ 2 -Spin’ representations ∆ . Combined with the G-Spinc structure P these yield G-equivariant ‘spinor bundles’

± ± S = P c ∆ ×Spin (2n) 1 on X. It follows from the particulars of the 2 -Spin representations (see [LM89]) that the endomorphism bundle End(S) is naturally (graded) isomorphic to the Cliﬀord bundle Cl(E), and therefore there is a G-equivariant ‘Cliﬀord multiplication’

ρ : E , End(S, S). → In particular this multiplication restricts to give a map

ρ : E , Hom(S+, S−) → which associates an isomorphism to each non-zero v E. ∈ E We’ll now use this data to form an element of the group KG(X ). To begin with, let p = p : XE X be the obvious projection. Then by pulling the spinor bundles S± back E → over p we obtain bundles H± = p∗S± on XE. At the same time, the pullback p∗E admits an obvious G-invariant section, namely s(v,t)= v; when combined with the Cliﬀord multiplication this gives a G-invariant section ρ s of Hom(H+, H−) which consists of isomorphisms away from ◦ the ‘poles’ (~0, 1) XE. In particular ρ s is an isomorphism when restricted to the equator ± ∈ ◦ S(E) S(E 1), and thus we can use it to glue the restrictions H+ and H− along their ⊂ ⊕ |D+ |D− common boundary; we shall denote the resulting G-vector bundle by H.

18 Deﬁnition 2.5.2. The Bott element associated to the G-Spinc vector bundle E is the class β = [H∗] [H−∗] K (XE). E − ∈ G

Remarks 2.5.3. (a) This definition differs from that used by Baum and Douglas in [BD82]; they chose to work with vector bundles rather than K-theory classes, and thus only used the bundle H∗. On the other hand, the element defined here is far better behaved. Observe that there are natural sections i : X XE given by sending a point x X to the pole (0, 1) E 1 ± → ∈ ± ∈ x ⊕ that lies above it. It follows immediately from the definition that i∗ β =0 K (X), and − E ∈ G this vanishing will be essential in later arguments. (b) The suspicious reader might question the sudden appearance of duals in the definition of

βE. This is a classic (and annoyingly clever) trick that makes a key index calculation (Corollary 3.5.6) turn out nicely. (c) Finally, although it is not clear from our notation, the Bott element depends on the choice of G-Spinc-structure on the vector bundle E.

Note that although we have pursued this discussion under the assumption that G is a countable discrete group acting on a cocompact G-space, it applies just as well to the case of a compact group G acting on a compact G-space. One example of this is the G-action on a single point. In this context a Euclidean G-vector bundle is simply a representation G SO(n), and an → equivariant Spinc-structure is just a lifting ρ : G Spinc(n). Meanwhile the associated sphere → bundle is nothing more than the ordinary n-sphere Sn Rn+1. ⊂ Paralleling our work above, suppose we focus on the even-dimensional case. Then associated c to the lift G Spin (2n) we have an element β KG(S2n), where KG denotes the equiv- → ρ ∈ ariant K-theory for compact groups deﬁned by Segal in [Seg68]. The standard representation Spinc(2n) SO(2n) gives a universal example of this construction. The identity homomorphism c → c 2n Spin (2n) Spin (2n) is a lifting, and thus there is a natural class β K c (S ); given → 2n ∈ Spin (2n) a homomorphism ρ : G Spinc(2n) we then recover β as the restriction res β . → ρ ρ 2n With this slight digression out of the way we are now in a position to explain a principal bundle construction which will underlie many of the arguments in the next two chapters. As usual, let X be a proper cocompact G-space. Continuing our notation from the last section we shall let G be a topological group and P a principal (G, G)-bundle P over a X. Given any compact G-space Z we can form the space P G Z; note that conditions on X and Z conspire × to make P G Z a proper cocompact G-space. × We have already seen one example of this construction earlier in the section. Indeed, suppose Z = Sn and let G = Spinc(n) act via

Spinc(n) SO(n) , SO(n + 1). → If E X is a real G-vector bundle with a given G-Spinc-structure (P, σ) then we can form → n the space P c S ; a moment’s thought about the structures involved reveals that this is ×Spin (n) simply the space M E = S(E 1). ⊕ 19 As the following lemma shows, this principal bundle construction gives a natural generalization of induction (compare with Lemma 2.2.2).

Lemma 2.5.4. There are natural semigroup homomorphisms

C C ind : VectG(Z) Vect (P G Z) P → G × R R ind : VectG(Z) Vect (P G Z) P → G × given by sending a G-vector bundle E Z to the induced vector bundle ind (E) = P G E. → P × When G is compact this induces a natural ring homomorphism

ind : KG(Z) K (P G Z). P → G × Applying this to the aforementioned case of the Spinc(n)-action on Z = Sn we obtain the following corollary.

Corollary 2.5.5. Let E X be an even-dimensional real G-vector bundle with G-Spinc struc- → E E n ture (P, σ) and let M = S(E 1). Then M = P c S and there is a natural ring ⊕ ∼ ×Spin (n) homomorphism 2n E ind : K c (S ) K (M ). P Spin (2n) → G which in particular sends the element β2n to βE.

As we shall see in the next chapter, this observation will allow us to deduce many results c 2n about G-Spin structures and their Bott elements from explicit computations in KSpinc(2n)(S ).

20 Chapter 3

Kasparov’s Equivariant KK -Theory

3.1 C∗-Algebras

For a large part of this chapter we shall make a slight break from our usual conventions and work with an arbitrary locally compact, second countable topological group G. In these ﬁrst few sections we shall give a very brief survey of C∗-algebras and equivariant KK -theory; the reader who is interested in details is advised to consult [Dav96] and [Bla98]. Recall that a C∗-algebra is a Banach -algebra A satisfying the ‘C∗-identity’ ∗ a∗a = a 2 a A. k k k k ∀ ∈ From the C∗-algebra A we can form the group Aut(A) of continuous -automorphisms of A; when ∗ given the strong topology Aut(A) is a topological group. A G-C∗-algebra is simply a C∗-algebra A together with a given continuous homomorphism from G to Aut(A).

Examples 3.1.1. (a) Given a Hilbert space the operator norm on linear operators satisﬁes the C∗-identity. H Thus the Banach algebra B( ) of bounded operators is a C∗-algebra, as is the closed subal- H gebra K = K( ) of compact operators. Finally, a (norm continuous) unitary representation H of G on turns both of these algebras into G-C∗-algebras. H (b) Let X be a locally compact topological space and C0(X) the algebra of (complex-valued) continuous functions on X which vanish at inﬁnity. Then the norm

f = sup f(p) k k p∈X | |

∗ ∗ satisﬁes the C -identity and makes C0(X) a commutative C -algebra. A continuous action ∗ of G on X induces a corresponding action on C0(X) and with this action C0(X)isa G-C - algebra. Conversely, a classical theorem of Gelfand says that every commutative C∗-algebra

A is of the form C0(X), and it follows that every G-action on A arises from a G-action on X. (c) Elaborating on the previous example, suppose that in addition to a locally compact space ∗ X we also have C -algebra A. Then one can consider the algebra A(X) = C0(X, A) of continuous A-valued functions on X which vanish at inﬁnity. Analogous to the previous example we see that the norm f = sup f(p) k k p∈X k k

21 satisﬁes the C∗-identity; under this norm the algebra A(X) is a C∗-algebra. Finally, given a G-action on both X and A there is a natural choice of G-action on A(X) making it into a G-C∗-algebra.

While it would be possible to develop Kasparov’s equivariant KK -theory without the language of graded C∗-algebras, we shall find that it allows a far more natural approach to some problems. A graded C∗-algebra is a C∗-algebra A with a decomposition A = A(0) A(1) into closed - ⊕ ∗ invariant subspaces such that A(m) A(n) A(m+n). An element of A(m) is said to be homogeneous · ⊆ of degree m; it is standard to denote the degree of a homogeneous element a by ∂a. A graded G-C∗-algebra is a graded C∗-algebra with an action of G by graded -automorphisms. ∗ Example 3.1.2. Given a finite-dimensional Hermitian vector space V one can combine its various tensor powers into the tensor algebra T V = V ⊗n. Note that T V is naturally Z-graded, and ⊕n inherits a Hermitian inner product from the inner product on V . Let V T V be the ideal generated by all elements of the form I ≤ u v + v u 2 u, v , u, v V. ⊗ ⊗ − h i ∀ ∈ The complex Clifford algebra Cl is the quotient TV/ V . The Z-grading on T V descends to give V I a Z2-grading on ClV , and the Hermitian inner product on T V yields a similar inner product on the quotient. Letting ClV act on itself by left multiplication allows us to realize it as an algebra of bounded operators on its underlying Hilbert space, and this endows ClV with the structure of a C∗-algebra. If in addition there is a continuous unitary action of G on V then there will be a corresponding ∗ action of G on the Clifford algebra, making ClV into a graded G-C -algebra.

We shall let Cln denote the Cliﬀord algebra of the standard Hermitian inner product on Cn V ∼= . Remark 3.1.3. In operator theory one is often concerned with properties of various commutators [a,b]= ab ba. In the graded context this must be modiﬁed slightly; given homogeneous elements − a and b of degrees ∂a and ∂b respectively, one sets [a,b]= ab ( 1)∂a∂bba. − −

3.2 Hilbert modules

Deﬁnition 3.2.1. Let B be a graded C∗-algebra. A graded pre-Hilbert B-module is a graded right B-module E = E(0) E(1) together with a B-valued inner product , : E(m) E(n) B(m+n) ⊕ h· ·i × → which satisﬁes

, is sesquilinear, • h· ·i x,ya = x, y a for any x, y E and a A, • h i h i ∈ ∈ x, y = y, x ∗ for any x, y E, and • h i h i ∈ x, x 0, with equality iﬀ x = 0. • h i≥ 1 For x E we set x = x, x, 2 ; if E is complete in this norm it is called a graded Hilbert ∈ k k kh ik B-module.

22 Remark 3.2.2. A Hilbert C-module is nothing more than a graded Hilbert space, and in general Hilbert B-modules can be viewed as a continuous family of graded Hilbert spaces over the ‘space’ represented by B.

Examples 3.2.3. (a) B itself is a Hilbert B-module, with inner product a,b = a∗b, as is any closed right ideal h i of B. (b) Given a collection E of Hilbert B-modules, the inner product x , y = x ,y { i} h⊕ i ⊕ ii h i ii makes the direct sum E a pre-Hilbert B-module. It follows that E = Bn is also a ⊕ i P Hilbert B-module, as is any closed B-invariant subspace. In particular, for any projection p M (B) the projective B-module pBn Bn is a Hilbert B-module. ∈ n ⊆ (c) Let = B∞ denote the completion of the direct sum of countably many copies of B. HB Then it is easily checked that consists of all sequences (b ) such that b∗ b converges HB n n n in B. P Deﬁnition 3.2.4. Let E be a Hilbert B-modules. B(E) is the set of all module maps T : E E → for which there exists an ‘adjoint’ T ∗ : E E satisfying → Tx,y = x, T ∗y , x, y E. h i h i ∀ ∈ As with Hilbert spaces the existence of an adjoint is enough to ensure that T is a bounded operator on the Banach spaces E. The converse, however, is not true — a bounded B-linear map between Hilbert B-modules need not have an adjoint.

Proposition 3.2.5. B(E) is a C∗-algebra with respect to the operator norm. Moreover, the decomposition B(E)= B(0)(E) B(1)(E) given by ⊕ B(m) = T B(E) : T (E(n)) E(m+n) { ∈ ⊆ provides B(E) with the structure of a graded C∗-algebra.

It will be important to have an analogue of compact operators on a Hilbert module. Note that there are certain obvious ‘rank one’ operators on E — for x, y E we let θ be the operator ∈ x,y given by θ (z)= x y,z . x,y h i

Deﬁnition 3.2.6. K(E) is the closed linear span of the θx,y in B(E).

∗ K ∗ B Since θx,y = θy,x it follows that (E)isa C -subalgebra of (E). Moreover, as Tθx,y = θTx,y for any T B(E) we see that K(E) is an ideal of B(E); the grading on B(E) makes K(E) a ∈ graded ideal.

n n Example 3.2.7. B(B ) is the multiplier algebra M(Mn(B)), while K(B ) is simply the algebra M (B) itself. More generally, for any Hilbert space one has K(B ) = B K( ). n H ⊗ H ∼ ⊗ H

Note 3.2.8. Given graded Hilbert B-modules E1 and E2, deﬁnitions similar to those above yield Banach spaces (E , E ) and (E , E ). B 1 2 K 1 2 23 ∗ Let B1 and B2 be two graded C -algebras with Hilbert modules E1 and E2 respectively. Suppose in addition that we have a -homomorphism φ : B B(E ); then we can construct a ∗ 1 → 2 tensor product E E as follows. First, form the (graded) algebraic tensor product E ˆ E . 1 ⊗φ 2 1 φ 2 On this tensor product there is an obvious B2-valued inner product given by

u u , v v = u , φ( u , v )v h 1 2 1 2i h 2 h 1 1i 2i for any choice of u , v E . Completing the algebraic tensor product in the associated norm j j ∈ j (and possibly quotienting out any elements of norm zero) yields the Hilbert B -module E E . 2 1 ⊗φ 2 Deﬁnition 3.2.9. Let f : B B be a -homomorphism and let E be a Hilbert B -module. 1 → 2 ∗ 1 Then the pushforward f E is the Hilbert B -module E B . ∗ 2 ⊗f 2

n n Example 3.2.10. The pushforward of the Hilbert B1-module B1 is B2 . More generally, given any projection p M (B ) the pushforward of pBn is the Hilbert B -module f(p)Bn. ∈ n 1 1 2 2 Of course, the previous discussion neglected the possible presence of a group action on the C∗- algebra B; if B is a G-C∗-algebra we shall ask for a bit more structure on our Hilbert B-modules in the form of a G-action.

Deﬁnition 3.2.11. Let B be a graded G-C∗-algebra, and E a Hilbert B-module. A G-action on E consists of a representation of G on E by bounded even linear transformations such that

g (xb) = (g x)(g b) · · · and g x, g y = g x, y, h · · i · h i for all g G, b B, and x, y E. A Hilbert B-module with such a G-action will be called a ∈ ∈ ∈ Hilbert (B, G)-module.

Example 3.2.12. When B is a G-C∗-algebra the Hilbert modules in Example 3.2.3 all possess natural G-actions which make them Hilbert (B, G)-modules. This list of examples can be further expanded by tensoring with various unitary representations of G, the most important such example being the Hilbert (B, G)-module G = L2(G). HB HB ⊗ It must be emphasized that unless the action of G on B is trivial the G-action on E cannot be given by B-linear maps. Nonetheless, conjugation by an element of G preserves both B(E) and K(E); it is an unfortunate technicality that the action of G on these two algebras is generally only strongly continuous, and not norm continuous.

Deﬁnition 3.2.13. Let B be a G-C∗-algebra with Hilbert B-module E. Given g G and ∈ T B(E) we shall let g T = gTg−1. An operator T B(E) is said to be G-continuous if the ∈ · ∈ map G B(E) given by g g T is norm-continuous. → 7→ ·

24 3.3 Kasparov’s KK -Theory

From this point on we will assume that all C∗-algebras in sight are both separable and σ-unital. This will not cause any problems with our intended applications, and will spare us an inordinate number of technical digressions.

Deﬁnition 3.3.1. Let A and B be graded G-C∗-algebras. A Kasparov G-module for (A, B)is a triple (E, φ, F ) consisting of a (countably generated) graded Hilbert (B, G)-module E, a graded -homomorphism φ : A B(E) and a G-continuous operator F B(1)(E) such that ∗ → ∈ [F, φ(a)], (F 2 1)φ(a), (F F ∗)φ(a) and (g F F )φ(a) − − · − lie in K(E) for all a A and g G. The collection of all Kasparov G-modules will be denoted ∈ ∈ by G(A, B). E

Examples 3.3.2. (a) Let f : A B be a G-equivariant -homomorphism between two graded G-C∗-algebras. → ∗ Then associated to f we have the element (B,f, 0) G(A, B). More generally, suppose ∈E H is a unitary Hilbert space representation of G, and let f : A B K( )bea G-equivariant → ⊗ H -homomorphism. Then we have a corresponding element (B , f, 0) G(A, B). ∗ ⊗ H ∈E q (b) An extension 0 B D A 0 of G-C∗-algebras is G-semisplit if there is an → → −→ → equivariant completely positive contraction σ : A D splitting the surjection onto A. → Since B is an ideal of D the map σ induces an equivariant completely positive contraction σ : A M(B), and by a slight elaboration on the Generalized Stinespring Theorem → [Kas80a] this dilates to an equivariant -homomorphism ρ : A B(B E) for some ∗ → ⊕ Hilbert (G,B)-module E. It is then rather straightforward to check that the triple δq = 1 0 ((B E) ˆ Cl1,ρ 1, ˆ i1) gives an element of G(A, B ˆ Cl1). ⊕ ⊗ ⊗ 0 1 ! ⊗ E ⊗ − Note that Kasparov modules are naturally contravariant in their ﬁrst variable: given an equivariant -homomorphism f : A A and a Kasparov module (E, φ, F ) G(A ,B) we can ∗ 1 → 2 ∈E 2 compose φ with f to obtain the triple f ∗(E, φ, F ) = (E, φ f, F ) G(A ,B). On the other hand, ◦ ∈E 1 extending the pushforward of Hilbert modules to Kasparov modules makes them covariant in the second variable. Speciﬁcally, given an equivariant -homomorphism g : B B and a Kasparov ∗ 1 → 2 module (E, φ, F ) G(A, B ) we can form the triple g (E, φ, F ) = (E B , φ 1, F 1) ∈ E 1 ∗ ⊗g 2 ⊗ ⊗ ∈ G(A, B ). E 2 Unfortunately, despite this functoriality the collections G(A, B) are far two large to be E practical. Furthermore, although there is a simple addition operation on G(A, B) given by E the direct sum of Hilbert modules, it is easy to see that the resulting semigroup has no inverses. Thus, to obtain a useful theory we need to impose some form of equivalence relation on Kasparov modules.

Deﬁnitions 3.3.3.

25 (a) Two Kasparov G-modules (E , φ , F ) G(A, B) are unitarily equivalent if there is j j j j=1,2 ∈ E an even unitary in B(E1, E2) which intertwines the φj and Fj . The relation of unitary equivalence will be denoted by . ≈u (b) Two Kasparov G-modules (E , φ , F ) G(A, B) are homotopic if there is a Kasparov j j j j=1,2 ∈E G-module (E, φ, F ) G(A, B[0, 1]) such that ev (E, φ, F ) (E , φ , F ), where ev ∈ E j∗ ≈u j j j j denotes the -algebra map coming from evaluation at j [0, 1]. ∗ ∈

Deﬁnition 3.3.4. The equivariant KK -theory group KK G(A, B) is the set of homotopy equivalence classes in G(A, B). E With this notion of equivalence it is straightforward to show that KK G(A, B) is actually a group — a standard rotation trick shows that the additive inverse of the element (E, φ, F ) is just (Eop, φ , F ), where Eop denotes the Hilbert module E with the opposite grading and ◦ A A is the grading operator on A. Furthermore, note that the functoriality of G(A, B) carries over E to the quotient KK G(A, B), so that it is contravariant in the ﬁrst variable and covariant in the second.

Deﬁnition 3.3.5.

0 KK G(A, B)= KK G(A, B) 1 KK G(A, B)= KK G(A, B ˆ Cl ) ⊗ 1 G G Remark 3.3.6. If X and Y are two (locally compact) -spaces we shall simply write KK ∗ (X, Y ) ∗ for KK G(C0(X), C0(Y )).

1 For a reader familiar with algebraic topology it may seem strange to deﬁne the KK G groups using the tensor product B ˆ Cl . As the following result shows, we could just as well have used ⊗ 1 the algebraic suspension B(0, 1); however the simpler structure of the algebra Cl1 often makes it the better choice to work with.

Proposition 3.3.7 (Bott Periodicity). There are natural isomorphisms

1 0 0 KK G(A, B) ∼= KK G(A, B(0, 1)) ∼= KK G(A(0, 1),B). As we saw earlier, every equivariant -homomorphism f : A B provides an element of the 0 ∗ → KK -group KK G(A, B), and for this reason the KK -groups are often thought of as consisting of ‘generalized homomorphisms’ between C∗-algebras. Of course, in order for this to be a worthwhile analogy there must be some way of composing two KK -theory elements.

Theorem 3.3.8 (Kasparov Products). For any choice of graded G-C∗-algebras A, B and D there is an associative composition product

i j i+j : KK G(A, D) KK G(D,B) KK G (A, B). ◦ ⊗ → In particular one has

(E , φ , F ) (E , φ , 0)=(E otimesˆ E , φ ˆ 1, F 1). 1 1 1 ◦ 2 2 1 φ2 2 1⊗ 1 ⊗ 26 ∗ Similarly, for any graded G-C -algebras Aj , Bj (j =1, 2) there is an external product

i j i+j : KK G(A ,B ) KK (A ,B ) KK (A ˆ A ,B ˆ B ), × 1 1 ⊗ G 2 2 → G 1⊗ 2 1⊗ 2 which is compatible with the composition product to the extent that

(α β ) (α β ) = (α α ) (β β ). 1 × 1 ◦ 2 × 2 1 ◦ 2 × 1 ◦ 2 Proposition 3.3.9. For any equivariant -homomorphism f : A A the induced maps ∗ 1 → 2 ∗ ∗ ∗ f : KK G(A ,D) KK G(A ,D) 2 → 1 ∗ ∗ f : KK G(D, A ) KK G(D, A ) ∗ 1 → 2 0 correspond to left and right composition with the element [f] KK G(A , A ). ∈ 1 2 The existence of the composition product is particularly useful when it comes to describing the long exact sequences in KK -theory.

q Proposition 3.3.10. Let 0 J A A/J 0 be a G-semisplit extension. Then for any → → −→ → graded G-C∗-algebra D there are natural six-term exact sequences

0 0 0 KK G(D, J) / KK G(D, A) / KK G(D, A/J) O

δq δq 1 1 1 KK G(D, A/J) o KK G(D, A) o KK G(D, J) and 0 0 0 KK G(A/J, D) / KK G(A, D) / KK G(J, D) . O

δq δq 1 1 1 KK G(J, D) o KK G(A, D) o KK G(A/J, D)

1 Moreover the boundary maps are given by composition with the element δ KK G(A/J, J) q ∈ associated to the extension (see Example 2).

Remark 3.3.11. In this context naturality means that given a map of G-semisplit extensions

q 0 / J / A / A/J / 0

fJ fA fA/J q0 0 / J 0 / A0 / A0/J 0 / 0 there is a corresponding commutative diagram involving the six-term exact sequences of the two extensions. The only non-trivial aspect of this is at the boundary maps, where the commutativity of the diagram is equivalent to the fact that

1 0 δ [f ] = [f ] δ 0 KK G(A/J, J ). q ◦ J A/J ◦ q ∈

27 Finally, we have the following relation with the classical K-theory groups deﬁned by Atiyah, Hirzebruch and Segal.

Proposition 3.3.12. Let G be a compact topological group. Then for any G-C∗-algebra A there is a natural isomorphism G ∼= 0 K (A) KK G(C, A) 0 −→ given by sending a projection p A End(V ), V a ﬁnite dimensional representation on G, to the ∈ ⊗ Kasparov (A, G)-module (p(A V ), µ, 0), where µ is simply scalar multiplication. In particular ⊗ for any G-space X there is an natural isomorphism

=∼ G KG(X) KK (pt,X). −→ It should be noted that this isomorphism relies heavily on the compactness of the group G; when G is inﬁnite discrete the aforementioned map fails to exist, let alone be an isomorphism. Nonetheless, as we shall see in the next section, one can still map the equivariant K-theory (albeit non-isomorphically) to an appropriate Kasparov group.

3.4 Vector Bundles and KK G(X, X)

Having briefly surveyed the landscape of KK -theory we can now relate it to the equivariant K- theory groups discussed in the last chapter. Recall that G is assumed to be a countable discrete group, and let V be a proper (though not necessarily cocompact) G-space. C We begin by considering a G-vector bundle E Vect (V ). Then according to Lemma 2.2.7 ∈ G we can find a G-invariant Hermitian structure q on E. Define a C0(V )-valued inner product on the space Γ0(E) by

u, v (x)= q(u(x), v(x)) u, v Γ (E), x V. h i ∀ ∈ 0 ∈

A quick check of the various properties shows that with this inner product Γ0(E) is a Hilbert

C0(V )-module. In fact since q is G-invariant even more is true - Γ0(E) is actually a Hilbert

(G, C0(V ))-module.

Lemma 3.4.1. Let End(E) denote the complex endomorphism bundle of E. Then

K(Γ0(E))=Γ0(End(E)).

Proof. Note that Γ0(End(E)) sits as a closed subalgebra of the bounded operators B(Γ0(E)). However, since the rank-one operators θ : u, v Γ (E) all lie in Γ (End(E)) it follows that { u,v ∈ 0 } 0 K(Γ (E)) Γ (End(E)). Conversely a partition of unity argument shows that every compactly 0 ⊆ 0 supported element of Γ (End(E)) gives a ﬁnite rank operator on Γ (E), and hence Γ (End(E)) 0 0 0 ⊆ K(Γ0(E)).

Now consider the triple [[E, q]] = (Γ (E), φ, 0), where µ : C (V ) B(Γ (E)) is given by 0 0 → 0 pointwise multiplication. We would like to show that [[E, q]] gives an element of KK G(V, V ),

28 and for this it suﬃces to check that the various elements given in Deﬁnition 3.3.1 all lie in

K(Γ0(E)). But since each of these elements involves multiplication bya C0 function, this follows immediately from the previous lemma. Note that while the Kasparov G-module [[E, q]] depends heavily on the choice of the Hermitian structure q, by Lemma 2.4.2 any other choice of Hermitian structure gives a unitarily equivalent Hilbert module. We thus obtain a single well-deﬁned element [[E]] KK G(V, V ) associated to ∈ the vector bundle E. By virtue of its simplicity this construction is extraordinarily well-behaved. For instance, while it is clear that [[E E ]] = [[E ]] + [[E ]], we also have [[E E ]] = [[E ]] [[E ]]. Indeed, 1 ⊕ 2 1 2 1 ⊗ 2 1 ◦ 2 because the operator F in all our Kasparov G-modules is zero we see that

[[E ]] [[E ]] = (Γ (E ), µ , 0) (Γ (E ), µ , 0) 1 ◦ 2 0 1 1 ◦ 0 2 2 = (Γ (E ) Γ (E ), µ 1, 0) 0 1 ⊗µ2 0 2 1 ⊗ = (Γ (E E ), µ, 0). 0 1 ⊗ 2 Combining these observations with the universal property of the Grothendieck group then yields the following result.

Proposition 3.4.2. Let V be a proper G-space. Then the construction outlined above deﬁnes a semiring homomorphism C Vect (V ) KK G(V, V ), G → where the multiplication in KK G(V, V ) is given by the composition product. In particular this implies that if X is a proper cocompact G-space and V X is a G-invariant subspace then there ⊆ is a ring homomorphism K (X) KK G(V, V ) G → given by sending E E K (X) to [[E ]] [[E ]]. 0 1 ∈ G 0|V − 1|V Remark 3.4.3. The exact same arguments serve to deﬁne a ring homomorphism

G KG(X) KK (X,X) → for any compact topological group G and compact G-space X — where, as usual, KG(X) denotes the Atiyah-Segal equivariant K-theory. Composing on the left with the element [π] G ∈ KK (pt,X) associated to the map π : X pt then produces the isomorphism → =∼ G KG(X) KK (pt,X). −→ Of course, one does not expect the various KK G-elements coming from a given complex G-vector bundle to be completely unrelated.

Lemma 3.4.4. Let E V be a complex G-vector bundle over a proper G-space V . → (a) For any proper G-map f : Z V one has → [[E]] [f] = [f] [[f ∗E]] KK G(V,Z). ◦ ◦ ∈

29 (b) For any open G-invariant subset U V one has ⊆ [[E ]] [i] = [i] [[E]] KK G(U, V ), |U ◦ ◦ ∈ where [i] KK G(U, V ) denotes the class associated to the -homomorphism C (U) , ∈ ∗ 0 → C0(V ).

Proof. Since F = 0 in all these Kasparov modules it is trivial to compute the products (3.3.8). As a result one ﬁnds that in the ﬁrst equality one has

[[E]] [f]=(Γ (E) C (Z), µ 1, 0); ◦ 0 ⊗C0(V ) 0 ⊗ since Γ (E) C (Z) = Γ (f ∗E) this reduces to 0 ⊗C0(V ) 0 ∼ 0 [[E]] [f]=(Γ (f ∗E), µ f, 0). ◦ 0 ◦ On the other hand this last triple is precisely the value of [f] [[f ∗E]]. ◦ As for the second equality, observe that

[[E ]] [i]=(Γ (E ) C (V ), µ 1, 0) |U ◦ 0 |U ⊗C0(V ) 0 ⊗ [i] [[E]] = (Γ (E), µ i, 0). ◦ 0 ◦ However the Hilbert (G, C (V ))-module (Γ (E ) C (V ) is simply the Hilbert (G, C (U))- 0 0 |U ⊗C0(V ) 0 0 module Γ (E ) viewed as a Hilbert (G, C (V ))-module, and so 0 |U 0 [[E ]] [i]=(Γ (E ), µ, 0). |U ◦ 0 |U Hence, unlike the first case, we obtain different Kasparov modules for the two products, and must construct a homotopy between them. So let = σ : [0, 1] Γ (E) σ(0) Γ (E ) . It H { → 0 | ∈ 0 |U } is a simple exercise to see that the element ( , µ i, 0) (C (U), C (V ) C[0, 1]) defines a H ◦ ∈ EG 0 0 ⊗ homotopy between the two products.

Proposition 3.4.5. Let E V be a complex G-vector bundle over a proper G-space V , Z V → G ⊆ a closed G-invariant subspace and U = V Z its complement. Then if δ KK 1 (Z,U) denotes i \ q ∈ the class of the extension 0 C (U) C (V ) C (Z) 0 one has → 0 −→ 0 −→ 0 → [[E ]] δ = δ [[E ]] KK G(Z,U). |Z ◦ ◦ |U ∈ 1 Unfortunately, unlike the previous identities the correct homotopy between these two products is by no means obvious. The following results allow for a more high-brow approach to the problem, and ultimately would allow one to write out a (relatively) explicit homotopy.

Theorem 3.4.6 (Proper Stabilization Theorem). Let V be a proper G-space and let E be a countably generated Hilbert (G, C0(V ))-module. Then there is an equivariant isomorphism of

Hilbert (G, C0(V ))-modules E G = G . ⊕ HC0(V ) ∼ HC0(V )

30 Proof. Theorem 2.9 of [Phi89].

In particular the theorem implies that for any complex G-vector bundle E V there is G G → an isomorphism Γ0(E) = , and thus we can find an invariant projection PE ⊕ HC0(V ) ∼ HC0(V ) ∈ G B( ) whose range is isomorphic to Γ0(E). When combined with the right action of C0(V ) HC0(V ) this gives an equivariant -homomorphism ∗ G ρE : C0(V ) B( ) → HC0(V ) f fP . 7→ E G G Note that this last map actually lands inside K( ) = C0(V ) K( C ). Indeed, the inclusion C0(V ) ∼ G H ⊗ H of Γ0(E) in induces -homomorphisms HC0(V ) ∗ G B(Γ0(E)) B( ) → HC0(V ) K(Γ (E)) K( G ) 0 → HC0(V ) T P T P , 7→ E E and the map ρ is simply the result of applying these maps to C (V ) K(Γ (E)). E 0 ⊆ 0 The net result of this discussion is that we have yet another way to associate an element of G KK (V, V ) to the G-vector bundle E — namely we can form the Kasparov G-module [ρE] = G ( ,ρE, 0). HC0(V ) Lemma 3.4.7. [ρ ] = [[E]] KK G(V, V ) E ∈ Proof. We need only construct a homotopy between the elements [ρ ] = ( G ,ρ , 0) and E C0(V ) E G H [[E]] = (Γ0(E), µ, 0). So let = σ : [0, 1] σ(0) ImPE ; it is easily verified that H { → HC0(V ) | ∈ } the element ( ,ρ , 0) (C (V ), C (V ) C[0, 1] gives the necessary homotopy. H E ∈EG 0 0 ⊗ Lemma 3.4.8. Let E X be a complex G-vector bundle, Z X be a closed G-invariant → ⊆ G subspace and U = X Z its complement. Let ρ : C (X) C (X) K( C ) be the equivariant \ E 0 → 0 ⊗ H -homomorphism defined above. Then there are canonical equivariant -homomorphisms ∗ ∗ G ρ : C (U) C (U) K( C ) E|U 0 → 0 ⊗ H G ρ : C (X) C (Z) K( C ) E|Z 0 → 0 ⊗ H such that

[ρ ] = [[E ]] KK G(U,U) E|U |U ∈ [ρ ] = [[E ]] KK G(Z,Z) E|Z |Z ∈ and the diagram

i q 0 / C0(U) / C0(X) / C0(Z) / 0

ρE|U ρE ρE|Z G i⊗1 G q⊗1 G 0 / C (U) K( C ) / C (X) K( C ) / C (Z) K( C ) / 0 0 ⊗ H 0 ⊗ H 0 ⊗ H commutes.

31 G Proof. The key is to realize that the choice of a projection PE B( ) whose range is Γ0(E) ∈ HC0(V ) G G also provides us with a canonical choice of projections PE|U B( and PE|Z B( ) ∈ HC0(U) ∈ HC0(Z) whose ranges are Γ (E ) and Γ (E ) respectively. 0 |U 0 |Z G To obtain the projection PE|U , we must simply observe that is actually a Hilbert HC0(U) G G (G, C0(V ))-submodule of ; in fact, it is precisely C0(U). It follows that PE HC0(V ) HC0(V ) · G restricts to give a projection PE|U on whose range is Γ0(E) C0(U) = Γ0(E U ). The HC0(U) · ∼ | map ρ (f)= fP then has all the requisite properties — that [ρ ] = [[E ]] follows from E|U E|U E|U |U Lemma 3.4.7, while the construction of P guarantees that fP = fP for any f C (U), E|U E|U E ∈ 0 and hence the the relevant square commutes.

Now for the projection PE|Z . Let q : C0(V ) C0(Z) denote the restriction map. The G G G pushforward q∗ = C (V ) C0(Z) naturally identiﬁes with ; let PE|Z be the HC0(V ) HC0(V ) ⊗ 0 HC0(Z) image of P under the induced -homomorphism E ∗ q : B( G ) B( G ) HC0(V ) → HC0(Z) T T 1 b 7→ ⊗ G and let ρE|Z : C0(Z) K( ) denote the -homomorphism associated to the projection → HC0(Z) ∗ P . By construction we have ImP = q ImP = q Γ (E) = Γ (E ), and so by Lemma 3.4.7 E|Z Z|E ∗ E ∗ 0 ∼ 0 |Z we can identify [ρ ] with [[E ]]. Furthermore, since q(fP ) = q(f)P for any f C (V ) E|Z |Z E E|Z ∈ 0 we see that the rightmost square commutes. b Proof of Proposition 3.4.5. When applied to the map of extensions in Lemma 3.4.8 the naturality of the boundary map in the six-term exact sequence is equivalent (3.3.11) to the condition that

[ρ ] δ = δ [ρ ] KK G(Z,U). E|Z ◦ ◦ E|U ∈ 1 The result now follows from the fact that the classes [ρ ] and [ρ ] identify with [[E ]] and E|Z E|U |Z [[E ]]. |U

3.5 Dirac Operators and KK G(M, pt) ∗ Let G be a discrete or compact Lie group and M an open n-dimensional G-Spinc manifold; if G is discrete we shall make the further assumption that the G-action is proper. Under these assumptions the presence of the G-Spinc structure on M allows one to construct a G-invariant self-adjoint first-order elliptic differential operator D called the Dirac operator on M. Indeed, 6 M we have already seen that the existence of the G-Spinc structure allows one to construct the spinor bundle S; in particular this complex G-vector bundle comes equipped with an equivariant bundle map µ : TM S S. Now suppose that :Γ∞(S) Γ∞(TM S) is a G-equivariant ⊗ → ∇ → ⊗ connection on S. Then the composition D = µ : Γ∞(S) Γ∞(S) defines a G-invariant 6 M ◦ ∇ → first-order elliptic differential operator on M, and a careful choice of connection ensures that it is self-adjoint. Unfortunately any discussion of the theory of such operators and their relationship to the G Kasparov groups KK ∗ (M, pt) would require a rather substantial digression, and so we shall have

32 to simply content ourselves with stating a few of the relevant properties. The interested reader should consult [HR00], where the non-equivariant case is worked out in some detail; because of our assumptions on the group G and its action on M the proofs carry directly over to the equivariant case.

Proposition 3.5.1.

(a) Associated to the diﬀerential operator D there is a corresponding class 6 M G [D ] KK (M, pt). 6 M ∈ n (b) Let U M be an open subset. Then under the map j : C (U) , C (M) the class [D ] is ⊆ 0 → 0 6 M sent to [D ]. 6 U Now suppose that W is a n+1-dimensional G-Spinc manifold with boundary ∂W . Then under our assumption about the action ∂W has an equivariant tubular neighborhood — that is, there is a G-invariant neighborhood ∂W U with a G-equivariant diﬀeomorphism U = ∂W [0, 1). ⊂ ∼ × Thus T W ∂W is equivariantly isomorphic to T ∂W 1, and so by Proposition 2.4.10 ∂W inherits | ⊕ G a canonical G-Spinc structure from W , and thus a class [D ] KK (∂W, pt). 6 ∂W ∈ n G G Proposition 3.5.2. The boundary map KK (W˚ , pt) KK (∂W, pt) associated to the short n+1 → n exact sequence 0 C (W˚ ) , C (W ) C (∂W ) 0 sends [D ] to [D ]. → 0 → 0 0 → 6 W˚ 6 ∂W Let us conclude this section with a short example which we will put to use in the next section. Consider the sphere Sn. One usually thinks of Sn as consisting of the unit vectors of Rn+1, but when considering actions on the sphere it is often more useful to view it as the homogeneous space SO(n + 1)/SO(n). In fact, as the following lemma shows, this view of the sphere provides a bit of geometric information as well.

Lemma 3.5.3. The orthonormal frame bundle F +(Sn) of TSn is naturally isomorphic to the principal (SO(n + 1), SO(n))-bundle SO(n) , SO(n + 1) Sn. → Proof. Recall that a point in F +(Sn) consists of a point x Sn together with an oriented n ∈ orthonormal basis (v1, . . ., vn) for the tangent space TxS . Then

F +(Sn) SO(n + 1) → (x, (v ,...,v )) (v . . . v x) 1 n 7→ 1| | n| deﬁnes a map of principal (SO(n + 1), SO(n))-bundles over Sn, and is therefore automatically an isomorphism.

This identification makes it trivial to define a Spinc(n + 1)-equivariant Spinc-structure on Sn; specifically, we have

Spinc(n) , Spinc(n + 1) Spinc(n + 1)/Spinc(n) = Sn. → ∼ Thus by applying the ideas from earlier in the section we obtain a Spinc(n + 1)-equivariant Dirac c operator D on Sn and a corresponding element [D] KK Spin (n+1)(Sn, pt). 6 6 ∈ n 33 Lemma 3.5.4. Let G Spinc(n + 1) be a compact subgroup, ρ a complex representation of G, ≤ 2n and Vρ the trivial G-equivariant complex vector bundle over S associated to ρ. Then

G [V ] [D] = IndexD V =0 KK (pt, pt). ρ ◦ 6 6 ⊗ ρ ∈ 0 Proof. The antipode map α : S2n S2n interchanges the spinor bundles S± and thus converts → D into D∗; on the other hand, since the bundle V is trivial we have α∗V = V . Thus 6 6 ρ ρ ρ IndexD V = IndexD α∗V 6 ⊗ ρ 6 ⊗ ρ = Indexα D V ∗ 6 ⊗ ρ = IndexD∗ V 6 ⊗ ρ = IndexD V , − 6 ⊗ ρ and hence IndexD V = 0. 6 ⊗ ρ ± c ± 2n Proposition 3.5.5. Let S = Spin (2n+1) Spinc(2n) ∆ be the complex spinor bundles on S , × c S± 2n Spin (2n+1) 2n and let [ ] denote the corresponding elements of KSpinc(2n+1)(S ) ∼= KK 0 (pt,S ). Then

Spinc(2n+1) [S+] [D] = IndexD S+ = ( 1)ndet KK (pt, pt) = R(Spinc(2n + 1)) ◦ 6 6 ⊗ − 2n+1 ∈ 0 ∼ and

c [S−] [D] = IndexD S− = ( 1)n+1det KK Spin (2n+1)(pt, pt) = R(Spinc(2n + 1)), ◦ 6 6 ⊗ − 2n+1 ∈ 0 ∼ where detn denotes the 1-dimensional complex representation associated to the quotient map Spinc(n) S1. → Proof. The result follows by expanding on an argument given by Atiyah [Ati68, 6] for the real § case in dimensions 8k. The essential point is that since S2n is a homogeneous Spinc(2n+1)-space and all objects in sight are Spinc(2n + 1)-equivariant one can make use of Bott’s result [Bot65] on homogeneous elliptic operators to relate the desired indices to the indices of the (equivariant) Euler and signature operators. Speciﬁcally one can perform explicit character computations (some of which are done for the Spin(2n) case in [AB68, 8]) to show that § (∆+ ∆−)2 = ( 1)n( ( 1)iΛi) det − − − 2n (∆+ + ∆−)(∆+ ∆−)=(Λn ΛXn ) det − + − − 2n as elements of R(Spinc(2n)), where the Λi are the exterior powers of the standard representation Spinc(2n) SO(2n) , U(2n), while Λn are the irreducible components of Λn. → ± Combining these formulas with Bott’s result then allows one to conclude that

IndexD (S+ S−)= χ(S2n) det = ( 1)n 2 det 6 ⊗ 2n+1 − 2n+1 IndexD (S+ S−) = Sign(S2n) det =0, 6 ⊗ ⊕ 2n+1 and solving for this system for the indices of the individual spinor bundles yields the desired result.

34 2n Corollary 3.5.6. Let β K c (S ) denote the Bott element constructed in Section 2.5. 2n ∈ Spin (2n) Then c β [D]=1 KK Spin (2n)(pt, pt). 2n ◦ 6 ∈ 0 Proof. Recall that β = [H∗] [H−∗], where in this case H± = S2n ∆± and H is obtained by 2n − ∼ × gluing H± along the equator. Thus it follows from Lemma 3.5.4 that [H−∗] [D]= 0, and so it ◦ 6 only remains to show that [H∗] [D]=1. ◦ 6 Note that when considered as merely a Spinc(2n)-equivariant bundle the spinor bundle S+ is isomorphic to H. Indeed, a little thought shows that Spinc(2n)-equivariant bundles on S2n are characterized by their restrictions to the ﬁxed points, and it is straightforward to check that these restrictions agree for S+ and H. Moreover an examination of the image of center of Spinc(2n) 1 under the 2 -Spin representations shows that

∆+ det∗ if n = 0 (mod 2) ∆+∗ = ⊗ 2n ∼ ∼ − ∗ ( ∆ det2n if n = 1 (mod 2). ) ⊗ ∼ It follows that S+ det∗ if n = 0 (mod 2) H∗ = S+∗ = ⊗ 2n ∼ ∼ ∼ S− ∗ ( det2n if n = 1 (mod 2) ) ⊗ ∼ as Spinc(2n)-equivariant vector bundles over S2, and combining this with the previous proposition yields the desired result.

3.6 Principal Induction and the Thom Isomorphism

Let G be a countable discrete group, G be a compact topological group; while neither of these restrictions on the groups is strictly necessary, they suﬃce for our purposes and will spare us some technicalities. Additionally, let P X be a principal (G, G)-bundle. Then given a G-C∗- → ∗ G algebra A, we shall let indP (A) denote the C -algebra C0(P, A) , where G acts on C0(P, A) via (g f)(p)= g f(pg). Note that ind (A) is a G-C∗-algebra via the action · · P (γ f)(p)= f(γ−1p). ·

Similarly, given a Hilbert (G,B)-module E, we deﬁne indP (E) to be the Hilbert (G, indP (B))- module C (P, E)G. Note that for any representation φ : A B(E) there is a corresponding 0 → representation φ˜ : ind (A) B(ind (E)). P → P Lemma 3.6.1. Let A and B be two G-C∗-algebras. Then there is an induction map

ind : G(A, B) (ind (A), ind (B)) P E →EG P P given by sending the Kasparov module (E, φ, F ) to (ind (E), φ,˜ F˜). Here F˜ B(ind (E)) is P ∈ P given by the formula F˜(p)= (g F ) dg; G · Z note that if F is G-invariant than this simply amounts to applying F pointwise.

35 Proof. This is simply a matter of verifying that the various compactness conditions are still satisfied for the induced module. We shall only verify that [F˜ , φ˜(ã)] K(ind (E)) for any ∈ P a˜ ind (A), but checking the other conditions is similar. ∈ P K K First, note that indP ( (E)) ∼= (indP (E)), and hence it suffices to compute the commutator pointwise. Expanding the definitions of F˜ and φ˜ then yields that

[F˜ , φ˜(˜a)](p)= [g F, φ(˜a(p))] dg. G · Z Generally such an integral would only be convergent in the strong topology, but as the operator F is required to be G-continuous this integral will also converge in the norm topology. Thus since each commutant

[g F, φ(ã(p))] = g [F,g−1 φ(ã(p))] = g [F, φ(ã(pg))] · · · · is compact, the desired integral is as well.

Proposition 3.6.2. Induction gives a well-deﬁned homomorphism

ind : KK G(A, B) KK (ind A, ind B) P → G P P which is functorial in A and B and compatible with the composition product. In particular, for any f : A B one has ind [f] = [ind (f)]. → P P

Proof. Since indP (B[0, 1]) ∼= indP (B)[0, 1] it is easy to see that the induction map sends homotopies to homotopies and thus the induction map descends to the Kasparov groups. The com- patibility with composition products is also straightforward, since induction sends connexions to connexions. Finally, the claim about indP [f] is just a matter of unraveling the deﬁnitions.

Now consider the case when A = C0(Z) for some G-space Z. Thanks to the freeness of the G ∗ G -action on P Z the induced C -algebra indP (A) ∼= C0(P Z) can be identiﬁed with the C0- × × c functions on P G Z). In particular, suppose E M is a2n-dimensional G-Spin vector bundle × → with principal (G, Spinc(n))-bundle P M, and let Spinc(2n) act on S2n through SO(2n). Then → 2n E indP (C(S )) ∼= C0(M ) and C indP ( ) ∼= C0(M). Lemma 3.6.3. Given the principal (G, Spinc(2n)-bundle described above, the diagram

2n Spinc(2n) 2n 2n KSpinc(2n)(S ) / KK (S ,S )

indP indP E G E E KG(M ) / KK (M ,M ) commutes. In particular we have [[β ]] = [[ind β ]] = ind [[β ]] KK G(M E,M E). E P 2n P 2n ∈

36 Proof. This is just a matter of chasing through the deﬁnitions of the various induction maps in order to see that both paths in the diagram produce the same Kasparov module for any Spinc(2n)-equivariant vector bundle on S2n.

Continuing with the same objects E and P one can also apply the induction map indP to c Spin (2n) 2n the class of the equivariant Dirac operator [D 2n ] KK (S , pt). This produces a new 6 S ∈ 0 class [D ] KK G(M E,M) which represents the fibrewise Dirac operator on M E — that is, 6 E ∈ the (non-elliptic) differential operator on M E constructed from the Dirac operators along the spherical fibres.

Corollary 3.6.4. Let M be a smooth G-manifold with boundary, and let E˚ denote the restriction ˚E˚ ˚ of E to the interior of M. Moreover let pE˚ : M M denote the natural projection. Then

[p ] [[β ]] [D ]=1 KK G(M,˚ M˚). E˚ ◦ E˚ ◦ 6 E˚ M˚ ∈ Proof. Let P˚ M˚ denote the restriction of the principal (G, Spinc(2n))-bundle to the interior → of M. Note that every term in the given product arises as an element induced by P˚. Namely, we have

[pE˚] = indP˚[p]

[[βE˚]] = indP˚[[β2n]] [D ] = ind [D ]] 6 E˚ P˚ 6 2n where p : S2n pt is the obvious map. Thus the result would hold so long as [p] [[β ]] [D ]= 2n 6 2n Spin→c ◦ ◦ 1pt KK (pt, pt). But as noted earlier, the product [p] [[β2n]] is simply the element ∈ c ◦ [β ] KK Spin (2n)(pt,S2n). The result now follows immediately from Corollary 3.5.6. 2n ∈ Note that if the manifold M is G-Spinc and E is a smooth vector bundle then the sphere bundle M E will possess a natural G-Spinc structure as well. Indeed, stabilizing the tangent bundle of T (M E) yields

T (M E) 1 = T E 1 = p∗ TM p∗ E 1, ⊕ ∼ ⊕ ∼ E ⊕ E ⊕ where p : M E M is the natural projection. But since TM, E and the trivial bundle all E → possess G-Spinc structures the two-out-of-three lemma endows T (M E) with a G-Spinc structure.

Proposition 3.6.5. Let M be a G-Spinc manifold, and E a smooth 2n-dimensional G-Spinc vector bundle over M. Then

[D ] [D ] = [D E ] 6 E ◦ 6 M 6 M G E in KK ∗ (M , pt). Sadly the proof of this proposition is beyond the scope of the meager summary of KK -theory given here, as it involves the technical machinery behind the composition product; it, as with most of the results of this section, is ultimately a special case of a more general theorem regarding the functoriality of the so-called ‘shriek’ maps. Regardless, by combining the last two results we are left with the following corollary.

37 Corollary 3.6.6. G ˚ [p ] [[β ]] [D E˚ ] = [D ] KK (M, pt) E˚ ◦ E˚ ◦ 6 M˚ 6 M˚ ∈ To conclude the section we shall look at one other, more explicit, application of the induction construction. Let G be a countable discrete group and H G a ﬁnite subgroup. Then by letting ≤ G G = H and P = G we obtain induction maps which we shall denote by indH .

Lemma 3.6.7. Let B be a G-C∗-algebra; by restriction B can also be viewed as an H-C∗-algebra. Then as indG (B) = C (G/H) B as G-C∗-algebras. H ∼ 0 ⊗ Proof. For the purposes of the proof it will be more convenient to view C (G/H) B as 0 ⊗ C0(G/H,B). Then

indG (B) C (G/H,B) H → 0 f gH g f(g) 7→ { 7→ · } gives the desired isomorphism of G-C∗-algebras.

Thus when B is a G-C∗-algebra there will be some measure of triviality in the induction map

indG : KK ∗ (A, B) KK ∗ (indG (A), C (G/H) B). H H → G H 0 ⊗ 0 C With a little eﬀort this triviality can be removed : within KK G(C0(G/H), ) we have the element 2 2 (` (G/H), µ, 0) given by the natural representation of C0(G/H) on ` (G/H), and composing it with the induction map produces a map

iG : KK ∗ (A, B) KK ∗ (indG (A),B). H H → G H Remark 3.6.8. The Kasparov module (`2(G/H), µ, 0) arises naturally as the composition of the morphism µ : C (G/H) , K(`2(G/H)) with the G-Morita equivalence of K(`2(G/H)) and C. 0 → Explicitly computing the eﬀect of the Kasparov product results in the following lemma.

G 2 H ˜ ˜ Lemma 3.6.9. iH sends the Kasparov module (E, φ, F ) KK H (A, B) to (` (G, E) , φ, F ) G ∈ ∈ KK G(indH (A),B).

The following proposition is essential for computing the equivariant KK -theory of proper G-spaces; the proof parallels that given in [GHT00] for equivariant E-theory.

Proposition 3.6.10. Let G be a countable discrete group, H G a ﬁnite subgroup. Then the ≤ induction map iG : KK ∗ (A, B) KK ∗ (indG (A),B) H H → G H is an isomorphism for any H-C∗-algebra A and G-C∗-algebra B.

38 Chapter 4

Topological Equivariant KK -Theory

4.1 Topological Cycles and Bordism

As usual, throughout this chapter G will always denote a countable discrete group. However, while the majority of results will only be stated for such groups it will be convenient to note that much of the discussion (especially in this section) applies equally well when G is replaced by a compact Lie group G.

Deﬁnition 4.1.1. Given a proper cocompact G-pair (X, A) and a compact G-space Y , a topological KK G-cycle for (X, A; Y ) is a triple (M,ξ,f) consisting of

a proper cocompact G-Spinc manifold M, • an element ξ (M Y ), and • ∈ KG × a G-map f : (M,∂M) (X, A). • → Two topological KK G-cycles for (X, A; Y ) are isomorphic if there is a diﬀeomorphism between their G-Spinc manifolds which identiﬁes the other components of the triples. The set of all isomorphism classes of topological KK G-cycles for (X, A; Y ) will be denoted by t G(X, A; Y ). E

Remarks 4.1.2. (a) While the G-map f : (M,∂M) (X, A) does not necessarily induce a map from M˚ = → M ∂M to X A, it does provide a map f ∗ : C (X A) C (M˚) between the corresponding \ \ 0 \ → 0 G-C∗-algebras; this will prove essential later on. (b) We will generally play a little fast and loose in our use of G-Spinc structures on smooth G-vector bundles and assume that any G-Spinc structure on such a bundle is ’smooth’ — that is, the associated principal bundle is a smooth G-manifold and the map to the frame bundle is a smooth map. Unfortunately many of our earlier constructions only guarantee the existence of topological G-Spinc structures; this gap is bridged in Appendix B, where it is shown that every topological G-Spinc structure on a smooth G-vector bundle can be smoothed, and that such smoothings are unique up to the obvious notion of equivalence.

Deﬁnitions 4.1.3. (a) For any (M,ξ,f) t G(X, A; Y ) the opposite cycle is the triple ( M,ξ,f) t G(X, A; Y ), ∈ E − ∈ E where M denotes the manifold M endowed with the opposite G-Spinc structure. − (b) Given two cycles (M , ξ ,f ) in t G(X, A; Y ) their sum is the cycle (M M , ξ i i i i=0,1 E 0 ∪ 1 0 ∪ ξ ,f f ) t G(X, A; Y ). 1 0 ∪ 1 ∈ E 39 Note that addition of cycles makes t G(X, A; Y ) a commutative monoid, where the identity E given by the trivial cycle (∅, , ). Moreover since any G-manifold splits into a direct union of − − even dimensional and odd dimensional components there is a natural decomposition

t G(X, A; Y )= t G(X, A; Y ) t G(X, A; Y ), E E0 ⊕ E1 where t G(X, A; Y ) denotes those cycles whose manifolds only have components of dimension En congruent to n mod 2. As is these monoids are quite clearly not groups — adding two cycles will never reduce the number of components of the G-Spinc manifold — but this can easily be remedied by introducing a suitable equivalence relation.

Deﬁnition 4.1.4. A regular domain D on a manifold M is a closed subset of M with non-empty interior such that if p ∂D then p has a coordinate chart φ : U Rn such that φ is an open ∈ → map, φ(p) is the origin, and

φ(U D)= x φ(U) : x 0 . ∩ { ∈ n ≥ } If M is a G-manifold, then a G-invariant regular domain will simply be called a regular G-domain.

Note that in particular every regular domain is a regular submanifold; furthermore every regular domain is disjoint from the boundary (if any) of the ambient manifold.

Deﬁnition 4.1.5. A G-bordism for (X, A; Y ) is a quadruple (W,M,η,F ) consisting of

a proper cocompact G-Spinc manifold W , • a regular G-domain M ∂W , • ⊆ an element η (W Y ), and • ∈ KG × a G-equivariant map F : W X such that F (∂W M˚) A. • → \ ⊆

Remarks 4.1.6. (a) The conditions on the map F ensure that we can extract a unique cycle from any G-bordism. The regularity condition on M implies that M˚ is an open G-submanifold of ∂W ; as such M inherits a natural G-Spinc structure from its inclusion in ∂W , and restricting the other components of the bordism to M yields the boundary cycle (M, η , F ) t G(X, A; Y ). |M |M ∈ E (b) When A is trivial the contents of a G-bordism can be reduced to the triple (W, η, F ). Indeed, in this case ∂W M˚ = ∅, and thus the manifold M will simply be ∂W . \ Deﬁnition 4.1.7. An element of t G(X, A; Y ) bounds if it is isomorphic to the boundary cycle E of a G-bordism. Two elements of t G(X, A; Y ) are bordant if their diﬀerence bounds. E

Remarks 4.1.8. (a) By the previous remark one ﬁnds that when A is trivial this reduces to what one expects —

namely, that two cycles (Mi, ξi,fi)i=0,1 are bordant if and only if there is a triple (W, η, F )

40 such that ∂W = M M , η = ξ and F = f . The presence of the regular G- 0 ∪− 1 |Mi i |Mi i domain M in the general case is simply a trick from bordism theory [Con79] that allows one to extend the deﬁnition to pairs while avoiding the use of manifolds with corners.

(b) Bordism clearly respects the decomposition of t G(X, A; Y ) into even- and odd-dimensional E cycles. Indeed, any G-bordism (W,M,η,F ) naturally decomposes as a disjoint union of even- and odd-dimensional G-bordisms, and this decomposition mirrors the decomposition of the boundary cycle.

As we’ll see, most of the properties of bordism will follow from gluing constructions. It is an unfortunate complication that when A is non-trivial the results of these gluings are usually not smooth manifolds. However the problems with the gluing are usually limited to codimension one subsets of the boundary, in which case the following lemma will allow us to correct matters (see Appendix A for more details).

Lemma 4.1.9 (Straightening the Angle). Let W n be a topological G-manifold and M n−2 ⊂ W a closed G-submanifold (without boundary). Suppose further that

W M has a differentiable structure, and • \ there exists a G-invariant neighborhood U of M in W and a G-equivariant homeomorphism • φ : U M R R which identifies M with M 0 0 and is a diffeomorphism on → × + × + ×{ }×{ } U M. \ Then there is a differentiable structure on W which extends that on W M and makes W a smooth \ G-manifold.

Corollary 4.1.10. n n n−1 (a) Let W1 and W2 be two proper G-manifolds, and let M be a regular G-domain in both their boundaries. Then the gluing W = W W can be given a G-equivariant smooth 1 ∪M 2 structure such that T W = T W T W as real G-vector bundles. ∼ 1 ∪M 2 (b) Let M1 and M2 be two smooth proper G-manifolds with boundary. Then the product M = M M can be given a G-equivariant smooth structure such that TM = π∗TM π∗TM 1 × 2 ∼ 1 1 ⊕ 2 2 as real G-vector bundles.

Remark 4.1.11. In both cases the description of the tangent bundle guarantees that if the two components each have G-Spinc-structures, so will the resulting gluing/product space.

Lemma 4.1.12. The sum of a cycle with the corresponding opposite cycle always bounds.

Proof. Given any cycle (M,ξ,f) one can construct a G-bordism (W, M M, ξ 1,f π), where ∪− ◦ W is the smooth G-Spinc manifold homeomorphic to M [0, 1] which is obtained by straightening × the angle. But the boundary cycle of this bordism is (M M, ξ ξ,f f). ∪− ∪ ∪

Lemma 4.1.13. Bordism deﬁnes an equivalence relation on t G(X, A; Y ). ∼b E

41 Proof. Symmetry and reﬂexivity are obvious; the only real issue is transitivity. Let (Mi, ξi,fi), i 0, 1, 2 , be three cycles in t G(X, A; Y ) and let (W ,M M , η , F ), j 0, 1 , be two ∈{ } E j j ∪− j+1 j j ∈{ } G-bordisms connecting (M0, ξ0,f0) to (M1, ξ1,f1) and (M1, ξ1,f1) to (M2, ξ2,f2) respectively. Then we can construct W = W W by gluing W and W over their common submanifold 0 ∪M1 1 0 1 M1. By Corollary 4.1.10 and the subsequent remark we see that W can be given the structure of a c smooth G-Spin -manifold, and thanks to Lemma 2.3.1 we can also glue the classes ηj along their common restriction ξ to obtain an element η K (W Y ). Likewise, the functions F combine 1 ∈ G × j to give F : (W, ∂W ) (X, A). Together these combine to give a G-bordism (W, M M ,η,F ) → 0 ∪− 2 connecting (M0, ξ0,f0) to (M2, ξ2,f2).

Let ΩG(X, A; Y )= t G(X, A; Y ) / denote the bordism classes of topological KK G-cycles E ∼b for (X, A; Y ); the decomposition of t G(X, A; Y ) then yields a corresponding decomposition E ΩG(X, A; Y )=ΩG(X, A; Y ) ΩG(X, A; Y ). 0 ⊕ 1 Lemma 4.1.14. Let t G(X, A; Y ) t G(X, A; Y ) denote the semigroup of boundary cycles. D∗ ≤ E∗ Then ΩG(X, A; Y ) = t G(X, A; Y )/t G(X, A; Y ). ∗ ∼ E∗ D∗ Proof. Since every element of t G(X, A; Y ) vanishes in ΩG(X, A; Y ) the natural surjection from D∗ ∗ t G(X, A; Y ) onto ΩG(X, A; Y ) factors through t G(X, A; Y )/t G(X, A; Y ). Now suppose that E∗ ∗ E∗ D∗ (M , ξ ,f ) (M , ξ ,f ), so that (M , ξ ,f ) + ( M , ξ ,f ) t G(X, A; Y ). Then in the 0 0 0 ∼b 1 1 1 0 0 0 − 1 1 1 ∈ D quotient semigroup t G(X, A; Y )/t G(X, A; Y ) one has E∗ D∗ (M , ξ ,f ) = (M , ξ ,f ) + [(M , ξ ,f ) + ( M , ξ ,f )] 1 1 1 1 1 1 0 0 0 − 1 1 1 = [(M , ξ ,f ) + ( M , ξ ,f )] + (M , ξ ,f ) 1 1 1 − 1 1 1 0 0 0 = (M0, ξ0,f0)], where in the last step we have used Lemma 4.1.12 to conclude that (M , ξ ,f ) + ( M , ξ ,f ) 1 1 1 − 1 1 1 ∈ t G(X, A; Y ). Thus two elements of t G(X, A; Y ) are bordant if and only if they give the same D∗ E∗ element in the quotient semigroup, and the result follows.

G Z Corollary 4.1.15. Ω∗ (X, A; Y ) is a 2-graded group, where the inverse of the cycle (M,ξ,f) is the opposite cycle ( M,ξ,f). − Proof. It is clear from the preceding lemma that ΩG(X, A; Y ) is a semigroup. Moreover Lemma 4.1.12 shows that (M,ξ,f) + ( M,ξ,f) is a boundary cycle, and hence is zero in ΩG(X, A; Y ). − It follows that ( M,ξ,f) is the additive inverse of (M,ξ,f) and hence that every element has − an inverse.

Using the constructions from the previous chapter we can associate a number of Kasparov modules to the components of a topological KK G-cycle. First there is the element [D ] 6 M˚ ∈ KK G(M,˚ pt) associated to the G-Spinc structure on M. Next for any subset U M Y there n ⊆ × is the element [[ξ ]] KK G(U,U). Finally there are the more pedestrian classes associated |U ∈ 0 42 to maps of G-C∗-algebras. By stringing these various elements together we obtain a natural transformation

Φ : t G(X, A; Y ) KK G(X A, Y ) E∗ → ∗ \ (M,ξ,f) [f] [π ] [[ξ ]] ([D ] 1 ), 7→ ◦ M˚ ◦ |M˚×Y ◦ 6 M˚ × Y where [f] KK G(X A, M˚) is the class associated to f ∗ : C (X A) C (M˚) and [π ] ∈ 0 \ 0 \ → 0 M˚ ∈ KK G(M,˚ M˚ Y ) is that associated to the natural projection π : M˚ Y M˚. 0 × M˚ × → Proposition 4.1.16. The map Φ descends to give a natural transformation of groups

Φ:ΩG(X, A; Y ) KK G(X A, Y ). ∗ → ∗ \ Proof. Given all we know about Dirac operators this essentially comes down to a diagram chase. First note that by Lemma 4.1.14 it suﬃces to show that every boundary cycle is sent to zero. So let (W,M,η,F ) be a G-bordism for (X, A; Y ) and suppose the components of W all have dimension congruent to (n + 1) mod 2, so that the boundary cycle lies in t G t G. Consider Dn ≤ En the commutative diagram

G ˚ ∂ G G ˚ KK n+1(W , Y ) / KK n (∂W, Y ) / KK n (M, Y ) (??)

∂ KK G (∅, Y ) / KK G(X, Y ) / KK G(X A, Y ). n+1 n n \ The commutativity of the ﬁrst square comes from the naturality of the boundary map in the six-term exact sequence when applied to the map of extensions

0 / C0(W˚ ) / C0(W ) / C0(∂W ) / 0 O O O ∗ ∗ F F |∂W

0 / C0(X) / C0(X) / 0, while the commutativity of the second square comes from the commutativity of the diagram

∗ F |∂W C0(X) / C0(∂W ) O O

∗ F |M C0(X A) / C (M˚). \ 0 G ∅ Note that since KK n+1( , Y ) ∼= 0 the bottom left corner of the diagram (??) vanishes, and hence every element of KK G (W˚ , Y ) is sent to zero in KK G(X A, Y ). n+1 n \ The result is now a consequence of the following lemma.

Lemma 4.1.17. Let (W,M,η,F ) be a G-bordism as in the previous proof and let (M,ξ,f) ∈ t G(X, A; Y ) be the associated boundary cycle. Consider the element [π ] [[η ]] ([D ] Dn W˚ ◦ |W˚ ×Y ◦ 6 W˚ × 1 ) KK G (W˚ , Y ). Then in the diagram (??) this element is carried to Φ(M,ξ,f). Y ∈ n+1

43 Proof. Let δ KK G(∂W, W˚ ) denote the class associated to the extension ∈ 1 0 C (W˚ ) C (W ) C (∂W ) 0 → 0 → 0 → 0 → and let i : C (M˚) , C (∂W ). Then the map of interest from KK G (W˚ , Y ) to KK G(X A, Y ) 0 → 0 n+1 n \ is just composition with [f] [i] δ. ◦ ◦ Now observe that we have the following identities regarding δ :

δ [π ] = [π ] (δ 1 ) • ◦ W˚ ∂W ◦ × Y This is exactly the naturality of the boundary in the six-term when applied to the extensions

0 / C0(W˚ ) / C0(W ) / C0(∂W ) / 0

0 / C (W˚ Y ) / C0(W Y ) / C0(∂W Y ) / 0. 0 × × × δ [[η ]] = [[η ]] (δ 1 ) • ◦ |W˚ ×Y |∂W ×Y ◦ × Y This follows from Proposition 3.4.5. (δ 1 ) ([D ] 1 ) = [D ] 1 • × Y ◦ 6 W˚ × Y 6 ∂W × Y This follows from Theorem 3.5.2.

Combining the three identities then yields that

δ [π ] [[η ]] ([D ] 1 ) = [π ] [[η ]] ([D ] 1 ). ◦ W˚ ◦ |W˚ ×Y ◦ 6 W˚ × Y ∂W ◦ |∂W ×Y ◦ 6 ∂W × Y Next we have a second batch of identities, this time involving [i] :

[i] [π ] = [π ] ([i] 1 ) • ◦ ∂W M˚ ◦ × Y This follows from the commutativity of the diagram

∗ π∂W C0(∂W ) / C0(∂W Y ) O O ×

i i×1Y π∗ M˚ C (M˚) / C (M˚ Y ). 0 0 × ([i] 1 ) [[η ]] = [[ξ ]] ([i] 1 ) • × Y ◦ |∂W ×Y |M˚×Y ◦ × Y This follows from Lemma 3.4.4. ([i] 1 ) ([D ] 1 ) = [D ] 1 • × Y ◦ 6 ∂W × Y 6 M˚ × Y This follows from Proposition 3.5.1.

Together this last array of identities shows that

[i] [π ] [[η ]] ([D ] 1 ) = [π ] [[ξ ]] ([D ] 1 ). ◦ ∂W ◦ |∂W ×Y ◦ 6 ∂W × Y M˚ ◦ |M˚×Y ◦ 6 M˚ × Y Finally, precomposing this last element with the class [f] yields exactly the Kasparov module for Φ(M,ξ,f).

44 4.2 Vector Bundle Modiﬁcation

Unfortunately, the homomorphism Φ defined in the previous section is far from being an isomorphism — while we shall ultimately find that the map is surjective, it is never injective. Indeed, starting from any (M,ξ,f) t G(X, A; Y ) we can form new cycles (M S2n, ξ β ,f π ) ∈ En × n ◦ M G which, though yielding distinct classes in Ωn (X, A; Y ), are all mapped to the same element of KK G(C (X A), C(Y )). n 0 \ In fact, we can easily produce even more cycles which are mapped to Φ(M,ξ,f). Specifically, given any even-dimensional smooth G-Spinc-vector bundle E on M we can form the sphere bundle M E = S(E 1), which as we saw in Section 3.6 will again be a G-Spinc manifold. At the same ⊕ time, the G-Spinc structure of E also provides M E with a natural ‘Bott element’ β K (M E) E ∈ G (see Section 2.5).

Deﬁnition 4.2.1. Given (M,ξ,f) t G(X, A; Y ) and an even-dimensional smooth G-Spinc- ∈ En vector bundle E on M, the vector bundle modiﬁcation of (M,ξ,f) by E is the cycle

(M,ξ,f)E = (M E, (p Id )∗ξ (β 1 ),f p ) t G(X, A; Y ), E × Y ⊗ E Y ◦ E ∈ En where p : M E M is the natural projection. E → Lemma 4.2.2. Let ξ K (M Y ), and consider the commutative diagram ∈ G × ˚E˚ o ˚ E˚ / E M π ˚ M Y M Y M˚E × × p E˚ pE˚×IdY pE ×IdY M˚ o π M˚ Y / M Y. M˚ × × Then ∗ [p ] [π E˚ ] [[((p 1 ) ξ) E˚ ]] = [π ] [[ξ ]] [p Id ] E˚ ◦ M˚ ◦ E × Y |M˚ ×Y M˚ ◦ |M˚×Y ◦ E˚ × Y in KK G(M,˚ M˚E˚ Y ). × Proof. Note that since the diagram commutes we have

[p ] [π E˚ ] = [π ] [p Id ]; E˚ ◦ M˚ M˚ ◦ E˚ × Y ∗ ∗ combining this with the fact that ((p 1 ) ξ) ˚ = (p Id ) (ξ ) yields E × Y |M˚E ×Y E˚ × Y |M˚×Y ∗ ∗ [p ] [π ˚] [[((p 1 ) ξ) ˚ ]] = [π ] [p Id ] [[(p Id ) (ξ )]]. E˚ ◦ M˚E ◦ E × Y |M˚E ×Y M˚ ◦ E˚ × Y ◦ E˚ × Y |M˚×Y Finally, by Lemma 3.4.4 we see that

[p Id ] [[(p Id )∗(ξ )]] = [[ξ ]] [p Id ], E˚ × Y ◦ E˚ × Y |M˚×Y |M˚×Y ◦ E˚ × Y and the result follows.

Proposition 4.2.3. Let (M,ξ,f) t G(X, A; Y ) and let E be a smooth even-dimensional G- ∈ En Spinc-vector bundle on M. Then

Φ((M,ξ,f)E)=Φ(M,ξ,f) KK G(X A, Y ). ∈ n \

45 Proof. By deﬁnition we have

Φ((M,ξ,f)E)=Φ(M E, (p Id )∗ξ (β 1 ),f p ) E × Y ⊗ E Y ◦ E ∗ = [f p ] [π ˚] [[(p Id ) ξ (β ˚ 1 )]] ([D ˚] 1 ) ◦ E˚ ◦ M˚E ◦ E × Y ⊗ E|M˚E Y ◦ 6 M˚E × Y ∗ = [f] [p ] [π E˚] [[(p Id ) ξ E˚]] ([[β E˚]] 1 ) ([D E˚] 1 ). ◦ E˚ ◦ M˚ ◦ E × Y |M˚ ◦ E|M˚ × Y ◦ 6 M˚ × Y But by Lemma 4.2.2 we can rewrite the middle of this expression to obtain

E Φ((M,ξ,f) ) = [f] [π ] [[ξ ]] [p Id ] ([[β E˚ ]] 1 ) ([D E˚] 1 ) ◦ M˚ ◦ |M˚×Y ◦ E˚ × Y ◦ E |M˚ × Y ◦ 6 M˚ × Y = [f] [π ] [[ξ ]] ([D ] 1 ), ◦ M˚ ◦ |M˚×Y ◦ 6 M˚ × Y where in the last line we have used Corollary 3.6.6 to reduce the product [p Id ] ([[β ˚ ]] E˚ × Y ◦ E|M˚E × 1 ) ([D E˚] 1 ) to ([D ] 1 ). Y ◦ 6 M˚ × Y 6 M˚ × Y

4.3 Topological KK -Theory

In light of Proposition 4.2.3 their is no chance that homomorphism

Φ:ΩG(X, A; Y ) KK G(X A, Y ) ∗ → ∗ \ could be an isomorphism. Nonetheless, one might still hope (perhaps na¨ıvely) that if we were to deﬁne a relation on t G(X, A; Y ) incorporating both bordism and vector bundle modiﬁcation E∗ then the resulting group would be isomorphic to KK G(X A, Y ). As with many statements in ∗ \ this chapter, this will ultimately be true, but proving it will be complicated by a number of technicalities.

Deﬁnition 4.3.1. Let denote the equivalence relation on t G(X, A; Y ) generated by bordism ∼ E∗ and vector bundle modiﬁcation. The topological KK G-theory of a G-pair (X, A; Y ) is

tKK G(X, A; Y )= t G(X, A; Y )/ , n 0, 1 . n En ∼ ∈{ } Proposition 4.3.2 (Direct Sum-Disjoint Union). Let M be a proper cocompact G-Spinc manifold with component dimensions congruent to n mod 2 and let f : (M,∂M) (X, A) be a → G-map. Then for any two elements ξ , ξ K (M Y ) one has 0 1 ∈ G × (M, ξ + ξ ,f) = (M, ξ ,f) + (M, ξ ,f) tKK G(X, A; Y ). 0 1 0 1 ∈ n Proof. By forming the vector bundle modiﬁcation by the trivial bundle M C2, we see that the × result would follow if

(M S2, (ξ + ξ ) β ,f π) (M S2, ξ β ,f π) + (M S2, ξ β ,f π), × 0 1 ⊗ 2 ◦ ∼b × 0 ⊗ 2 ◦ × 1 ⊗ 2 ◦ 2 c 2 where β2 is the Bott element in K(S ). Note that there is a clear Spin bordism W between S 2 2 and S S . Indeed, let a+ = (0, 0, 1) and a− = (0, 0, 1). Fix a value of r, 0

46 of R, R> 1+ r> 1, and let W denote the region inside the closed ball of radius R but not inside 2 the spheres S±.

Projecting radially outward from the points a± onto the outer boundary of W gives maps p : W S2. Let β and β denote the pullbacks of the class β K(S2) along p and p ± → + − 2 ∈ + − respectively. Then it is easily checked that β± S2 = β2 and β± S2 = 0. | ± | ∓ Now, after possibly straightening the angle, M W will a smooth G-Spinc manifold, and × ∂(M W ) will contain M ∂W as a regular G-domain. Together with the previous comments × × this yields a quadruple

(M W, M ∂W, ξ β + ξ β ,f π) × × 0 ⊗ + 0 ⊗ − ◦ which gives the desired bordism between the cycles.

Corollary 4.3.3. For any cycle (M,ξ,f) t G(X, A; Y ) one has ∈ En ( M,ξ,f) = (M, ξ,f) tKK G(X, A; Y ) − − ∈ n G Proof. Both are additive inverses of the cycle (M,ξ,f) in the group tKK n (X, A; Y ). Indeed, we have already seen that ( M,ξ,f) is an additive inverse. As for (M, ξ,f), note that − − (M,ξ,f) + (M, ξ,f) = (M, 0,f); − G but any cycle of the form (M, 0,f) is equivalent to 0 in tKK n (X, A; Y ), since there is an obvious G-bordism whose boundary is the vector bundle modiﬁcation

2 (M, 0,f)M×R = (M S2, 0,f π). × ◦ As mentioned at the start of the section, one would now like to show that the topological KK G-groups defined above agree with the Kasparov KK G-groups, at least in the case where (X, A) is a pair of finite G-CW complexes. This would be easy to achieve if we knew that the topological groups formed a homology theory, but unfortunately the implicit nature of the equivalence relation makes this rather difficult. Indeed, while the appropriate maps for a six-term sequence are all fairly obvious it is not at all clear that the sequence is exact, since the definition of fails to provide a concise condition for when a cycle is equivalent to zero. ∼ Thus we shall find the following proposition quite useful in the proof of the six-term exact sequence, though proving it will require a brief diversion.

Proposition 4.3.4. (M,ξ,f) t G(X, A; Y ) represents the zero cycle in tKK G(X, A; Y ) if and ∈ E only if there exists a G-Spinc ‘normal bundle’ ν for M such that (M,ξ,f)ν bounds.

4.4 A Technical Lemma

Note that when the underlying manifold is connected vector bundle modiﬁcation is actually a special case of a more general principal bundle construction. Indeed, suppose we have a compact

47 Lie group G and a smooth principal (G, G)-bundle P over M. Then given from any compact G- manifold Z we can form the product P G Z; one can easily check that this is a proper cocompact × G-manifold. Moreover if the original G-manifold Z is G-Spinc this product will possess a natural G-Spinc structure. Thus for any triple (P,ξ,f) consisting of a smooth principal (G, G)-bundle P over M, ξ ∈ K (M) and f : M X we can define an ‘induction’ map G → ind : t G(pt) t G (X, A; Y ) (P,ξ,f) E∗ → E∗+n ∗ (Z, η) (P G Z, (π Id ) ξ (ind (η) 1 ),f π), 7→ × × Y ⊗ P Y ◦ where π is the natural projection from P G Z onto M. × To see how this generalizes vector bundle modification, recall that an even-dimensional G- Spinc vector bundle E M has an associated principal (G, Spinc(2n))-bundle P . At the same → time the homomorphism Spinc(2n) SO(2n) SO(2n + 1) → → provides S2n R2n+1 with a action of Spinc(2n). But as was shown in Section 2.5, the nat- ⊆ ural Spinc structure for S2n is equivariant with respect to this action and there is a canonical 2n 2n Bott element β2n KSpinc(2n)(S ). Together these provides a ‘spherical’ class (S ,β2n) c ∈ ∈ t Spin (2n)(pt), and it follows from our previous discussion on the relationship between induction E0 and Bott elements that 2n E ind(P,ξ,f)(S ,β2n) = (M,ξ,f) . Lemma 4.4.1. For any (M,ξ,f) t G(X, A; Y ) and a smooth principal (G, G)-bundle P over ∈ E M the map ind(P,ξ,f) descends to give a well-defined graded homomorphism

ind :ΩG(pt) ΩG(X, A; Y ). (P,ξ,f) ∗ → ∗

Proof. By Lemma 4.1.14 it suffices to show that ind(P,ξ,f) sends boundary cycles to boundary G cycles. So let (W, η) be a G-bordism whose boundary cycle is (∂W, η ∂W ) t (pt). After | c ∈ E possibly straightening the angle, P GW will be a proper cocompact G-Spin manifold containing × P G ∂W as a regular G-domain in the boundary. But this results in a G-bordism × ∗ (P G W, P G ∂W, (π Id ) ξ (ind (η) 1 ),f π) × × × Y ⊗ P Y ◦ for (X, A; Y ) whose boundary cycle is easily seen to be ind (∂W, η ). (P,ξ,f) |∂W Before finally proving the main result of the section, we first need to establish a result about

2ni products of spherical classes. To be more specific, if we are given two classes (S ,β2ni ) c ∈ t Spin (2ni)(pt), i 0, 1 , we can form the direct product E0 ∈{ } c c (S2n0 S2n1 ,β β ) t Spin (2n0)×Spin (2n1)(pt). × 2n0 2n1 ∈ E0 However, this is not the only way to manufacture an element of this bordism group. By pulling Spinc(2(n +n )) the element (S2(n0+n1),β ) t 0 1 (pt) back along the homomorphism 2(n0+n1) ∈ E0 Spinc(2n ) Spinc(2n ) Spinc(2(n + n )) 0 × 1 → 0 1 Spinc(2n )×Spinc(2n ) we also obtain the element (S2(n0+n1),β ) t 0 1 (pt). 2(n0+n1) ∈ E0 48 Spinc(n )×Spinc(n ) Lemma 4.4.2. (S2n0 S2n1 ,β β ) (S2(n0+n1),β ) in t 0 1 (pt). × 2n0 2n1 ∼b 2(n0+n1) E0 Proof. To simplify notation, let G = Spinc(2n ) Spinc(2n ). Our first task is simply to construct 0 × 1 a G-equivariant bordism between the manifolds S2n0 S2n1 and S2(n0+n1). So fix a value of r, × 0

i : S2n0 S2n1 R2n0 R2n1 R r × → × × (~x ,t ) (~x ,t ) ((1 + rt ) ~x , r~x , (1 + rt ) t ) 0 0 × 1 1 7→ 1 · 0 1 1 · 0 which generalizes the standard embedding of S1 S1 into R3. Note that because the G-action × on S2n0 S2n1 factors through SO(n ) SO(n + 1) the map is G-equivariant. × i ⊂ i Now choose R, R> 1+ r> 1, and let DR denote the closed ball of radius R about the origin.

A simple computation then shows that the image of ir falls in the interior of DR, and thus the

2n0+2n1+1 desired bordism W is just the regular domain in R which lies between the image of ir

2(n0+n1) and ∂DR = S . All that remains is to define a class η KG(W ) which restricts to the appropriate classes on ∈ the boundary. To this end, consider the point z = (0, 0, 1) R2n0 R2n1 R. As a point in the ∈ × × interior of D W there exists a sufficiently small > 0 such that the open ball B = B (z) also R\ lies in D W (in fact, any r will do). Moreover, because z is a fixed point of the (isometric!) R\ ≤ G-action, B itself is preserved by the G-action. To finish things off, let V = D B. Then V is G-equivariantly diffeomorphic to S2(n0+n1) R\ × 2(n0+n1) G [0, 1], and so KG(V ) ∼= KG(S ). At the same time, W is a closed -submanifold of ∗ V . Define η KG(W ) to be the restriction of the class in KG(V ) corresponding to β ∈ 2(n0+n1) ∈ 2(n0+n1) 2(n0+n1) KG(S ). It is clear by construction that the restriction of η to S is β2(n0+n1), while a brief computation shows the restriction to the other boundary is precisely β2n0 β2n1 .

G Lemma 4.4.3. Let (M,ξ,f) t n (X, A; Y ). Then for any two smooth even-dimensional G- c ∈ E Spin vector bundles E0 and E1 over M one has

∗ (M,ξ,f)E0⊕E1 [(M,ξ,f)E0 ]p E1 ∼b in t G(X, A; Y ). En Proof. Because the vector bundle modification of a sum of cycles is isomorphic to the sum of the individual modifications, it suffices to consider the case when M is connected. In this case c dim Ei =2ni is globally constant, and each Ei has an associated smooth (G, Spin (2ni))-bundle

Pi. The key to the proof is then the observation that both cycles in the proposition arise from the same principal bundle construction. Namely, let G = Spinc(2n ) Spinc(2n ). Then the 0 × 1 ﬁber product P = P P is a smooth (G, G)-bundle over M, and 0 ×M 1

2(n0+n1) E0⊕E1 ind(P,ξ,f)(S ,β2(n0+n1)) = (M,ξ,f) ∗ ind (S2n0 S2n1 ,β β ) = [(M,ξ,f)E0 ]p E1 . (P,ξ,f) × 2n0 2n1 The result now follows by combining Lemmas 4.4.1 and 4.4.2.

49 4.5 Normal Bordism

One might reasonably wonder why we deﬁned the equivalence relation to be that generated by ∼ bordism and vector bundle modiﬁcation, rather than some more explicit condition. For instance, G one could declare that two cycles (Mi, ξi,fi) t n (X, A; Y ), i 0, 1 , are equivalent if and c ∈ E ∈ { } only if there are even-dimensional G-Spin vector bundles Ei over Mi such that

(M , ξ ,f )E0 (M , ξ ,f )E1 . 0 0 0 ∼b 1 1 1 Unfortunately, while this is clearly reﬂexive and symmetric, it is not at all clear that it is tran- G sitive. For suppose we are given three cycles (Mi, ξi,fi) t n (X, A; Y ), i 0, 1, 2 , along with c 0 ∈ E ∈{ } even-dimensional G-Spin vector bundles E0, E1, E1 and E2 such that

(M , ξ ,f )E0 (M , ξ ,f )E1 0 0 0 ∼b 1 1 1 0 (M , ξ ,f )E1 (M , ξ ,f )E2 . 1 1 1 ∼b 2 2 2 0 E1 E If we could somehow arrange for a bordism between the cycles (M1, ξ1,f1) and (M1, ξ1,f1) 1 then everything would be fine. Unfortunately it is quite clear for dimension reasons that this will usually not be possible. Thus we will need to vector bundle modify these two cycles in order to have a chance of finding a bordism. In light of the Technical Lemma one could modify the cycles 0 by E1 and E1 respectively; however this destroys the bordisms we already have, since there is no 0 guarantee that the bundles E1 and E1 can be extended across the bordisms. The upshot of this discussion is that to be sure that we have a transitive relation we need to limit the kinds of vector bundles which are used in vector bundle modifications — specifically, we need a collection (M) of G-Spinc vector bundles over M such that for any E, E0 (M) there C ∈C exist G-Spinc bundles F, F 0 such that E F = E0 F 0 and — more importantly — F, F 0 can ⊕ ∼ ⊕ be extended across any bordism containing M in the boundary. There is one obvious candidate already widely used in bordism theory — normal bundles. But how does one define a normal bundle when the manifolds involved need not have suitable finite dimensional embeddings? We begin by recalling some concepts from Chapter 2.

Deﬁnition 4.5.1. For any proper cocompact G-space Z a G-space over Z is simply a G-space X together with a G-map a : X Z. X →

Remarks 4.5.2. (a) It follows from Lemma 2.1.3 that any G-space over Z is automatically a proper G-space. (b) Every proper cocompact G-space is a G-space over itself.

Deﬁnition 4.5.3. Let X be a G-space over Z. A complex (resp. real, resp. G-Spinc) G-vector bundle on X is Z-trivial if it is isomorphic to the pullback over aX of a complex (resp. real, resp. G-Spinc) G-vector bundle on Z.

Lemma 4.5.4. Let X be a G-space over Z, and let E be a G-Spinc vector bundle over X. Then there exists a G-Spinc vector bundle F over X such that E F is a Z-trivial G-Spinc vector ⊕ bundle.

50 Proof. Consider the complexiﬁcation E C. It follows from Corollary 2.2.10 that there is a ⊗ complex G-vector bundle F such that (E C) F is a Z-trivial complex G-vector bundle. But 0 ⊗ ⊕ 0 as a G-Spinc vector bundle E C = E E\, and so one can simply let F = F E\. ⊗ ∼ ⊕ 0 ⊕ Deﬁnition 4.5.5. Let M be a smooth G-Spinc manifold over Z. A Z-normal bundle for M is a smooth G-Spinc vector bundle ν such that TM ν is a Z-trivial G-Spinc bundle. ⊕ Lemma 4.5.6. Every smooth G-Spinc manifold M over Z has a Z-normal bundle.

Proof. Apply Lemma 4.5.4 to obtain a topological G-Spinc bundle and then smooth it.

Lemma 4.5.7. Let M be a smooth G-Spinc manifold over Z. Then any two Z-normal bundles for M are stably isomorphic; that is, for any two Z-normal bundles ν0, ν1 for M there exist smooth Z-trivial G-Spinc bundles , such that ν = ν as G-Spinc vector bundles 0 1 0 ⊕ 0 ∼ 1 ⊕ 1 over M.

Proof. Let = TM ν and = TM ν . Then ν = TM ν ν . 0 ⊕ 1 1 ⊕ 0 k ⊕ k ∼ ⊕ 0 ⊕ 1 Remark 4.5.8. If X is a G-space over Z and (M,ξ,f) is a topological KK G-cycle for (X, A; Y ), then M is naturally a G-manifold over Z with map a = a f. M X ◦ Deﬁnition 4.5.9. Let X be a G-space over Z.

(a) A cycle (M,ξ,f) t G(X, A; Y ) is said to Z-normally bound if there exists an even- ∈ En dimensional Z-normal bundle ν for M such that (M,ξ,f)ν bounds. (b) Two cycles (M , ξ ,f ) t G(X, A; Y ), i 0, 1 , are said to be Z-normally bordant if their i i i ∈ En ∈{ } diﬀerence normally bounds; in other words there exist even-dimensional Z-normal bundles

νi for the Mi such that (M , ξ ,f )ν0 (M , ξ ,f )ν1 . 0 0 0 ∼b 1 1 1

Remarks 4.5.10.

(a) When the G-space Z and G-map aX are clear we shall often simply refer to ‘normal bundles’ and say that two cycles are ‘normally bordant’. (b) Any two normally bordant cycles are equivalent under . ∼ Lemma 4.5.11. Let X be a G-space over Z. Then normal bordism deﬁnes an equivalence relation on t G(X, A; Y ). ∼n En Proof. The symmetry of normal bordism is obvious, while reﬂexivity follows from the fact that every G-Spinc manifold in a cycle has a Z-normal bundle. So the only real obstacle is transitivity. G So suppose we are given three cycles (Mi, ξi,fi) t n (X, A; Y ), i 0, 1, 2 , along with 0 ∈ E ∈ { } Z-normal bundles ν0, ν1, ν1 and ν2 such that

(M , ξ ,f )ν0 (M , ξ ,f )ν1 0 0 0 ∼b 1 1 1 0 (M , ξ ,f )ν1 (M , ξ ,f )ν2 . 1 1 1 ∼b 2 2 2

51 Now, because all Z-normal bundles for M1 are stably isomorphic there exist two Z-trivial c ∗ 0 ∗ 0 0 0 G-Spin bundles 1 = a Z and = a on M1 such that such that ν1 1 = ν as G- M1 1 M1 Z ∼ 1 1 c ∗ ∗ 0 ⊕ ⊕ Spin bundles. Let 0 = aM0 Z and 2 = aM2 Z . Modifying the entire bordisms by (smoothings 0 of) the pullbacks of Z and Z results in bordisms

∗ ∗ [(M , ξ ,f )ν0 ]p0 0 [(M , ξ ,f )ν1 ]p1 1 0 0 0 ∼b 1 1 1 ν0 p∗ 0 ν p∗ [(M , ξ ,f ) 1 ] 10 1 [(M , ξ ,f ) 2 ] 2 2 . 1 1 1 ∼b 2 2 2 and so

∗ (M , ξ ,f )ν0⊕0 [(M , ξ ,f )ν0 ]p0 0 0 0 0 ∼b 0 0 0 ∗ [(M , ξ ,f )ν1 ]p1 1 ∼b 1 1 1 (M , ξ ,f )ν1⊕1 ∼b 1 1 1 0 0 (M , ξ ,f )ν1⊕1 ∼b 1 1 1 ν0 p∗ 0 [(M , ξ ,f ) 1 ] 10 1 ∼b 1 1 1 ∗ [(M , ξ ,f )ν2 ]p2 2 ∼b 2 2 2 (M , ξ ,f )ν2⊕2 . ∼b 2 2 2

The result follows, since ν and ν are Z-normal bundles for M and M respectively. 0 ⊕ 0 2 ⊕ 2 0 2

Deﬁnition 4.5.12. Let X be a G-space over Z. The topological KK G|Z -theory of the G-pair (X, A; Y ) is tKK G|Z(X, A; Y )= t G(X, A; Y )/ , n 0, 1 . n En ∼n ∈{ }

Lemma 4.5.13. Let (X, A; Y ) be a G-pair with X a G-space over Z. Then bordant topological KK G-cycles for (X, A; Y ) are normally bordant.

Proof. Note that it suﬃces to show that boundary cycles normally bound. Let (W,M,η,F ) be a G-bordism with boundary cycle (M,ξ,f) in t G(X, A; Y ). Choose D a Z-normal bundle ν for W . Then (W,M,η,F )ν⊕1 is a G-bordism whose boundary cycle is (M,ξ,f)ν|M ⊕1. But TM ν 1 = T W ν = (T W ν) is a Z-trivial bundle on M, ⊕ |M ⊕ ∼ |M ⊕ |M ∼ ⊕ |M and thus ν 1 is a Z-normal bundle for M. So (M,ξ,f) normally bounds. |M ⊕

Lemma 4.5.14. Let X be a G-space over Z. If (M,ξ,f) t G(X, A; Y ) and E is a smooth ∈ E∗ even-dimensional G-Spinc vector bundle on M, then (M,ξ,f)E is normally bordant to (M,ξ,f).

Proof. Obviously if we are to prove a normal bordism, our ﬁrst task is to ﬁnd a normal bundle for M E. To this end, note that if p : M E M is the natural projection, then T (M E) 1 = E → ⊕ ∼ T E 1 = p∗ (TM E) 1. Now let ν be a Z-normal bundle to M and Ec a Z-complement to ⊕ ∼ E ⊕ ⊕ E, so that TM ν = and E Ec = for some Z-trivial G-Spinc vector bundles and ⊕ ∼ M ⊕ ∼ E M 52 E. Then

T (M E) p∗ (Ec ν 1) = T (M E) 1 p∗ (Ec ν) ⊕ E ⊕ ⊕ ∼ ⊕ ⊕ E ⊕ = p∗(TM E) 1 p∗(Ec ν) ∼ ⊕ ⊕ ⊕ ⊕ = p∗(TM ν) p∗(E Ec) 1 ∼ ⊕ ⊕ ⊕ ⊕ = 1 ∼ M ⊕ E ⊕ and thus p∗(Ec ν 1) is a Z-normal bundle for M E. ⊕ ⊕ However, modifying by this bundle and applying Lemma 4.4.3 we see that

E p∗ (Ec⊕ν⊕1) E⊕Ec⊕ν⊕1 [(M,ξ,f) ] E (M,ξ,f) ∼b (M,ξ,f)ν⊕E ⊕1, ∼b and since ν 1 is a Z-normal bundle for M this shows that the two cycles are normally ⊕ E ⊕ bordant.

G Proposition 4.5.15. Let X be a G-space over Z. Two elements of t n (X, A; Y ) are normally G E bordant if and only if they represent the same element of tKK n (X, A; Y ). In other words, the natural homomorphism G|Z G tKK ∗ (X, A; Y ) tKK ∗ (X, A; Y ) is an isomorphism.

Proof. It’s clear from the deﬁnition of normal bordism that any two normally bordant cycles G represent the same element of tKK n (X, A; Y ). On the other hand, the converse follows from Lemmas 4.5.13 and 4.5.14, since is generated by bordism and vector bundle modiﬁcation. ∼ G Corollary 4.5.16. Let X be a G-space over Z. Then a cycle (M,ξ,f) t n (X, A; Y ) represents G ∈ E the zero element in tKK n (X, A; Y ) if and only if it normally bounds.

4.6 The Eilenberg-Steenrod Axioms

Let g : (X, A) (X0, A0) be a G-map between proper cocompact G-pairs. Then there is an → obvious map g : tKK G(X, A; Y ) tKK G(X0, A0; Y ) ∗ ∗ → ∗ given by sending a topological cycle (M,ξ,f) to (M,ξ,g f). If X0 is a G-space over Z then the ◦ G-map a = a 0 g makes X a G-space over Z as well; there is then a map X X ◦ g : tKK G|Z(X, A; Y ) tKK G|Z (X0, A0; Y ) ∗ ∗ → ∗ together with a commuting diagram

g G|Z ∗ G|Z 0 0 tKK ∗ (X, A; Y ) / tKK ∗ (X , A ; Y )

=∼ =∼

g G ∗ G 0 0 tKK ∗ (X, A; Y ) / tKK ∗ (X , A ; Y ).

53 Lemma 4.6.1 (Homotopy Invariance). If h : (X, A) [0, 1] (X0, A0) is a G-equivariant × → homotopy between proper cocompact G-pairs. Then

h = h : tKK G(X, A; Y ) tKK G(X0, A0; Y ). 0∗ 1∗ ∗ → ∗ Proof. Let (M,ξ,f) be an element of t G(X, A; Y ), and consider the product space W = M E∗ × [0, 1]. After straightening the angle W is a proper cocompact G-Spinc manifold, and its boundary contains (M 1) ( M 0) as a regular G-domain. Then (W, (M 1) ( M 0), ξ 1,h × ∪ − × × ∪ − × ◦ (f Id ) deﬁnes a G-bordism for (X0, A0; Y ) whose boundary is precisely (M,ξ,h f)+ × [0,1] 1 ◦ ( M,ξ,h f), from which it follows that h (M,ξ,f)= h (M,ξ,f). − 0 ◦ 0∗ 1∗ Lemma 4.6.2. Let V M be a regular G-domain in a proper cocompact G-Spinc manifold M. ⊆ If f : M X is a G-map with f(M V˚) A then [M,ξ,f] = [V, ξ ,f ] tKK G(X, A; Y ). → \ ⊆ |V ×Y |V ∈ Proof. As in the previous lemma we form the product W = M [0, 1]. After straightening the × angle W is a proper cocompact G-Spinc manifold, and its boundary contains (M 1) ( V 0) × ∪ − × as a regular G-domain. Then (W, (M 1) ( V 0), ξ 1,f π )isa G-bordism for (X, A; Y ) × ∪ − × ◦ M whose boundary is (M,ξ,f) + ( V, ξ ,f ). − |V ×Y |V Deﬁnition 4.6.3. Let M be a G-Spinc manifold with boundary. The thickening of M is the G-Spinc manifold τ(M) = M (∂M [0, 1]). For any Z M let τ(Z) τ(M) be the set ∪∂M × ⊆ ⊂ Z (Z ∂M [0, 1]). Note that M is natural included in τ(M) as a regular G-domain. ∪Z∩∂M ∩ × Lemma 4.6.4. Let (M,ξ,f) be a topological cycle for (X, A; Y ) and let c : τ(M) M be the → G-map given by collapsing the collar ∂M [0, 1] onto ∂M. Set ξ = c∗ξ and f = f c. Then × ◦ (τ(M), ξ, f) is a new topological cycle for (X, A; Y ), and b b G b b [τ(M), ξ, f] = [M,ξ,f] tKK ∗ (X,Z; Y ). ∈ Proof. Since M is a regular G-domainb b in τ(M) this is simply a special case of Lemma 4.6.2.

Lemma 4.6.5. Let P and Q be two closed disjoint G-invariant subsets of a proper cocompact G-manifold M. Then there exists a regular G-domain V in τ(M) with P V˚ and V τ(Q)= ∅. ⊆ ∩ Remark 4.6.6. The use of τ(M) will only be essential when P ∂W = ∅. ∩ 6 Proof. Let Q0 = τ(Q) ∂τ(M) τ(M). Then P and Q0 are disjoint closed G-invariant subsets, ∪ ⊂ and hence we can apply Urysohn’s lemma to the subsets P/G and Q0/G in τ(M)/G to obtain a continuous G-invariant function σ : τ(M) [ 1, 1] such that σ(P )= 1 and σ(Q) = 1. Fix an → − − , 0 << 1, and let σ∞ denote a G-invariant smooth -approximation to σ. Then

σ (P ) < 1+ < 1 < σ (Q) ∞ − − ∞ and so by Sard’s Theorem σ has a regular value t [ 1+ , 1 ]. Let V = σ−1(( ,t]). ∞ ∈ − − ∞ −∞ Lemma 4.6.7 (Excision). Let U X be an open set such that U A˚. Then the inclusion ⊆ ⊂ i : (X U, A U) , (X, A) induces an isomorphism \ \ → ∼ i : tKK G(X U, A U; Y ) = tKK G(X, A; Y ). ∗ ∗ \ \ −→ ∗

54 Proof. Let (M,ξ,f) be a topological cycle for (X, A; Y ), and set P = f −1(X A˚), Q = f −1(U). \ Then by Lemma 4.6.5 there is a regular G-domain V τ(M) such that P V˚ and V τ(Q)= ⊂ ⊆ ∩ ∅. It follows that f(∂V ) A U, and thus we have a topological cycle (V, ξ , f ) for ⊆ \ |V ×Y |V (X U, A U; Y ). By Lemma 4.6.2 we see that i∗[V, ξ V ×Y , f V ] = [τ(M), ξ, f], which by Lemma \ \ b | | b b 4.6.4 is equals the original cycle. It follows that i∗ is surjective. b b b b The same argument applied to G-bordisms shows that if the image of a topological cycle normally bounds (using, say, Z = X) then G-bordism can be replaced with the image of a G-bordism for (X U, A U; Y ), and thus the map is injective. \ \ Proposition 4.6.8 (Six-term Exact Sequence). For any proper cocompact G-pair (X, A) and compact G-space Y there is an exact sequence

G G G tKK 0 (A; Y ) / tKK 0 (X; Y ) / tKK 0 (X, A; Y ) O ∂ ∂ G G G tKK 1 (X, A; Y ) o tKK 1 (X; Y ) o tKK 1 (A; Y ) where the boundary maps send a cycle (M,ξ,f) in t G(X, A; Y ) to (∂M,ξ ,f ) in En |∂M×Y |∂M t G (A; Y ). En−1 Proof. It’s straightforward to check that the composition of successive maps is zero; the real issue is whether the image of each map surjects onto the kernel of its successor. However thanks to G|X Proposition 4.5.15 we can instead work with the groups tKK ∗ , for which the arguments are almost exactly the same as those used in [Con79] for oriented bordism. G|X EXACTNESS AT tKK n (A; Y ) Suppose (M,ξ,f) tKK G|X (A; Y ) maps to the zero element of tKK G|X (X; Y ). Then there is ∈ n n an X-normal bundle ν for M such that (M,ξ,f)ν bounds in t G(X; Y ). Note that if (W, η, F ) En represents the bordism, then ∂W = M ν . But then F (∂W ) A, and so (W, η, F ) can be viewed ⊆ G|X ν as an element of tKK n+1 (X, A; Y ); its image under the boundary map is (M,ξ,f) = (M,ξ,f) G|X ∈ tKK n (A; Y ). G|X EXACTNESS AT tKK n (X; Y ) Suppose (M,ξ,f) tKK G|X (X; Y ) maps to the zero element of tKK G|X (X, A; Y ). Then there ∈ n n is an X-normal bundle ν for M such that (M,ξ,f)ν bounds in t G(X, A; Y ). Note that if En (W, M ν ,η,F ) represents the bordism, then since M ν is without boundary one must have ∂W = M ν M 0, where F (M 0) A. But now we can view (W, η, F ) as a bordism between (M,ξ,f)ν ∪− ⊆ 0 and (M , η 0 , F 0 ), and therefore |M |M ν 0 G|X (M,ξ,f) = (M,ξ,f) = (M , η 0 , F 0 ) tKK (X; Y ). |M |M ∈ n 0 G|X But because F (M ) A this last cycle is the image of an element in tKK n (A; Y ). ⊆ G EXACTNESS AT tKK n (X, A; Y ) Suppose (M,ξ,f) tKK G|X (X, A; Y ) maps to the zero under the boundary map. Then there ∈ n is an X-normal bundle ν for ∂M such that (∂M,ξ ,f )ν bounds in t G (A; Y ). Note that |∂M |M En−1 if (W, η, F ) represents the bordism, then ∂W = (∂M)ν . Because all normal bundles for ∂M

55 are stably isomorphic there exists an X-normal bundle ν for M and an X-trivial G-Spinc vector ν bundle on ∂M such that ν ∂M = ν . If pν : ∂W = (∂M) ∂M denotes the natural projec- | ⊕ p∗ → ν p∗ ν⊕ tion then it follows that (∂W, η , F ) ν = [(∂M,ξ ,f ) ] ν (∂M,ξ ,f ) = |∂W |∂W |∂M |M ∼b |∂M |M (∂M,ξ ,f )ν in t G (A; Y ). Thus if represents the extension of the X-trivial bundle p∗ |∂M |∂M En−1 ν from ∂W to the whole of W then the boundaries of (W, η, F ) and (M,ξ,f)ν are bordant. Hence G we can ’glue’ these two cycles together via the bordism to obtain a cycle in tKK n (X; Y ) whose ν G|X image (according to Lemma 4.6.2) is equal to (M,ξ,f) (M,ξ,f) tKKn (X, A; Y ). ∼n ∈ Putting all the preceding results together leaves us with the following theorem.

Theorem 4.6.9. tKK G( ; Y ) is a generalized G-equivariant homology theory on the category ∗ − of proper cocompact G-pairs and G-maps.

Proposition 4.6.10. The map Φ : tKK G(X, A; Y ) KK G(X A, Y ) is a natural transforma- ∗ → ∗ \ tion of homology theories.

Proof. All that is really needed to check that Φ is compatible with the boundary maps in the six-term exact sequences. So let (M,ξ,f) be a topological KK G-cycle for (X, A; Y ). Then

∂Φ(M,ξ,f)= δ [f] [π ] [[ξ ]] ([D ] 1 ) ◦ ◦ M˚ ◦ |M˚×Y ◦ 6 M˚ × Y which by the naturality of the boundary map in KK G can be changed to

δΦ(M,ξ,f) = [f ] δ [π ] [[ξ ]] ([D ] 1 ) |∂M ◦ M ◦ M˚ ◦ |M˚×Y ◦ 6 M˚ × Y = [f ] [π ] (δ 1 ) [[ξ ]] ([D ] 1 ) |∂M ◦ ∂M ◦ M × Y ◦ |M˚×Y ◦ 6 M˚ × Y where δ represents the class of the extension 0 C (M˚) C (M) C (∂M) 0. But by M → 0 → 0 → 0 → Proposition 3.4.5 and Theorem 3.5.2 this can be further rewritten as

δΦ(M,ξ,f) = [f ] [π ] [[ξ ]] (δ 1 ) ([D ] 1 ) |∂M ◦ ∂M ◦ |∂M×Y ◦ M × Y ◦ 6 M˚ × Y = [f ] [π ] [[ξ ]] ([D ] 1 ) |∂M ◦ ∂M ◦ |∂M×Y ◦ 6 ∂M × Y = Φ(∂M,ξ ,f ). |∂M |∂M G Thus to conclude that the topological groups tKK ∗ (X, A; Y ) agree with those produced by Kasparov’s bivariant theory it remains only to prove that Φ is an isomorphism on the proper equivariant ‘points’ G/H, where H G is a ﬁnite subgroup. ≤

4.7 Induction and the Dimension Axiom

Let H G be a ﬁnite subgroup, X a compact H-space and Y a compact G-space. Our ﬁrst goal ≤ in this section is to construct an induction map tKK H (X; Y ) tKK G(G X; Y ) analogous ∗ → ∗ ×H to the KK -theory induction map obtained in Section 3.6. We shall do so by applying the various induction constructions that have littered the previous chapters; in order to make this work we shall need the following trivial observation.

56 Lemma 4.7.1. Let H G be a subgroup, M an H-space and Y a G-space. Then the map ≤ G (M Y ) (G M) Y ×H × → ×H × (γ,m,y) (γ,m,γ y) 7→ · is a G-equivariant homeomorphism.

As a result the K-theory induction map indG : K∗ (M Y ) K∗ (G (M Y )) can H H × → G ×H × instead be viewed as a homomorphism K∗ (M Y ) K∗ ((G M) Y ). H × → G ×H × Definition 4.7.2. Let H G be a finite subgroup, X a compact H-space and Y a compact ≤ G-space. For any (M,ξ,f) t H (X; Y ) define ∈ E∗ iG (M,ξ,f) = (G M, indG ξ, indG f) t G(G X; Y ). H ×H H H ∈ E∗ ×H

G Lemma 4.7.3. iH descends to give a well-deﬁned homomorphism

iG : tKK H (X; Y ) tKK G(G X; Y ). H ∗ → ∗ ×H Proof. The induction of a bordism is a bordism between the inductions, while the fact that G G E G indH E iH ((M,ξ,f) ) ∼= [iH (M,ξ,f)] shows the vector bundle modiﬁcations are sent to vector bundle modiﬁcations.

Proposition 4.7.4. Let H G be a ﬁnite subgroup, X a compact H-space and Y a compact ≤ G-space. Then the diagram

G iH tKK H (X; Y ) / tKK G(G X; Y ) ∗ ∗ ×H Φ Φ G iH KK H (X, Y ) / KK G(G X, Y ) ∗ ∗ ×H commutes.

Proof. Let (M,ξ,f) be a topological KK G-cycle for (X; Y ). Then

Φ(iG (M,ξ,f)) = Φ(G M, indG ξ, indG f) H ×H H H = [indG f] [π ] [[indG ξ]] ([D ] 1 ) H ◦ G×H M ◦ H ◦ 6 G×H M × Y while

iG (Φ(M,ξ,f)) = iG ([f] [π ] [[ξ]] ([D ] 1 )) H H ◦ M ◦ ◦ 6 M × Y = indG ([f]) indG ([π ]) indG ([[ξ]]) (iG ([D ]) 1 ). H ◦ H M ◦ H ◦ H 6 M × Y A straightforward computation using the formulas from the end of Section 3.6 then reveals that the corresponding terms of the products are identical.

57 Proposition 4.7.5. Let H G be a ﬁnite subgroup, X a compact H-space and Y a compact ≤ G G-space. Then iH provides a natural isomorphism

iG : tKK H (X; Y ) tKK G(G X; Y ). H ∗ → ∗ ×H Proof. Naturality is obvious; we shall only worry about proving isomorphism. Let (M,ξ,f) be a topological KK G-cycle for (G X; Y ) and set M = f −1(1 X). Note that M is an ×H H × H H-Spinc submanifold of M; moreover the manifold M, together with its G-Spinc structure, can be recovered as G M . At the same time, the restriction ξ = ξ gives an element of ×H H H |MH ×Y K (M Y ) such that indG ξ = ξ. Finally, by letting f = f we obtain a ‘compression’ H H × H H H |MH map

compG : t G(G X; Y ) t H (X; Y ) H E ×H → E (M,ξ,f) (M , ξ ,f ). 7→ H H H The same arguments applied to G-bordisms and vector bundle modiﬁcations show that the compression map descends to give a homomorphism

compG : tKK G(G X; Y ) tKK H (X; Y ), H ×H → and it should be clear from its construction that this is an inverse to the induction map.

Proposition 4.7.6. Let H be a ﬁnite group and Y a compact H-space. Then the maps

K0 (Y ) tKK H (pt; Y ) H → 0 [ξ] (pt, ξ) 7→ and

K1 (Y ) = K˜ 0 (S1 Y ) tKK H (pt; Y ) H ∼ H × → 1 [ξ] (S1,ξ,π : S1 pt) 7→ → are natural surjections.

0 Proof. We shall only deal with the case of KH , the odd case is similar. Let (M, ξ) be an element of t H (pt; Y ). Since H is compact we can find an even-dimensional Spinc-representation ρ of H E0 and an H-equivariant embedding of M in Vρ; note that under these conditions the normal bundle ν of the embedding is naturally an even-dimensional H-Spinc bundle. Furthermore the sphere bundle M ν can be realized as an embedded submanifold of V 1. Let D V 1 denote the ρ ⊕ R ⊂ ρ ⊕ closed ball of radius R about the origin; since M ν is compact one can find a sufficiently large R> 0 such that M ν D˚ . Let W D be the regular H-domain whose boundary consists of ⊂ R ⊂ R ∂D = S(V 1) and M ν . A variation (see the remark below) on the excision/clutching argument R ρ ⊕ of Baum & Douglas [BD82] shows that the class (p Id )∗ξ (β 1 ) K0 (M ν Y ) can ν × Y ⊗ ν Y ∈ H × be realized as the restriction of a class η K0 (W Y ); moreover the class η is such that ∈ H × η = 0 K (Y ). From this last condition it follows that η is of the form |(~0,−R)×Y ∈ H |S(Vρ⊕1)×Y 0 0 ν 0 Vρ βρ ξ KH (S(Vρ 1) Y ), and thus (M, ξ) b (pt, ξ ) . As a result we see that every ∈ H ⊕ × ∼ element of tKK 0 (pt; Y ) is in the image KH (Y ).

58 Remark 4.7.7. Let (X, A) be a compact H-pair. Then to any pair of complex H-vector bundles ∼ E , E X and equivariant isomorphism φ : E = E the constructions of Baum & 0 1 → 0|A −→ 1|A Douglas associate the complex H-vector bundle E E X X obtained by identifying 0 ∪φ 1 → ∪A E with E via φ. To obtain a K-theory element η with the properties asserted in the 0|A 1|A proof of the proposition one must instead replace this vector bundle with the formal difference E E E E . In particular note that the restriction of this formal difference to the 0 ∪φ 1 1 ∪Id 1 subset X = A X X X is equal to zero in K0 (X); this is ultimately why the restriction ∪A ⊆ ∪A H of η to (~0, R) Y vanishes in the proof above. − × Proposition 4.7.8. Let H be a finite group and Y an H-space. Then Φ induces an isomorphism

∼ tKK H (pt; Y ) = KK H (pt, Y ) ∗ −→ ∗ and thus induces isomorphisms

∼ tKK G(G/H; Y ) = KK G(G/H, Y ). ∗ −→ ∗ Proof. The result of composing the map K∗ (Y ) tKK H (pt; Y ) from the previous proposition H → ∗ H H ∗ with Φ : tKK ∗ (pt; Y ) KK ∗ (pt, Y ) is easily seen to be the standard isomorphism KH (Y ) H → → KK∗ (pt, Y ). Combining this with the earlier propositions results in a commutative diagram

∼ H = G tKK ∗ (X; Y ) / tKK ∗ (G H X; Y ) qq8 8 × qqq qqq qqq ∗ KH (Y ) Φ Φ MMM MM ∼= MMM MMM & ∼= KK H (X, Y ) / KK G(G X, Y ). ∗ ∗ ×H A short diagram chase then yields that every map in the diagram is an isomorphism.

Corollary 4.7.9. Φ is an isomorphism on any pair of ﬁnite proper G-CW complexes (X, A).

Proof. Any natural transformation of equivariant cohomology theories which induces an isomorphism for the equivariant points G/H, H G ﬁnite, induces an isomorphism for all ﬁnite proper ≤ G-CW pairs.

59 Part II

Equivariant Bivariant Homology

60 Chapter 5

Homological Algebra

5.1 Abelian Categories

The most intimidating aspect of homological algebra is the rather large amount of categorical machinery that it employs. However since it is quite diﬃcult to precisely formulate many of the fundamental results without this language, we shall start with a survey of the essential concepts.

Definition 5.1.1. An additive category is a category such that C for every pair of objects (X, Y ) in the family of morphisms Hom (X, Y ) is given the • C C structure of an abelian group, the composition law induces a homomorphism • Hom (X, Y ) Hom (Y,Z) Hom (X,Z), and C ⊗ C → C possesses finite products and a zero object. • C Definition 5.1.2. An additive functor is a functor F : 0 between additive categories such C→C that each of the map Hom (X, Y ) Hom 0 (F (X), F (Y )) is a group homomorphism. C → C Let be an additive category. Then End(X) = Hom (X,X) is a ring for every X Obj( ); C C ∈ C in particular, additive categories with only a single object are in one-to-one correspondence with rings. Thus additive categories can be viewed as a category-theoretic generalization of rings, in much the same way that groupoids provide a generalization of groups. The fact that the morphisms between two objects form an abelian group allows us to extend many ideas from group theory to additive categories. The most important example of this is that one can make sense of the kernel and cokernel of a morphism.

Deﬁnitions 5.1.3. (a) A morphism i : W X is called the kernel of f if → i f 0 Hom (A, W ) ∗ Hom (A, X) ∗ Hom (A, Y ) → C −→ C −→ C is an exact sequence for every A Obj( ). ∈ C (b) A morphism q : Y Z is called the cokernel of f if → q∗ f ∗ 0 Hom (Z, A) Hom (Y, A) Hom (X, A) → C −→ C −→ C is an exact sequence for every A Obj( ). ∈ C

61 Remark 5.1.4. Kernels and cokernels need not exist; however if they do exist they are unique up to isomorphism.

As we shall see momentarily it is not always reasonable to assume that every morphism in C has both a kernel and cokernel; however, suppose for the moment that this is actually the case. Some playing around with kernels and cokernels then reveals that we are in the rather awkward position of having two reasonable ways of deﬁning the ‘image’ of a morphism.

Deﬁnition 5.1.5. Let f : X Y be a morphism in . The image of f is deﬁned to be → C Imf = Ker(Cokerf), while the coimage of f is Coimf = Coker(Kerf).

In general there is no reason to believe that the image and coimage of a given morphism should be isomorphic; however it follows quite easily from the deﬁnitions of kernel and cokernel that there is always a canonical morphism Coimf Imf. → Deﬁnition 5.1.6. An abelian category is an additive category such that C every morphism has a kernel and cokernel, and • the canonical morphism from the coimage to the image is always an isomorphism. • The fact that an abelian category has a single notion of image means that a number of ’obvious’ statements are actually true, not the least of which is the following lemma.

Lemma 5.1.7. Let be an abelian category, and let f : X Y be a morphism with trivial C → kernel and cokernel. Then f is an isomorphism.

Proof. One easily ﬁnds that the image of f is the identity morphism on X, the coimage of f is the identity morphism on Y , and the morphism between them is f itself. But since is abelian, C this implies that f is an isomorphism.

The existence of kernels and images also allows one to develop the machinery of exact sequences in the general context of abelian categories.

Deﬁnitions 5.1.8. f g (a) A sequence of morphisms X Y Z is exact if g f = 0 and the natural morphism −→ −→ ◦ Imf Kerf is an isomorphism. → (b) An additive functor F : 0 between abelian categories is left (resp. right) exact if for C→C any short exact sequence 0 X Y Z 0 → → → → the induced sequence 0 F (X) F (Y ) F (Z) → → → (resp. F (X) F (Y ) F (Z) 0) is exact. → → → (c) An additive functor is exact if it is both left and right exact.

62 Remark 5.1.9. It can be shown that a left exact functor preserves kernels, while a right exact functor preserves cokernels. As a result one ﬁnds that an additive functor F : is exact iﬀ C→C for every exact sequence X Y Z → → the induced sequence F (X) F (Y ) F (Z) → → is exact.

Example 5.1.10. Let A Obj( ). Then the functor Hom (A, ) : Groups is left exact. ∈ C C − C → (This follows immediately from the deﬁnition of kernel.)

5.2 Complexes and Homotopy

Throughout this section shall denote an abelian category. C Our goal in this chapter and the next is to deﬁne a bivariant cohomology for G-spaces, and to be truly worthy of being called a bivariant cohomology this theory should be deﬁned in terms of (cochain) complexes.

Deﬁnitions 5.2.1. (a) A complex in consists of a family Xn Z of objects from together with morphisms C { }n∈ C d : Xn Xn+1 satisfying d2 = 0 for all n Z. X → X ∈ (b) A chain map between two complexes X and Y is a sequence of morphisms f n : Xn Y n → such that d f n = f n+1 d for all n Z. Y ◦ ◦ X ∈ Together the complexes in and their chain maps form an abelian category which we shall C denote by Ch( ). For later results it will often be necessary to place limits on the non-trivial C entries of our complexes, hence the following deﬁnitions.

Deﬁnition 5.2.2. A complex X is bounded below (resp. bounded above) if there exists a natural number a (resp. b) such that Xn = 0 for all nb). A complex is bounded if it is bounded both above and below.

The full subcategories of bounded below, bounded above and bounded complexes will be denoted by Ch+( ), Ch−( )) and Chb( ) respectively. C C C Remark 5.2.3. Given two complexes X and Y we shall often have need to work with sequences of morphisms f n : Xn Y n which are not chain maps; we shall denote the family of such → sequences by HomC(X, Y ).

Deﬁnition 5.2.4. Let X be a complex in Ch( ) and set C Zk(X) = Ker(d : Xk Xk+1) X → Bk(X) = Im(d : Xk−1 Xk). X →

63 Then there is a natural morphism Bk(X) Zk(X), and the kth cohomology of X is deﬁned as → Hk(X) = Coker(Bk(X) Zk(X)). → Deﬁnition 5.2.5. The shift functor T : Ch( ) Ch( ) is the automorphism given by sending C → C a complex X to the complex

T (X)n = Xn+1 d = d , T (X) − X and sending a morphism f : X Y to the morphism → T (f)n = f n+1 : Xn+1 Y n+1. →

Remarks 5.2.6. (a) This is only one of many possible deﬁnitions of the shift functor; fortunately the particular choice of shift functor has no impact on any of the results in this section. (b) As is traditional, we shall often write X[k] and f[k] in place of T k(X) and T k(f).

The following lemma is now a trivial consequence of the deﬁnitions.

Lemma 5.2.7. Let X and Y be two complexes and let f Hom (X, Y ) be a sequence of ∈ C morphisms. Then the sequence of morphisms

[d, f]n = d f n f n+1d : Xn Y n+1 Y − X → deﬁnes a chain map from X to Y [1].

Deﬁnition 5.2.8. Two chain maps f ,f : X Y are chain homotopic if there exists an 0 1 → h Hom (X, Y [ 1]) such that ∈ C −

f f = [d, h] HomCh (X, Y ). 0 − 1 ∈ (C) It follows quite easily from this definition that chain homotopy defines an equivalence relation on chain maps which is compatible with composition. The homotopy category (of complexes) is then defined to be the category K( ) whose objects are the same as Ch( ), but whose morphisms C C are chain homotopy classes of chain maps; the corresponding full subcategories consisting of bounded below, bounded above and bounded complexes will be denoted by K+( ), K−( ) and C C Kb( ) respectively. C Remark 5.2.9. Recall that the opposite category of a given category is the category op with C C the same objects as but with C

HomCop (X, Y ) = HomC(Y,X).

Note that when the category is abelian then the same is true of the opposite category, and C thus it makes sense to consider the various categories Ch∗( op) and K∗( op). C C 64 Now consider the operation which reverses the indexing of a complex, sending Xn to X−n . { } { } An examination of the resulting boundary maps shows that this exchanges objects of Ch( ) and C Ch( op); further consideration of its eﬀect on morphisms reveals that it induces a contravariant C functor Ch( ) Ch( op), or alternatively a covariant functor Ch( )op Ch( op). As a result C → C C → C one obtains categorical equivalences

Ch( )op Ch( op) K( )op K( op) C → C C → C Ch±( )op Ch∓( op) K±( )op K∓( op) C → C C → C Chb( )op Chb( op) Kb( )op Kb( op). C → C C → C While seemingly a lot of abstract flimflam, these identifications are actually rather useful for quickly proving ’dual’ results. With the homotopy category firmly in hand we are finally in a position to formulate our first attempt at defining a bivariant homology for complexes; specifically, one might hope to set

k HH C(X, Y ) = HomK(C)(X, Y [k]). Lest this seem completely oﬀ the wall, it should be noted that these groups have another, perhaps more familiar, formulation. Lemma 5.2.10. For any two complexes X and Y in Ch( ) one has C k Π p q HomK(C)(X, Y [k]) ∼= H (Tot HomC(X , Y )), where TotΠ signiﬁes that we use direct products to form the total complex. What then is the problem with using these groups as our bivariant theory? Anything that might reasonably be called a bivariant homology theory on complexes should at the very least enjoy a few fundamental properties, not the least of which is that given any short exact sequence

0 X Y Z 0 → → → → in Ch( ) one should have long exact sequences C HH k−1(D,Z) HH k (D,X) HH k (D, Y ) HH k (D,Z) HH k+1(D,X) ···→ C → C → C → C → C →··· HH k+1(X,D) HH k (Z,D) HH k (Y,D) HH k (X,D) HH k−1(Z,D) ···→ C → C → C → C → C →··· for every complex D in Ch( ). Unfortunately there is no reason to expect that this should be the k C case when HH C(X, Y ) = HomK(C)(X, Y [k]). One might hope that fact that Hom’s are left exact could somehow be brought to bear on the problem, were it not for some rather fatal diﬃculties. For instance, while our short exact sequence is exact in Ch( ), we are taking the Hom in the C quotient category K( ), where it could very well fail to be exact. In fact the whole issue of exact C sequences in the homotopy category is rather muddled, since while K( ) is easily seen to be an C additive category it is generally not abelian. In particular, most morphisms in K( ) do not have C kernels, a failing which makes short exact sequences rather hard to come by. (Exercise: Show that the chain map Z Z has no kernel in K(Z Mod).) 2 − Dealing with this shortage of short exact sequences in the homotopy category unfortunately requires yet another journey through categorical nonsense.

65 5.3 Triangulated Categories

Let be an additive category endowed with an automorphism T : . Then a triangle in K K → K K is just a sequence of morphisms X Y Z T (X); a morphism between two triangles takes → → → the form of a commutative diagram

X / Y / Z / T (X)

u T (u) X0 / Y 0 / Z0 / T (X0)

Deﬁnition 5.3.1. A triangulated category consists of

an additive category endowed with an automorphism T : , and • K K → K a distinguished family of triangles, called exact triangles, • such that

(TR 0) every triangle isomorphic to an exact triangle is exact,

(TR 1) the triangle X Id X 0 T (X) is exact for every X Obj( ), −→ → → ∈ K

f (TR 2) every morphism f : X Y embeds in some exact triangle X Y Z T (X), → −→ → →

f g g −T (f) (TR 3) X Y Z h T (X) is exact if and only if Y Z h T (X) T (Y ) is exact, −→ −→ −→ −→ −→ −−−−→

(TR 4) every commutative diagram

X / Y / Z / T (X)

u X0 / Y 0 / Z0 / T (X0)

involving two exact triangles extends to a morphism of triangles, and

(TR 5) for every three exact triangles of the form

f X / Y / Z0 / T (X) g Y / Z / X0 / T (Y ) g◦f X / Z / Y 0 / T (X)

there exists a fourth exact triangle

Z0 / Y 0 / X0 / T (Z0)

66 making the diagram

X x FF f xx FF g◦f xx FF xx g FF {xx F# Y + Z vv: AA vv AA vv AA vv AA v T −1(X0) X0 H > H } H } H } H } # 0 0 Z EY [ ] _ a c e2 Y EE yy EE yy EE yy E" |yy T (X) commute.

Remarks 5.3.2. (a) The final property is often referred to as the ‘octahedral axiom’, since if one identifies X with T (X) and T −1(X0) with X0 one obtains the 1-skeleton of an octahedron. For our present purposes (TR 5) will go largely unused; however it plays an essential rôle in the more general applications of triangulated categories. (b) The opposite category op of a triangulated category is naturally triangulated; one K K simply uses T −1 as the translation automorphism and associates an exact triangle

op gop f op T (X) h Z Y X −−→ −−→ −−→ in op to each exact triangle K f g X Y Z h T (X) −→ −→ −→ in . K Deﬁnition 5.3.3. An additive functor F between between triangulated categories is triangulated if F T T F and F sends exact triangles to exact triangles. ◦ ' ◦ While exact triangles are a much weaker notion than exact sequences, they do have at least some properties in common. f g Lemma 5.3.4. Let X Y Z T (X) be an exact triangle in . Then g f =0. −→ −→ → K ◦ Proof. By (TR 1) the triangle X Id X 0 T (X) is exact. It then follows from (TR 4) that −→ → → we can complete the diagram

Id X / X / 0 / T (X)

Id f f g X / Y / Z / T (X) into a morphism of triangles, from which it follows that g f = 0. ◦ 67 Deﬁnition 5.3.5. Let be an abelian category. An additive functor F : is cohomological C K→C if for any exact triangle X Y Z T (X) the induced sequence F (X) F (Y ) F (Z) is → → → → → exact.

In light of Remark 5.1.9, one can view cohomological functors as being the analog of exact functors for a triangulated category.

Proposition 5.3.6. The functor Hom (D, ) : Groups is cohomological for every D K − K → ∈ Obj( ). K f g Proof. Let X Y Z T (X) be an exact triangle in . It follows from Lemma 5.3.4 that −→ −→ → K the composition f g Hom (D,X) ∗ Hom (D, Y ) ∗ Hom (D,Z) K −→ K −→ K is zero; thus all that needs to be shown is that if v : D Y is a morphism such that g v = 0, → ◦ then there is a morphism u : D X such that v = g u. But the fact that g v = 0 means that → ◦ ◦ we can form a commutative diagram

Id D / D / 0 / T (D)

v f g X / Y / Z / T (X).

The combination of (TR 3) and (TR 4) then allows us to ﬁll in the diagram with the desired morphism u.

Corollary 5.3.7. Let X Y Z T (X) be an exact triangle in . Then there is a long → → → K exact sequence

Hom (D,T k−1(Z)) Hom (D,T k(X)) Hom (D,T k(Y )) ···→ K → K → K → Hom (D,T k(Z)) Hom (D,T k+1(X)) K → K →··· for every D Obj( ). ∈ K Remark 5.3.8. Applying the corollary to the opposite category op results in a similar exact K sequence involving Hom (T k( ),D). K − Now let us return to the homotopy category of complexes K( ) and show that it is a trian- C gulated category. While the choice of automorphism T : K( ) K( ) is clear, there is still the C → C small matter of the exact triangles.

Deﬁnition 5.3.9. Let be an abelian category, and let f : X Y be a morphism in Ch( ). C → C The mapping cone of f is the complex M(f) given by

M(f)n = Xn+1 Y n ⊕ d (x ,y ) = ( d (x ), d (y )+ f n+1(x )). f n+1 n − X n+1 Y n n+1

68 Note that the mapping cone of a morphism f : X Y is deﬁned so that it naturally forms → an exact sequence q 0 Y i M(f) X[1] 0; → −→ −→ → combining the morphisms in the exact sequence with the original morphism then produces a f q triangle X Y i M(f) X[1] which we shall call the mapping triangle of f. −→ −→ −→ Deﬁnition 5.3.10. A triangle in K( ) is an exact triangle if and only if it is isomorphic to the C mapping triangle of some morphism f : X Y . → Proposition 5.3.11. With this choice of exact triangles K( ) becomes a triangulated category, C and cohomology becomes a cohomological functor.

Proof. Verifying the axioms (TR 0) – (TR 5) is a straightforward, if tedious, exercise — see [KS90], [Wei94]. It should be noted, however, that it is essential that we are working in the category K( ), and not in Ch( ); many of the necessary diagrams can only be made to commute C C up to homotopy. f Proving that cohomology is a cohomological functor requires one to show that Hk(X) ∗ g f g −→ Hk(Y ) ∗ Hk(Z) is exact for any exact triangle X Y Z X[1]. However in light −→ −→ −→ p→ of axiom (TR 3) it suﬃces to consider the case of Y i M(f) X[1] Y [1], and since p −→ −→ → 0 Y i M(f) X[1] 0 is a short exact sequence in Ch( ) this follows from the existence → −→ −→ → C of the long exact sequence in cohomology.

Corollary 5.3.12. Let X Y Z X[1] be an exact triangle in K( ). Then there is a long → → → C exact sequence

HomK (D,Z[k 1])) HomK (D,X[k]) HomK (D, Y [k]) ···→ (C) − → (C) → (C) → HomK (D,Z[k]) HomK (D,X[k + 1]) (C) → (C) →··· for every D Obj(K( )). ∈ C f g In particular, given a short exact sequence 0 X Y Z 0 one can form the exact → −→ −→ → triangle Y Z M(g) Y [1] and obtain a corresponding long exact sequence → → →

HomK (D,M(g)[k 1])) HomK (D, Y [k]) HomK (D,Z[k]) ···→ (C) − → (C) → (C) → HomK (D,M(g)[k]) HomK (D, Y [k + 1]) . (C) → (C) →··· Still, in spite of all this work we still don’t have the long exact sequence we were looking for in the last section. There is room for hope, however. Note that because X is the kernel of f the morphisms f n+1 0 : Xn+1 Y n+1 Zn combine to give a chain map X[1] M(g); moreover ⊕ → ⊕ → the composition X[1] M(g) Y [1] is simply f[1]. Thus we can form a commutative diagram → →

HomK(C)(D, Y ) / HomK(C)(D,Z) HomK(C)(D,X[1])

HomK(C)(D, Y ) / HomK(C)(D,Z) / HomK(C)(D,M(g)).

69 The problem with this diagram is that it is easy to ﬁnd examples where the map X[1] M(g) → fails to induce an isomorphism HomK (D,X[1]) HomK (D,M(g)), and it is precisely the (C) → (C) lack of such an isomorphism which keeps HomK ( , ) from giving a bivariant cohomology. (C) − − This could be corrected, however, if we were to modify the category K( ) to ensure an inverse C to this morphism.

5.4 Localization of Categories

Deﬁnition 5.4.1. Let be a category. A multiplicative system in is a wide1 subcategory C C S satisfying

(Ore Condition) For every g : X Z in and t : Y Z in S there is a commutative diagram → C → W / Y

s t g X / Z

(Cancellation) For any two morphisms f,g : X Y in the following conditions are → C equivalent:

(i) sf = sg for some s : Y Y 0 in S, and → (ii) ft = gt for some t : X0 X in S. → Deﬁnition 5.4.2. Let be a category, a multiplicative system in . The localization of at C S C C S is the category −1 given by S C Obj( −1 ) = Obj( ) S C C and s f Hom −1 (X, Y )= X W Y : s S / , S C { ←− −→ ∈ } ∼ s f s0 f 0 where X W Y is equivalent to X W 0 Y iﬀ there exists a commutative diagram ←− −→ ←− −→ W O B || BB f || BB ||s BB }|| B! Xo V Y aB t > BB f 0 || BB || 0 BB || s B || W 0 with t in S.

Remarks 5.4.3. 1That is, a subcategory containing the same objects, but possibly fewer morphisms.

70 f (a) The diagram X s W Y will often simply be written as fs−1; for this reason we shall ←− −→ usually refer to morphisms in −1 as (right) fractions. S C (b) While it is relatively straightforward to check that the stated relation on fractions is in fact an equivalence relation, proving that it’s transitive is a bit of a notational quagmire. We shall therefore only note that both the Ore condition and cancellation are needed in the proof, and leave the diagram-building to the reader. (c) To compose two fractions fs−1 : X W Y and gt−1 : Y W 0 Z one ﬁrst uses the ← → ← → Ore condition to construct a diagram

g V / W 0 / Z

t f Xo s W / Y

and then defines the composition to be X V Z. ← → (d) Continuing with our recurring theme of opposite categories, note that a multiplicative system in simultaneously gives a multiplicative system op in op; the result of localizing S C S C op at op is simply the opposite category of −1 . C S S C (e) Finally, it should be noted that there are some set-theoretic difficulties here which we have chosen to sweep under the rug. In particular, when one first defines categories one typically demands that the morphisms between any two objects should be a set; this need not be true of the localization constructed above. However this difficulty can be avoided if the multiplicative system is ‘locally small’ (see [Wei94]), and this holds true in every case of interest.

Example 5.4.4. Let R be a commutative ring and the corresponding single-object category C with End( ) = R. Then any multiplicatively closed subset S R gives a multiplicative system ∗ ∼ ⊆ in , and −1 is simply the single-object category associated to the localization S−1R. S C S C In light of this example one might expect that the localization of any additive category should again be additive. The only obstacle to this is that it is not immediately clear that the Hom-sets −1 −1 of the localization actually form abelian groups. So let f1s1 and f2s2 be two morphisms in

Hom −1 (X, Y ). By applying the Ore condition to s and s we can ﬁnd morphisms g and S C 1 2 ∈C t such that s−1 = gt−1s−1 in the localization, and with this in hand we simply let ∈ S 2 1 −1 −1 −1 −1 −1 −1 f1s1 + f2s2 = f1tt s1 + f2gt s1 −1 = (f1t + f2g)(s1t) .

That this is well-deﬁned follows from an application of the cancellation property, and leaves us with the following lemma.

Lemma 5.4.5. Let be an additive category, a multiplicative system in . Then −1 is also C S C S C an additive category, and the natural functor q : −1 is an additive functor. C → S C 71 Continuing along this line of inquiry, let us now consider a triangulated category with K multiplicative system and attempt to determine when −1 will be triangulated. There is S S K an obvious candidate for exact triangles in the localization — one simply declares a triangle in −1 to be exact if it is isomorphic to an exact triangle in . With this choice of exact triangles S K K most of the requisite properties follow immediately from the corresponding properties for exact triangles in . The only exceptions to this are (TR 4) and (TR 5); however it is easily checked K that for these exact triangles (TR 4) would imply (TR 5), and thus we need only worry about establishing the former. So let us look more closely at property (TR 4). We begin with a commutative diagram of the form X / Y / Z / T (X)

fs−1 gt−1 X0 Y 0 Z0 T (X0) where, without loss of generality, we can assume that the rows are given by exact triangles in . K Deciphering what it means for this diagram to commute shows that there exists a commutative diagram X / Y O `BB O BB s BB t u BB X o W / Y

f g X0 / Y 0 with u , and applying (TR 2) and (TR 4) for triangles in then allows us to form morphisms ∈ S K of triangles X / Y / Z / T (X) (??) O O O O u t v T (u)

W / Y / Z / T (W )

X0 / Y 0 / Z0 / T (X0).

Unfortunately this will generally not give a morphism of triangles in −1 , since the morphism S K v : Z Z need not lie in . However we do obtain the following proposition. → S Proposition 5.4.6. Let be a triangulated category and let be a multiplicative system in K S K such that every morphism of exact triangles

X / Y / Z / T (X)

u t v T (u) X0 / Y 0 / Z0 / T (X0)

72 with u,t must also have v . Then −1 is a triangulated category, and q : −1 is ∈ S ∈ S S K K → S K a triangulated functor.

Proposition 5.4.7. Let be a triangulated category, F : a cohomological functor, and K K→C the wide subcategory consisting of all morphisms in which map to isomorphisms under SF ⊆ K K F . Then is a multiplicative system in , −1 is a triangulated category, and q : −1 SF K SF K K → SF K is a triangulated functor.

Proof. First let us show that is a multiplicative system, starting with the Ore condition. SF Suppose g : X Z and t : Y Z are morphisms in and respectively. By applying (TR 2) → → C SF and (TR 4) we can construct a morphism of triangles

f h◦t W / Z / V / T (W )

s t T (f)

g h X / Y / V / T (X), which after applying F gives a morphism of long exact sequences. Since F (t) and F (Id) both induce isomorphisms the Five Lemma implies the same is true of F (s), and thus s . An ∈ SF identical argument works to prove the same statement with the arrows reversed. The cancellation property is a bit trickier. Note that because we are in an additive category it suﬃces to prove that ft = 0 iﬀ sf = 0. So let f : X Y and t : X0 X be morphisms in → → K and respectively such that ft = 0. Then by applying (TR 2) and (TR 4) we can construct a SF morphism of triangles t g X0 / X / W / T (X0)

f h Id 0 / Y / Y / 0, from which it follows that f = hg and F (W ) = 0. Now let W h Y s Y 0 T (W ) be an −→ −→ → embedding of h into an exact triangle. Then by Lemma 5.3.4 we see that sf = shg = 0, and since F (W ) = 0 we also ﬁnd that s . ∈ SF Finally we are left with the task of proving that the localization is triangulated. Consider a morphism

X / Y / Z / T (X)

u t v T (u) X0 / Y 0 / Z0 / T (X0) of exact triangles in , and suppose s,t . Then applying F results in a morphism of long K ∈ SF exact sequence in which F (s) and F (t) are isomorphisms, and so by the Five Lemma F (u) is an isomorphism as well. Applying the previous proposition now shows that −1 is triangulated. SF K

73 5.5 The Derived Category

Deﬁnition 5.5.1. A chain map f : X Y is a quasi-isomorphism if each of the induced maps → Hn(X) Hn(Y ) are isomorphisms. → Proposition 5.5.2. Let K( ) denote the wide subcategory whose morphisms are the quasi- Q⊆ C isomorphisms. Then is a multiplicative system, and the localization D( )= −1K( ), called Q C Q C the derived category of , is a triangulated category. C Proof. This is an immediate consequence of Proposition 5.4.7.

Remark 5.5.3. An identical construction using the category K+( ) yields the derived category C D+( ); note that one can also realize D+( ) as the full subcategory of D( ) whose objects are C C C the bounded below complexes. Similar considerations produce the derived category D−( ) (resp. C Db( )) from K−( ) (resp. Kb( )). The opposite category correspondence described in previous C C C sections then yields categorical equivalences

D( )op D( op) D±( )op D∓( op) Db( )op Db( op). C → C C → C C → C

Proposition 5.5.4. The groups HomD(C)(X, Y [ ]) form a bivariant cohomology theory on Ch( ) ∗ f g C — that is, given any short exact sequence 0 X Y Z 0 and complex D in Ch( ) one → −→ −→ → C has long exact sequences

HomD (X[k + 1],D) HomD (Z[k],D) HomD (Y [k],D) ···→ (C) → (C) → (C) → HomD (X[k],D) HomD (Z[k 1],D) (C) → (C) − →··· and

HomD (D,Z[k 1]) HomD (D,X[k]) HomD (D, Y [k]) ···→ (C) − → (C) → (C) → HomD (D,Z[k]) HomD (D,X[k + 1]) . (C) → (C) →··· Proof. Both results would follow from Corollary 5.3.7 if we could ﬁnd a morphism Z X[1] → in the derived category making X Y Z X[1] an exact triangle. Now, associated to → → → the chain map g we have a mapping triangle Y Z M(g) Y [1]; rotating this using (TR → → → 3) results in the exact triangle M(g)[ 1] Y Z M(g). Our earlier discussion following − → → → Corollary 5.3.12 resulted in the commutative diagram

X / Y / Z X[1]

M(g)[ 1] / Y / Z / M(g) − The chain map X M(g)[ 1] is generally not invertible in K( ), but since it is a quasi- → − C isomorphism it is invertible in D( ). One can therefore deﬁne the morphism Z X[1] via C → the composition Z M(g) X[1], and since the resulting triangle is isomorphic to an exact → ← triangle it too is exact.

74 th Remark 5.5.5. The group HomD(C)(X, Y [n]) is sometimes called the n hyperext group of X n and Y , and is denoted ExtC (X, Y ).

5.6 Derived Functors

Now that we have an abundance of derived categories one can begin to consider functors between them. In particular, given an additive functor F : there is a corresponding functor C → D K(F ) : K( ) K( ) on the homotopy category, and it is natural to wonder if this process can C → D be extended one step further to give a functor between derived categories.

Deﬁnition 5.6.1. Let F : be an additive functor between abelian categories. A right C → D derived functor for F consists of a triangulated functor RF : D( ) D( ) and a natural C → D transformation s : q K(F ) RF q which is universal among all such pairs. That is, given any ◦ → ◦ functor G : D( ) D( ) and natural transformation s0 : q K(F ) G q there is a unique C → D ◦ → ◦ natural transformation τ : RF G such that s0 = τ s. → ◦ Remark 5.6.2. By reversing the various arrows in this deﬁnition we obtain a notion of left derived functor.

Lemma 5.6.3. RF exists for any exact functor F : . C → D Proof. An exact functor sends quasi-isomorphisms to quasi-isomorphisms; one can therefore simply set RF (fs−1)= RF (f)RF (s)−1.

In general, the key to constructing a derived functor is to leverage the exact functor case — if we can ﬁnd a ‘big enough’ subcategory of on which F is exact then we can construct RF . C Proposition 5.6.4. Let be a full additive subcategory of with the property that for any X I C ∈C there exists a monomorphism X , I with I . Then for any complex X K+( ) there exists → ∈ I ∈ C a quasi-isomorphism X X0 with X0 K+( ). → ∈ I The proposition has the following immediate corollary.

Corollary 5.6.5. Let be as in the proposition. Then the inclusion of in induces an I I C equivalence of categories Q−1K+( ) D+( ), where Q = Q K+( ) denotes the multiplicative I I → C I ∩ I system of quasi-isomorphisms in K+( ). I Deﬁnition 5.6.6. A full additive subcategory of is F -injective if for every short exact I C sequence 0 J A Q 0 with J, A Obj( ) one has that → → → → ∈ I Q is also an object in , and • I the sequence 0 F (J) F (A) F (Q) 0 is exact. • → → → → The category is said to have enough F -injectives if there is an F -injective subcategory C I with the property that for any X there exists a monomorphism X , I with I . ∈C → ∈ I

75 Proposition 5.6.7. Let F : be a left-exact functor. If has enough F -injectives then C → D C there is a derived functor RF : D+( ) D+( ). C → D Proof. Let denote the F -injective subcategory. That F preserves kernels and is F -injective I I allows one to show that F sends acyclic complexes in K+( ) to acyclic complexes in K+( ); I D applying this to the mapping cone of a quasi-isomorphism in K+( ) yields that F sends quasi- I isomorphisms in K+( ) to quasi-isomorphisms in K+( ). From this it follows that one can use I D F to deﬁne a functor −1K+( ) D+( ). But −1K+( ) is equivalent to D+( ) by Corollary QI I → D QI I C 5.6.5, and one can easily check that the composition of this equivalence with the previous functor satisﬁes the requirements for a derived functor of F .

Deﬁnition 5.6.8. An object I in is injective if the functor Hom ( , I) is exact. The category C C − is said to have enough injectives if for every X there is a monomorphism X , I with C ∈ C → I injective. ∈C i q Lemma 5.6.9. Let I be an injective object in . Then any short exact sequence 0 I X C → −→ −→ Q 0 splits; that is, it is isomorphic to the short exact sequence 0 I I Q Q 0. → → → ⊕ → → Proof. Applying the functor Hom ( , I) to the short exact sequence 0 I i X Q 0 C − → −→ → → results in a short exact sequence

∗ 0 Hom (Q, I) Hom (X, I) i Hom (I, I) 0. → C → C −→ C → In particular the map i∗ is surjective, and so there exists a morphism j : X I such that → j i = Id . This in turn allows us to construct a morphism of short exact sequences ◦ I i 0 / I / X / Q / 0

j⊕q 0 / I / I Q / Q / 0. ⊕ But one can easily check that both the kernel and cokernel of the morphism X I Q are trivial, → ⊕ and hence by Lemma 5.1.7 we have constructed an isomorphism of short exact sequences.

Lemma 5.6.10. Let I be an injective object in . Then for any short exact sequence 0 I C → → X Q 0 and additive functor F one has a corresponding short exact sequence 0 F (I) → → → → F (X) F (Q) 0. → → Proof. This is trivially true for split short exact sequences, and by the previous lemma every short exact sequence starting with an injective splits.

In particular we can apply this last lemma to the functor Hom(D, ) for any object D in − C to ﬁnd that the sequence

0 Hom (D, I) Hom (D,X) Hom (D,Q) 0 → C → C → C → is exact.

76 Lemma 5.6.11. If 0 I0 I I00 0 is a short exact sequence in with I0, I injective, → → → → C then I00 is also injective.

Proof. Let 0 J A Q be a short exact sequence in . Then we can combine the two exact → → → C sequences into a diagram

0 0 0

0 0 0 0 / HomC (Q, I ) / HomC(A, I ) / HomC(J, I ) / 0

0 / HomC(Q, I) / HomC(A, I) / HomC(J, I) / 0

00 00 00 0 / HomC(Q, I ) / HomC(A, I ) / HomC(J, I ) / 0 (??)

0 0 0 where, with the possible exception of the starred row, every row and column is exact. It is now a simple exercise in homological algebra to see that this implies the last row is exact as well.

Putting the previous lemmas together results in the following proposition.

Proposition 5.6.12. The injective objects in are an F -injective subcategory for every additive C functor F .

Corollary 5.6.13. If has enough injectives, then every left exact functor F : has a C C → D derived functor RF : D+( ) D+( ). C → D

Remarks 5.6.14. (a) A more direct way to prove the corollary is to show that if contains enough injectives C then D+( ) = K+( ). The essential point is that any quasi-isomorphism in is actually C ∼ I QI invertible within K+( ); hence the localization −1K+( ) is simply K+( ). On the other I QI I I hand since has enough injectives it follows from Corollary 5.6.5 that −1K+( ) = D+( ); C QI I ∼ C combining these two results shows D+( ) is equivalent to K+( ). C I (b) Reversing the arrows in the deﬁnition of an F -injective subcategory gives the conditions for an F -projective subcategory. Carrying through the same discussion for F -projective subcategories allows one to show that if F is a right exact functor and has enough C F -projectives then there is a corresponding derived functor denoted LF .

5.7 Example : Modules

To make the abstract nonsense of the preceding sections somewhat more tangible we shall take a brief look at the category of R-modules, R a unital ring; this will also provide us with an

77 opportunity to illustrate some of the techniques used to show that an abelian category has enough injectives.

Proposition 5.7.1. The category of Z-modules has enough injectives.

For our purposes the proof of this proposition is not particularly enlightening — see [Wei94, 2.2 & 2.3] for the details. What is more interesting is way in which this proposition can be § leveraged to prove the same result for modules under an arbitrary ring; to this end the machinery of adjoint functors proves quite helpful.

Deﬁnition 5.7.2. Let and be additive categories. A pair of additive functors F : C D C → D and G : are adjoint if there is a natural isomorphism D→C φ : Hom (F (X), Y ) Hom (X, G(Y )) X,Y D → C for every X Obj and Y Obj . One calls F the left adjoint of G and G the right adjoint of ∈ C ∈ D F . Associated to every adjoint pair one has a unit and a counit of adjunction; these are the natural transformations η : Id G F and : F G Id given by the morphisms C → ◦ ◦ → D η = φ (Id ) : X GF (X) X X,F (X) F (X) → and = φ−1 (Id ) : F G(Y ) Y. Y G(Y ),Y G(Y ) → Lemma 5.7.3. Let and be abelian categories, and let G : be an additive functor C D D→C which is right adjoint to an exact functor. Then G sends injective objects of to injective objects D of . C Proof. Denote the left adjoint of G by F and let I be an injective object in . Our goal is to C prove that G(I) is also injective — that is, that Hom ( , G(I)) is an exact functor on . But C − C Hom ( , G(I)) is naturally isomorphic to Hom (F ( ), I), and since both F and Hom ( , I) C − D − D − are exact, so is their composition.

To bend these ideas to the task at hand, consider the functor

HomZ(R, ) : Z Mod R Mod. − − → − Given an R-module M and a Z-module A it is a simple matter to construct an natural isomorphism HomZ(M, A) ∼= HomR(M, HomZ(R, A)); in other words, HomZ(R, ) is the right adjoint of the forgetful functor − For : R Mod Z Mod. − → −

But the forgetful functor is exact, and thus HomZ(R, ) carries injective Z-modules to injective − R-modules.

78 Proposition 5.7.4. Let R be a unital ring. Then the category of R-modules has enough injectives.

Proof. Let M be an R-module. Since Z-Mod has enough injectives there exists an injective Z- module I and a Z-linear injection M , I. We can then apply the left-exact functor HomZ(R, ) → − to obtain an injection HomZ(R,M) , HomZ(R, I). Finally, there is an obvious inclusion of M → into HomZ(R,M) given by

M HomZ(R,M) → m r r m ; 7→ { 7→ · } composing this with the previous injection gives the desired embedding of M in an injective R-module.

Corollary 5.7.5. Every left exact functor on R-modules has a derived functor.

Remark 5.7.6. Chasing through the details reveals that inclusion of M into HomZ(R,M) used above is none other than the unit of adjunction. The proof can therefore be distilled to the following essentials points :

There is an exact functor F : R Mod Z Mod. • − → − There is a left-exact functor G : Z Mod R Mod. • − → − F is the left adjoint of G, and the unit of adjunction M GF (M) is injective for every • → R-module M.

While the categories of sheaves in the next chapter are substantially more general than the categories of modules discussed here, these exact same steps will allows us to conclude that they too have enough injectives.

79 Chapter 6

Equivariant Sheaf Theory

6.1 Sheaves

Let G be a countable discrete group and X be a G-space. Our ﬁrst task is to construct a category which encodes at least part of the equivariant topology of X. To this end, let p (X) be the O G category whose objects are the open subsets of X and for which

Mor(V,U) : γ G : γ−1 V U . { ∈ · ⊆ } Note that if γ Mor(W, V ) and γ Mor(V,U) then γ γ Mor(W, U), and we can W V ∈ V U ∈ W V V U ∈ therefore deﬁne γ γ = γ γ . V U ◦ W V W V V U Example 6.1.1. p (pt) is simply Gop, where we view G as the category with a single object O G such that End( )= G. In fact, whenever the G-action on X is trivial one can identify p (X) ∗ ∗ O G with p(X) Gop. O × Deﬁnition 6.1.2. Let k be a commutative ring. A G-presheaf of k-modules on X is a contravariant functor from p (X) to k-Mod. A morphism between two presheaves is simply a O G natural transformation of functors.

Example 6.1.3. The assignment (U)= C(U), where C(U) denotes the continuous complex- CX valued functions on U, deﬁnes a G-presheaf on X. Indeed, given two open sets U, V and an CX element γ G such that γ−1 V U one has the morphism C(U) C(V ) given by sending a ∈ · ⊆ → function f C(U) to (γ f) . ∈ · |V

We shall denote the category of G-presheaves of k-modules over X by PShG(X; k); a little work shows that it inherits the structure of an abelian category from k-Mod.

Example 6.1.4. In light of Example 6.1.1 we see that when X has a trivial G-action then

PShG(X; k) is equivalent to PSh(X; k[G]); in particular PShG(pt; k) is equivalent to the category of k[G]-modules.

Let F be a G-presheaf on X and let U be an open subset of X. An element s F (U) is called ∈ a section of F over U; one often writes Γ(U, F ) instead of F (U) for the k-module of sections over U. If V U is an open subset then one has a ’restriction’ morphism ρ : F (U) F (V ) ⊆ V U → corresponding to the fact that (Id)−1V = V U; given a section s F (U) one usually writes ⊂ ∈ s for the section ρ (s). |V V U For any point x X the stalk of F at x is given by limit ∈

Fx = lim F (U), →U

80 where U runs through all open neighborhoods of x and one considers only those morphisms coming from restriction; note that Gx acts on this system, so that Fx is naturally a k[Gx] module. If s Γ(U, F ) is a section of F over some neighborhood U of x then the image of s in ∈ Fx is called the germ of s at x, and is denoted by sx. The support of G-sheaf F on X is deﬁned to be the closed set supp(F )= x X : F =0 . Similarly, one deﬁnes the support of a section { ∈ x 6 } s Γ(U, F ) to be the set supp(s) = x U : s = 0 . Finally, if φ : F F 0 is a morphism ∈ { ∈ x 6 } → of sheaves then the universal property of limits yields a morphism φ : F F 0 which will be x x → x called the germ of the morphism φ at x. Unfortunately the stalks of a G-presheaf generally say very little about the G-presheaf itself. For example when X is non-trivial the presheaf given by

k if U = X, F (U)= 0 otherwise  will have only trivial stalks; from this we see that one cannot decide the triviality of a presheaf using only the local information contained in the stalks.

Deﬁnitions 6.1.5. (a) Let F be a G-presheaf on X and U be a collection of open subsets of X. A family { α}α∈I of sections s F (U ) is said to be compatible if it satisﬁes { α ∈ α } s = s α|Uα∩Uβ β|Uα∩Uβ for every α, β . ∈ I (b) A G-presheaf F on X is called a G-sheaf if for every collection U of open sets in X and { α} every compatible family of sections s F (U ) there exists a unique section s F ( U ) { α ∈ α } ∈ ∪ α with s = s . |Uα α

Let ShG(X; k) denote the full subcategory of PShG(X; k) consisting of G-sheaves, so that a morphism of G-sheaves is simply a morphism of the underlying G-presheaves. As a full subcategory of an additive category ShG(X; k) is necessarily additive; note that the same reasoning does not allow us to conclude that it is abelian.

Examples 6.1.6. (a) The G-presheaf of continuous functions deﬁned in Example 6.1.3 is a G-sheaf. CX (b) Let M be a k-module (or more generally a k[G]-module). The G-presheaf which assigns M to every open set is generally not a G-sheaf.

modules. For a proper subgroup H < G it is not the case that PShG(G/H; k) is equivalent

to the category of k[H]-modules; on the other hand one can identify ShG(G/H; k) with k[H]-Mod.

Lemma 6.1.7. Let E and F be two G-sheaves of k-modules over X, φ : E F a morphism → and U X an open set. ⊆ 81 (a) If φ is the zero morphism for all x U, then φ(U) : E(U) F (U) is the zero morphism. x ∈ → (b) If φ is injective for all x U, then φ(U) : E(U) F (U) is injective. x ∈ → (c) If φ is an isomorphism for all x U, then φ(U) : E(U) F (U) is an isomorphism. x ∈ → Remark 6.1.8. Note that there is no statement regarding surjections — this absence is ultimately what makes sheaf cohomology a non-trivial invariant.

Proof. We shall only establish the first assertion; the proofs of the other two statements are similar. Suppose φ is zero morphism for every x U and let s F (U) be a section over U. x ∈ ∈ Then since φ (s) is zero one can find a neighborhood V U of x such that φ(U)(s) = 0; as x ⊆ |V this can be done for every x U it follows from the sheaf property that φ(U)(s)=0. ∈ An essential consequence of Lemma 6.1.7 is that one can naturally identify F (U) with a particular submodule of Fx. Namely, let F (U) consist of those elements sx such that for x∈U S every x U there exists anQ open neighborhood V U and section t F (V ) withQ sv = tv for all ∈ ⊂ ∈ t V . It is easily checked that F is a G-sheaf; moreover the obvious morphism F F given ∈ S → S by sending s F (U) to s is a stalkwise isomorphism. It then follows immediately from the ∈ x lemma that F = F . ∼ S Q Note that with the exception of this final claim these arguments apply equally well when F is a G-presheaf. Indeed, we can construct the G-sheaf F just as above, and there will still be a S stalkwise isomorphism F F . Moreover, since a morphism of G-presheaves F F 0 induces → S → a natural morphism F F 0, we actually find ourselves with a functor : PSh (X; k) S → S S G → ShG(X; k). Proposition 6.1.9. is a left adjoint to the forgetful functor For : Sh (X; k) PSh (X; k); S G → G that is, the natural morphism i : E E induces an isomorphism E → S ∼ i∗ : Hom ( E, F ) = Hom (E, ForF ) E ShG(X;k) S −→ PShG(X;k) for any E PSh (X; k) and F Sh (X; k). ∈ G ∈ G ∗ Proof. We need only prove that the induced map iE is an isomorphism. Injectivity follows immediately from the commutative diagram

i∗ Hom ( E, F ) F / Hom (E, ForF ) ShG(X;k_ ) S PShG(X;k)

Hom (E , F ) Id Hom (E , F ), k[Gx] x x / k[Gx] x x x∈X x∈X Q Q since the leftmost arrow is injective by Lemma 6.1.7. As for surjectivity, consider the homomorphism j : Hom (E, ForF ) Hom ( E, F ) given by the composition E PShG(X;k) → ShG(X;k) S i−1 Hom (E, ForF ) S Hom ( E, F ) F ∗ Hom ( E, F ), PShG(X;k) −→ ShG(X;k) S S −−→ ShG(X;k) S where the last homomorphism relies on the fact that i : F F is an isomorphism for any F → S sheaf F . Chasing through the deﬁnitions yields that i j is the identity map, and it follows E ◦ E that iE is surjective.

82 Example 6.1.10. Let M be a k-module (or more generally a k[G]-module). The constant G- sheaf with stalk M is the G-sheaf MX associated to the G-presheaf which assigns M to every open set. Unlike the original G-presheaf, MX does not assign M to every open set; rather, it assigns one copy of M for every connected component of U.

As mentioned earlier, it is clear that ShG(X; k) is an additive category. What is unclear, however, is whether it is abelian — the kernels and cokernels of morphisms in ShG(X; k) might only exist in PSh (X; k). So let φ : E F be a morphism of sheaves in Sh (X; k) and let G → G KerPShφ and CokerPShφ denote the kernel and cokernel of φ when viewed as a map of presheaves.

If KerPSh and CokerPSh happen to be sheaves then everything is ﬁne — Sh(X; k) will be an abelian category. Unfortunately, while one can quite easily show that KerPShφ is a sheaf, the same is not true of Coker φ. However, thanks to the previous proposition the sheaf Coker φ will PSh S PSh serve just as well as a cokernel. Indeed, in order for Coker φ to be a cokernel in Sh (X; k) S PSh G we must show that

0 Hom ( Coker φ, A) Hom (F, A) Hom (E, A) → ShG(X;k) S PSh → ShG(X;k) → ShG(X;k) is exact for every A Sh (X; k). But by the previous proposition we have a commutative ∈ G diagram

0 / HomSh (X;k)( CokerPShφ, A) / HomSh (X;k)(F, A) / HomSh (X;k)(E, A) G S G G ∼= ∼= =∼ 0 / HomPShG(X;k)(CokerPShφ, A) / HomPShG(X;k)(F, A) / HomPShG(X;k)(E, A) where all the vertical maps are isomorphisms and the bottom row is exact (since CokerPShφ is a cokernel in PShG(X; k)).

The only remaining obstacle to ShG(X; k) being an abelian category is whether the morphism from coimage to image is always an isomorphism, the veriﬁcation of which we leave to the reader (it follows quickly from the fact that is a functor). This leaves us with the following proposition. S

Proposition 6.1.11. ShG(X; k) is an abelian category for every topological space X and commutative ring k.

Remark 6.1.12. Note that it follows from Lemma 6.1.7 that exactness in ShG(X; k) can be determined at the level of stalks; as a result the functor : PSh (X; k) Sh (X; k) is exact. S G → G + With this proposition in hand we can proceed to form the category ChG(X; k) of (bounded- + below) complexes of G-sheaves, and this in turn leads us to the homotopy category KG(X; k). Chasing through the abstract nonsense of the previous chapter then leaves us with the following deﬁnition.

Deﬁnition 6.1.13. Let F be a complex of G-sheaves on X. The nth cohomology sheaf is the G- sheaf n(F ) associated to the presheaf which assigns Hn(F (U)) to the open set U. A morphism H F F 0 between two complexes G-of sheaves over X is a quasi-isomorphism if the induced map → on cohomology sheaves is an isomorphism.

83 Remark 6.1.14. In particular the stalk n(F ) of the cohomology sheaf at x X is isomorphic H x ∈ n to the cohomology H (Fx), and thus quasi-isomorphisms are determined at the level of stalks.

Continuing in our application of ideas from the previous chapter we can now invert the quasi- + isomorphisms within the homotopy category to obtain the derived category DG(X; k).

Example 6.1.15. Let H G be a subgroup. As noted in earlier the category ShG(G/H; k) ≤ + is equivalent to the category of k[H]-modules, from which we can conclude that DG(G/H; k) is equivalent to D+(k[H] Mod). −

6.2 Direct and Inverse Images

Deﬁnition 6.2.1. Let f : X Y be a G-map between two G-spaces. →

(a) Let E be a G-sheaf on X. The direct image of E under f, denoted f∗F , is the G-sheaf on Y given by −1 (f∗E)(U)= E(f (U)).

(b) Let F be a G-sheaf on Y . The inverse image (or pullback) of F over f, denoted f ∗F , is the G-sheaf on X associated to the presheaf

V lim F (U), 7→ U where the limit ranges over those open sets U containing f(V ) and one considers only those morphisms coming from restriction.

Note that f and f ∗ extend in an obvious way to give functors f : Sh (X; k) Sh (Y ; k) ∗ ∗ G → G and f ∗ : Sh (Y ; k) Sh (X; k). G → G Example 6.2.2. Let X be a G-space and let a : X pt be the obvious G-map. Recall that X → we can identify ShG(pt; k) with k[G]-Mod.

∗ (a) Let M be a k[G]-module. Then the constant G-sheaf MX is isomorphic to aX M. (b) The global sections functor Γ( )=Γ(X, ) from G-sheaves to k[G]-modules is equivalent − − to the direct image functor aX∗. (c) For any point x X let i : G x X denote the inclusion of the corresponding orbit. ∈ x · → Then the functor i∗ : Sh (X; k) Sh (G x; k) = k[G ] Mod associates to each G-sheaf x G → G · ∼ x − F the stalk Fx.

Recall that a subset A X is locally closed if it is the intersection of an open and closed set ⊆ in X.

Deﬁnition 6.2.3. Let i : A X denote the inclusion of a G-invariant locally closed subset and → let F be a G-sheaf on X. The restriction of F to A is the G-sheaf F = i∗F . |A The following lemma is left as an exercise to the reader (alternatively, see [Ive86]).

84 Lemma 6.2.4. Let f and g be two composable G-maps. Then

(a) (f g)∗ = g∗ f ∗, ◦ ◦ (b) (f g) = f g , ◦ ∗ ∗ ◦ ∗ ∗ (c) f is the left adjoint of f∗,

(d) f∗ is left exact, and (e) f ∗ is exact.

Corollary 6.2.5. Let f : X Y be a G-map. Then the direct image functor f is sends → ∗ injectives to injectives.

Proof. This follows from the fact that f∗ is the right adjoint of the exact functor f∗ (see Lemma 5.7.3).

Corollary 6.2.6. ShG(X; k) has enough injectives.

Proof. Let F be a G-sheaf on X; we must construct an injective G-sheaf I on X together with a monomorphism F , I. Choose a point x X and let i : G x , X denote the inclusion → ∈ x · → of the corresponding orbit. Since the category of k[Gx]-modules has enough injectives we can ﬁnd a monomorphism F , I for some injective k[G ]-module I . By identifying k[G ]-Mod x → x x x x with Sh (G x; k) we can then apply the direct image functor i ; since i is left exact and G · x∗ x∗ preserves injectives it follows that we have a monomorphism i F , i I with i I is an x∗ x → x∗ x x∗ x injective G-sheaf on X. Finally let I = Ix. Then I is an injective G-sheaf on X and the [x]∈X/G composition Q F , i F , i I = I → x∗ x → x∗ x x∈YX/G x∈YX/G is the desired inclusion.

It follows from this last corollary that every left exact functor on ShG(X; k) has a derived functor; in particular there are derived functors Rf for every G-map f : X Y . On the other ∗ → hand since f ∗ is an exact functor we also have derived functors f ∗ = Rf ∗.

Deﬁnition 6.2.7. The sheaf cohomology of X with coeﬃcients in k is the Z-graded abelian group ∗ ∗ H (X; k)= H (RΓ(kX )).

Lemma 6.2.8. Let f : Y Z and g : X Y be two composable G-maps. Then → → R(f g) = Rf Rg . ◦ ∗ ∗ ◦ ∗ Proof. Let F be a complex of G-sheaves on X and let F I be an injective resolution of F ; → that is, a quasi-isomorphism from F to a complex of injective G-sheaves. Then Rg∗(F )= g∗(I).

Note that since g∗ sends injectives to injectives it follows that g∗(I) is a complex of injective G-sheaves on Y . Hence

Rf (Rg (F )) = Rf (g (I)) = f (g (I)) = (f g) (I) = R(f g) (F ). ∗ ∗ ∗ ∗ ∗ ∗ ◦ ∗ ◦ ∗

85 Lemma 6.2.9. Let f : X Y be a G-map. Then f ∗ : D+(Y ; k) D+(X; k) is the left adjoint → G → G of Rf∗; that is, there is a natural isomorphism

∗ ∼= HomD+ (X;k)(f F, E) HomD+ (Y ;k)(F, Rf∗E) G −→ G for every E D+(X; k) and F D+(Y ; k). ∈ G ∈ G Proof. Let I be an injective resolution of E, so that

∗ ∗ HomD+ (f F, E) = HomK+ (f F, I). G(X;k) ∼ G(X;k)

∗ Since f is the left adjoint to f∗ this can then be rewritten as

∗ HomD+ (f F, E) = HomK+ (F,f∗I), G(X;k) ∼ G(Y ;k) which because f∗ preserves injectives simply amounts to

∗ HomD+ (f F, E) = HomD+ (F,f∗I). G(X;k) ∼ G(Y ;k)

But Rf∗E is naturally isomorphic to f∗I, and the result follows.

Finally, let us conclude this section by using the direct and inverse image functors to understand the relationship between Sh (X; k) and Sh(X/G; k). Let q : X X/G denote the quo- G X → tient map. Because X/G is a trivial G-space the natural inclusion triv : k Mod , k[G] Mod G − → − can be used to obtain an inclusion triv : Sh(X/G; k) , Sh(X/G; k[G]) = Sh (X/G; k); com- G → G ∗ posing this with the inverse image functor qX results in

Λ = q∗ triv : Sh(X/G; k) Sh (X; k). X X ◦ G → G On the other hand one can also use the G-invariants functor inv : k[G] Mod k Mod G − → − to deﬁne a corresponding functor inv : Sh (X/G; k) = Sh(X/G; k[G]) Sh(X/G; k); applying G G → this after the direct image functor qX∗ yields

Q = inv q : Sh (X; k) Sh(X/G; k). X G ◦ X∗ G →

Lemma 6.2.10. ΛX is the left adjoint of QX .

∗ Proof. We have already seen that qX is the left adjoint of qX∗; it is a simple exercise to check that triv is the left adjoint to inv . It follows that q∗ triv is the left adjoint of inv q . G G X ◦ G G ◦ X∗ Proposition 6.2.11. If X is a free and proper G-space then the (co)units of adjunction

Id Q Λ and Λ Q Id Sh(X/G;k) → X ◦ X X ◦ X → ShG(X;k) are natural isomorphisms, and thus Q : Sh (X; k) Sh(X/G; k) is equivalence of abelian X G → categories.

86 Proof. Recall that in order to prove that a morphism of sheaves is an isomorphism it suﬃces to prove it is an isomorphism locally. But both QX and ΛX are well-behaved under restriction to G-invariant open sets; that is, if U X is a G-invariant open set then one has a commutative ⊆ diagram

QX ΛX ShG(X; k) / Sh(X/G; k) / ShG(X; k)

QU ΛU ShG(U; k) / Sh(U/G; k) / ShG(U; k).

Since X is free and proper it therefore suﬃces to consider only the case when X = G Z, for × which the result follows by explicit computation.

6.3 Borel Cohomology

Having worked so hard to develop the semi-abstract nonsense of the previous sections, let us begin to put it to work by deﬁning some invariants of G-spaces.

Deﬁnition 6.3.1. The Borel G-cohomology of X with coeﬃcients in k is the Z-graded abelian group

n HG(X; k) = HomD+ (kX , kX [n]) G(X;k) n = ExtShG(X;k)(kX , kX ).

As noted earlier the category of k-sheaves over a point is nothing more than the category of ∗ ∗ ∗ k[G]-modules, and so it follows that one can identify HG(pt; k) with Extk[G](k, k) ∼= Hgrp(G; k), where the last expression denotes the group cohomology of G. On the other hand, the following lemma shows that Borel cohomology reduces to sheaf cohomology whenever the group is trivial; in this way Borel cohomology serves to unify both group and sheaf cohomology into a single theory.

Lemma 6.3.2. Let ΓG : Sh (X; k) k Mod denote the functor which gives the G-invariant G → − global sections of a sheaf. Then

∗ ∗ G HG(X; k)= H (RΓ (kX )).

Proof. First observe that one can use the adjunction between the inverse and direct image functors to rewrite the deﬁnition of Borel cohomology as

n HG(X; k) = HomD+ (kX , kX [n]) G(X;k) ∗ = HomD+ (aX k, kX [n]) G(X;k)

= HomD+(k[G])(k, RaX∗(kX )[n])

= HomD+(k[G])(k, RΓ(kX )[n]).

87 Now suppose I is an injective resolution of kX ; then because Γ sends injectives to injectives one has

HomD+(k[G])(k, RΓ(kX )[n]) = HomD+(k[G])(k, Γ(I)[n])

= HomK+(k[G])(k, Γ(I)[n]) n G n G = H (Γ (I)) = H (RΓ (kX )).

Note that it follows immediately from the original deﬁnition that there is an associative ‘cup’ product Hp (X; k) Hq (X; k) Hp+q(X; k) G ⊗ G → G given by composition of morphisms in the equivariant derived category. What is not so obvious is that this cup product is actually graded-commutative; we shall defer the proof of this fact until the end of the section, when it shall follow from the analogous result in non-equivariant cohomology.

Lemma 6.3.3. Borel G-cohomology is a contravariant functor on G-spaces; that is, for any G-map f : X Y there is a corresponding ring homomorphism → f ∗ : H∗ (Y ; k) H∗ (X; k) G → G such that for any two composable G-maps f and g one has (f g)∗ = g∗ f ∗. ◦ ◦ Proof. As we saw earlier the functor f ∗ : Sh (Y ; k) Sh (X; k) is exact, and so descends to G → G give a functor f ∗ : D+(Y ; k) D+(X; k). This in turn induces a homomorphism G → G n ∗ ∗ HG(Y ; k) = HomD+(Y ;k)(kY , kY [n]) HomD+ (X;k)(f kY ,f kY [n]), G → G

∗ ∗ but since f kY = kX this last expression is simply HG(X; k). The functorial nature of this homomorphism then ensures that it respects cup products, while the composition property follows from the analogous property for the inverse image functor.

Lemma 6.3.4. For any free and proper G-space X there is a natural ring isomorphism

∗ ∗ HG(X; k) ∼= H (X/G; k).

Proof. The equivalence of categories Q : Sh (X; k) Sh(X/G; k) given in 6.2.11 sends k to X G → X kX/G, and hence identiﬁes HomD+ (kX , kX [n]) with HomD+(X/G;k)(kX/G, kX/G[n]). G(X;k)

Proposition 6.3.5. For every G-space X there is a convergent ﬁrst-quadrant spectral sequence

Epq = Hp(G; Hq(X; k)) Hp+q(X; k). 2 ⇒ G Proof. In the proof of Lemma 6.3.2 we saw that

∗ ∗ HG(X; k) ∼= Extk[G](k, RΓ(kX )).

88 But because Γ( ) sends injectives to injectives this yields a Grothendieck spectral sequence − [Wei94, 10.8] § Epq = Extp (k, Hq(RΓ(k ))) Extp+q (k, RΓ(k )). 2 k[G] X ⇒ k[G] x q q p q But H (RΓ(kX )) is simply the sheaf cohomology H (X; k), while Extk[G](k, H (X; k)) is by deﬁnition the group cohomology Hp(G; Hq(X; k)).

Corollary 6.3.6 (Weak invariance). Let f : X Y be a G-map which induces an iso- → morphism on non-equivariant cohomology. Then the corresponding map on the level of Borel G-cohomology is also an isomorphism.

Proof. It follows from the assumptions that f induces an isomorphism between the E2-terms of the spectral sequences given by Proposition 6.3.5, after which a standard argument shows that the morphism between their limits is also an isomorphism.

∗ We are now ﬁnally in position to explain the reasoning behind calling the groups HG(X; k) the Borel G-cohomology. Recall that for every group G there is a unique (up to homotopy) contractible free G-CW complex EG.

Proposition 6.3.7. For every G-space X one has a natural ring isomorphism

∼ H∗ (X; k) = H∗(EG X; k). G −→ ×G

Remark 6.3.8. The process of converting a G-space X into the space EG X is known as ×G the ”Borel construction”, and the groups H∗(EG X; k) are what is commonly known as the ×G Borel equivariant cohomology of X. Note that since the cup product in sheaf cohomology is graded-commutative, the proposition shows that the same is true of Borel G-cohomology.

Proof. Since EG is contractible the projection EG X X is a homotopy equivalence; in ∼ × → particular it induces an isomorphism H∗(X; k) = H∗(EG X; k). It then follows from the weak −→ × ∼ invariance of Borel G-cohomology that this projection also induces an isomorphism H∗ (X; k) = G −→ H∗ (EG X; k), and since EG X is a free and proper G-space this last ring is isomorphic to G × × H∗(EG X; k). ×G

Corollary 6.3.9. For every proper G-space X one has a natural ring isomorphism

∗ C ∗ C HG(X; ) ∼= H (X/G; ).

Proof. Let π : EG X X/G be the obvious projection. In light of the previous proposition, it ×G → suffices to show that the corresponding ring homomorphism π∗ : H∗(X/G; C) H∗(EG X; C) → ×G is an isomorphism. But since the fiber of π over [x] X/G can be identified with the classifying ∈ space BGx (with Gx finite) one finds that the fibers of π are rationally acyclic. The Vietoris-Begle mapping theorem [Ive86, IV.1.2] then says that π∗ is an isomorphism. §

89 6.4 Proper Supports

Unless otherwise noted we shall henceforth assume all spaces to be locally compact.

Deﬁnition 6.4.1. Let f : X Y be a G-map and let F be a G-sheaf on X. The direct image → with proper supports of F under f is the G-sheaf f!F on Y given by

(f F )(U)= s F (f −1(U)) f : supp(s) U is proper . ! { ∈ | → }

Examples 6.4.2. (a) Let f : X Y be a proper G-map. Then f = f . → ! ∗ (b) Let X be a G-space, U X an open subset and F a G-sheaf on X. We shall say that a ⊆ section s Γ(U; F ) is compactly supported whenever supp(s) is compact; let Γ (U; F ) ∈ c ⊆ Γ(U; F ) denote the submodule of compactly supported sections over U. Then one has

Γc(X; F )= aX!F,

where a : X pt denotes the obvious G-map. X → Deﬁnition 6.4.3. Let F be a G-sheaf on X and i : A , X the inclusion of a G-invariant → locally closed subset; by pulling back and then (properly) pushing forward we obtain a G-sheaf ∗ FA = i!i F on X.

Remark 6.4.4. When dealing with a constant G-sheaf F = MX we shall use MXA to denote the G-sheaf FA.

∗ ∗ Note that when Z is a G-invariant closed subset of X one has FZ = i!i F = i∗i F ; as a result the unit of adjunction for the adjoint pair (i∗,i ) furnishes a map φ : F F . Chasing ∗ Z → Z through the deﬁnitions then yields that the germ φ : F F is an isomorphism for x Z, Zx x → Zx ∈ while F = 0 for x / Z. Zx ∈ On the other hand if U is a G-invariant open subset of X then

F (V )= s F (U V ) supp(s) is closed in V . U { ∈ ∩ | } But thanks to the sheaf property any section in F (U V ) whose support is closed in V has a ∩ unique extension by zero to a section in F (V ), and this yields an injection φ : F , F . A little U U → more work shows that the germ φ : F F is an isomorphism for x U, while F = 0 for Ux Ux → x ∈ Ux x / U; combining this with our previous observation regarding invariant closed subsets results ∈ in the following lemma.

Lemma 6.4.5. Let Z be a G-invariant closed subset of X and let U = X Z be its complement. \ Then there is a short exact sequence of G-sheaves

0 F F F 0. → U → → Z →

The explicit description of the sheaf FU given above also makes it clear that the functor F F is exact. 7→ U 90 Lemma 6.4.6. Let X be a G-space, U X a G-invariant open set. Then the functor ⊆ Sh (X; k) Sh (X; k) G → G F F 7→ U is exact.

The following two lemmas are left as exercises to the reader; the arguments are similar to those for the functor f∗.

Lemma 6.4.7. Let f and g be two composable G-maps. Then

(f g) = f g . ◦ ! ! ◦ ! Lemma 6.4.8. Let f : X Y be a G-map. Then f extends to a left exact functor f : → ! ! Sh (X; k) Sh (Y ; k). G → G It follows from the last lemma that there are derived functors Rf : D+(X; k) D+(Y ; k). ! G → G Unfortunately, unlike the ordinary direct image the functors f! do not generally send injectives to injectives. As a result, when working with direct images with proper support we will ﬁnd it more convenient to work with a slightly larger class of sheaves than the injectives.

Deﬁnition 6.4.9. A G-sheaf F on X is c-soft if the restriction morphism ρ : Γ(X; F ) KX → Γ(K; F ) is surjective for every compact K X. ⊆ Remark 6.4.10. Being c-soft has very little to do with the G-action on the sheaf; a G-sheaf is c-soft if and only if it is c-soft when viewed as a (non-equivariant) sheaf.

Examples 6.4.11. (a) When X is discrete every sheaf is c-soft. (b) is c-soft by the Tietze extension theorem; if X is a smooth G-manifold then the same CX applies to the G-subsheaf ∞ of smooth functions. More generally, the G-sheaf Ωn CX ⊆CX M of smooth diﬀerential n-forms is also c-soft.

Lemma 6.4.12. A G-sheaf F on X is c-soft if and only if the restriction morphism Γ (X; F ) c → Γ (Z; F ) is surjective for every closed Z X. c ⊆ Proof. Suppose the surjectivity condition holds; then in particular it holds when Z = K is compact. But in that case Γ (K; F ) = Γ(K; F ), and since Γ (X; F ) Γ(X; F ) surjects onto c c ⊆ Γ(K; F ), so must Γ(X; F ). As for the converse, suppose F is c-soft and let s Γ (Z; F ) be a section with compact ∈ c support K. Since X is locally compact we can ﬁnd a precompact neighborhood U of K. Deﬁne an extension s Γ(∂U (Z U); F ) of s by s = 0 and s = s . Since F is c-soft and ∈ ∪ ∩ |∂U |Z∩U |Z∩U ∂U (Z U) is closed this can be further extended to a global section on X; moreover since ∪ ∩ s = 0 thisb global extension can be chosenb with supportb inside U. |∂U b 91 Corollary 6.4.13. Let F be a c-soft G-sheaf on X.

(a) If Z is a locally closed G-invariant subset of X then F is c-soft. |Z (b) If f : X Y is a G-map then f F is c-soft. → ! (c) If Z is a locally closed G-invariant subset of X then FZ is c-soft.

Proof.

(a) If Z is open the result is trivial, so suppose Z is closed. Let K be a compact subset of Z;

then K is also a compact subset of X. Since F is c-soft the restriction ρKX is surjective;

however since ρKX = ρKZ ρZX then ρKZ must also be surjective. ◦ −1 (b) Let K be a compact subset of Y . Then Γc(K; f!F )=Γc(f (K); F ), while Γc(Y ; f!F ) = Γ (X; F ). But by Lemma 6.4.12 the restriction Γ (X; F ) Γ (f −1(K); F ) is surjective. c c → c (c) Since F = i (F ), this follows from (a) and (b). Z ! |Z

Proposition 6.4.14. Let 0 F 0 F F 00 0 be a short exact sequence of G-sheaves on X → → → → with F 0 c-soft. Then the sequence

0 Γ (X; F 0) Γ (X; F ) Γ (X; F 00) 0 → c → c → c → is exact.

Proof. The presence of a G-action is completely irrelevant to the conclusion, and so it suffices to consider the case when G is trivial. We shall begin by assuming that X is compact. Let s00 Γ(X; F 00); we must show that s00 ∈ lies in the image of Γ(X; F ). At the very least since the sheaf map F F 0 is locally surjective → 00 we can find a finite cover Ui and sections si Γ(Ui; F ) whose images agree with s . Choose ∈ |Ui a minimal such cover; we must show it contains only a single open set. Suppose not. Then the section s s gives an element of Γ(U U ; F 0), and since F 0 is c-soft s s is the restriction 1 − 2 1 ∩ 2 1 − 2 of some section s0 Γ(X; F 0). Replacing s with s + s0 we see that s = s on U U . We can ∈ 2 2 1 2 1 ∩ 2 therefore replace the two sections with a single section over U U , thereby contradicting the 1 ∪ 2 minimality of our cover.

Now suppose that X is non-compact. For any sheaf F on X one can realize Γc(X; F ) as lim Γc(U; FU ), where U ranges over the precompact open subsets of X. But given a short exact → sequence as in the proposition and a precompact open set U one obtains a corresponding short exact sequence 0 F 0 F F 00 0 → U → U → U → 0 with FU c-soft (6.4.6, 6.4.13). The argument for the compact case then shows that

0 Γ (U; F 0 ) Γ (U; F ) Γ (U; F 00) 0 → c U → c U → c U → is exact. Since direct limits are exact the result then follows.

92 To extend this result from Γc to f! one needs that following technical lemma; for a proof see [KS90, 2.5.2].

Lemma 6.4.15. Let X and Y be two topological spaces, f : Y X a continuous map and G a → sheaf on Y . Then for any x X the canonical morphism ∈ −1 α : (f G) Γ (f (x); G −1 ) ! x → c |f (x) is an isomorphism.

Corollary 6.4.16. Let 0 F 0 F F 00 0 be a short exact sequence of G-sheaves on X → → → → with F 0 c-soft. Then the sequence

0 f F 0 f F f F 00 0 → ! → ! → ! → is exact for any G-map f : X Y . → Proof. Again, the presence of a G-action is completely irrelevant to the conclusion, and so it suﬃces to consider the case when G is trivial. To prove the sequence exact we must show that the corresponding stalkwise sequences are exact; in light of Lemma 6.4.15 this means proving that

−1 0 −1 −1 00 0 Γ (f (x); F −1 ) Γ (f (x); F −1 ) Γ (f (x); F −1 ) 0 → c |f (x) → c |f (x) → c |f (x) → 0 is exact. But according to Corollary 6.4.13 the sheaf F −1 is c-soft, and so the result follows |f (x) by applying the previous proposition to the short exact sequence

0 00 0 F −1 F −1 F −1 0. → |f (x) → |f (x) → |f (x) → Lemma 6.4.17. Let 0 F 0 F F 00 0 be a short exact sequence of G-sheaves on X with → → → → F 0,F c-soft. Then F 00 is c-soft.

Proof. Let Z X be a closed subset, and consider the commutative diagram ⊆ 00 Γc(X; F ) / Γc(X; F )

α γ

β 00 Γc(Z; F ) / Γc(Z; F ).

Note that α is surjective since F is c-soft, while β is surjective by Proposition 6.4.14. It follows that γ is also surjective, and thus F 00 is c-soft.

Together the last two results have the following corollary.

Corollary 6.4.18. Let f : X Y be a G-map. Then the c-soft G-sheaves on X are an → f!-injective subcategory.

Of course, in order to be a useful f!-injective subcategory we must also know that there are enough c-soft G-sheaves on X.

93 Proposition 6.4.19. Let X be a G-space and F a G-sheaf over X. Then there exists a c-soft G-sheaf S over X and a monomorphism F , S. →

Remark 6.4.20. In other words ShG(X; k) has enough c-soft sheaves, and we can therefore compute the derived functors Rf! using c-soft sheaves.

Proof. Let S be the G-sheaf given by S(U) = x∈U Fx; there is an obvious monomorphism F , S given by sending a section s F (U) to s . The claim is that S is c-soft. Indeed, let K → ∈ Qx be a compact subset of X and let s Γ(K; S). Then s is the restriction of a section s S(U) ∈ Q U ∈ for some open set U containing K, and extending this by zero in those stalks outside of U gives a global section sX which restricts to s.

Lemma 6.4.21. Let f : Y Z and g : X Y be two composable G-maps. Then → → R(f g) = Rf Rg . ◦ ! ! ◦ ! Proof. Let F be a complex of G-sheaves on X and let S be a c-soft resolution of F . Then because g! sends c-soft to c-soft one has

(Rf Rg )(F ) = Rf (g (S)) ! ◦ ! ! ! = (f g )(S)) ! ◦ ! = (f g) (S) ◦ ! = R(f g) (F ). ◦ !

6.5 Equivariant Bivariant Cohomology

Deﬁnition 6.5.1. The equivariant bivariant cohomology of two G-spaces X and Y is the Z- graded abelian group

n HH G(X, Y ) = HomD+(C[G])(RΓc(kX ), RΓc(kY )[n]) n = ExtC[G](RΓc(kX ), RΓc(kY )).

Remark 6.5.2. Composition of morphisms in the derived category D+(C[G]) provides equivariant bivariant cohomology with a composition product

HH p (X, Y ) HH q (Y,Z) HH p+q(X,Z). G ⊗ G → G

Examples 6.5.3. (a) Let f : Y X be a proper G-map. Associated to f one has a morphism C f C = → X → ∗ Y f C of G-sheaves over X; applying RΓ : D+(X; k) D+(C[G]) turns this into a chain ! Y c G → map φ : RΓ (C ) RΓ (C ), thereby providing an element [f] = [φ ] HH 0 (X, Y ). f c X → c Y f ∈ G One can easily check that [f g] = [f] [g] for any two composable proper G-maps. ◦ ◦ 94 + C + C C C (b) Applying the functor RΓc : DG(X; ) D ( [G]) to HomD+(X;C)( X , X [n]) results in → G a homomorphism

C C C C HomD+ (X;C)( X , X [n]) HomD+(C[G])(RΓc( X ), RΓc( X )[n]); G → in other words, a ring homomorphism

H∗ (X; C) HH ∗ (X,X). G → G The simplicity of this construction is the main reason behind our rather non-standard deﬁnition of Borel cohomology.

Now let us show that HH G is actually a bivariant cohomology theory. The ﬁrst thing to check is (proper) homotopy invariance.

Proposition 6.5.4. Let f : Y [0, 1] X be a proper G-map. Then [f ] = [f ] HH 0 (X, Y ). × → 0 1 ∈ G

Remark 6.5.5. It follows that HH G is invariant under proper G-homotopy in both variables.

Proof. Let p : Y [0, 1] Y be the natural projection, and let φ : RΓ(C ) RΓ(C ) × → p Y → Y ×[0,1] denote the corresponding chain map. Note that on the level of cohomology φp is simply the homomorphism p∗ : H∗(Y ; C) H∗(Y [0, 1]; C); as such it follows that φ is a quasi-isomorphism, c → c × p and thus is invertible in the derived category. Now consider the inclusions i : Y Y [0, 1] given by i (y) = (y,t). Then p i = Id for t → × t ◦ t Y any t [0, 1], and thus [p] [i ] = Id HH 0 (Y, Y ) for all such t. Left composition with the ∈ ◦ t Y ∈ G inverse of φ then allows us to conclude that [i ] = [i ] for any s,t [0, 1]. p t s ∈ The result now follows from the fact that f = f i . t ◦ t Proposition 6.5.6. Let X be a G-space, Z X a closed G-invariant subset and U = X Z its ⊆ \ complement. Then for any G-space W one has long exact sequences

HH n+1(U, W ) HH n (Z, W ) HH n (X, W ) HH n (U, W ) HH n−1(Z, W ) ···→ G → G → G → G → G →··· and

HH n−1(W, Z) HH n (W, U) HH n (W, X) HH n (W, Z) HH n+1(W, U) . ···→ G → G → G → G → G →···

Proof. As a derived functor RΓc sends exact triangles to exact triangles, and thus applying RΓc to the short exact sequence 0 C C C 0 → XU → X → XZ → yields an exact triangle

RΓ (C ) RΓ (C ) RΓ (C ) RΓ (C )[1] c XU → c X → c XZ → c XU in the derived category D+(C[G]).

95 Let i : Z , X denote the inclusion of Z into X. Since C = i i∗C one can identify → XZ ! X ∗ RΓc(CXZ ) with RΓc(i CX ) = RΓc(CZ ); one can similarly identify RΓc(CXU ) with RΓc(CU ) to obtain an exact triangle

RΓ (C ) RΓ (C ) RΓ (C ) RΓ (C )[1] c U → c X → c Z → c U in the derived category D+(C[G]). The desired long exact sequences now follow from Corollary 5.3.7.

Corollary 6.5.7. For any two G-spaces X, Y one has natural isomorphisms

HH n (X R, Y ) = HH n+1(X, Y ) = HH n (X, Y R). G × ∼ G ∼ G × Proof. Apply the previous long exact sequences to the G-space X (0, 1] and closed G-invariant × subset X 1; note that HH ∗ (X (0, 1], W ) = 0. × G × ∼ Combining the last few results shows that the bivariant equivariant cohomology groups HH ∗ ( , ) are in fact a bivariant cohomology theory — that is, they are a homology theory in G − − the ﬁrst variable and a cohomology theory in the second.

Proposition 6.5.8. Let H G be a subgroup and let X, Y be two H-spaces. Then there is a ≤ natural induction morphism

indG : HH ∗ (X, Y ) HH ∗ (G X, G Y ) H H → G ×H ×H which is compatible with the composition product.

Proof. Let SX , SY be c-soft resolutions of CX and CY by H-sheaves. Then the induced G-sheaves G S and G S are c-soft resolutions of C = G C and C = G C . ×H X × Y G×H X ×H X G×H Y ×H Y Note that since C[G] is a free right C[H]-module the functor C[G] C : C[H] Mod ⊗ [H] − − → C[G] Mod is exact, and hence descends to the derived category. Combining this with the − resolutions above one obtains morphisms

n HH H (X, Y ) = HomD+(C[H])(Γc(SX ), Γc(SY )[n])

HomD+ C (C[G] C Γ (S ), C[G] C Γ (S )[n]) → ( [G]) ⊗ [H] c X ⊗ [H] c Y = HomD+ C (Γ (G S ), Γ (G S )[n]) ( [G]) c ×H X c ×H Y n = HH G(X, Y ). Finally, let f : Y X be a proper H-map and indG the induced proper G-map f : G Y → H ×H → G X. Then one easily ﬁnds that indG [f] = [indG f], and hence indG is natural. ×H H H H Now suppose Y is a G-space and let S be a c-soft resolution of Y by G-sheaves. Then by restricting the action to H we ﬁnd that S is also a c-soft resolution by H-sheaves; as a result G S is a c-soft resolution of C by G-sheaves. It follows that the natural G-equivariant ×H G×H Y chain map

Γ (G S)= C[G] C Γ (S) Γ (S) c ×H ⊗ [H] c → c a [γ] s a(γ s) ⊗ 7→ ·

96 defines an element µ HH ∗ (G Y, Y ). ∈ G ×H Remark 6.5.9. As noted earlier, when Y is a G-space the induced G-space G Y is G- ×H homeomorphic to (G/H) Y . When H is finite this point of view allows us to identify the chain × map µ with 1 , where : C[G/H] C denotes integration on G/H. G/H × Y G/H → Definition 6.5.10.R Let H GR be a subgroup, X an H-space and Y a G-space. Then one has ≤ a natural homomorphism

iG : HH ∗ (X, Y ) HH ∗ (G X, Y ) H H → G ×H given by sending φ HH ∗ (X, Y ) to indG (φ) µ HH ∗ (G X, Y ). ∈ H H ◦ ∈ G ×H Proposition 6.5.11. Let H G be a ﬁnite subgroup and Y be a G-space. Then ≤ iG : HH ∗ (pt, Y ) HH ∗ (G/H, Y ) H H → G is an isomorphism.

Proof. Let S be a c-soft resolution of CY as a G-sheaf; as noted above, restricting the action of G on S to the subgroup H G also gives a c-soft resolution of C as an H-sheaf. Next, ≤ Y since CG/H is c-soft one sees that RΓc(CG/H ) can be represented by the projective C[G]-module C[G/H]. Finally, note that because H is ﬁnite C is a projective C[H]-module. Putting all of this together then gives

n C HH H (pt, Y ) = HomD+(C[H])( , resH Γc(S)[n])

= HomK+(C[H])(C, resH Γc(S)[n])

= HomK+(C[G])(C[G/H], Γc(S)[n])

= HomD+(C[G])(C[G/H], Γc(S)[n]) n = HH G(G/H, Y ).

G But it is easily checked that the eﬀect of these identiﬁcations is precisely the map iH .

Finally, let us conclude this section by examining the interplay between the homomorphism H∗ (X) HH ∗ (X,X) and some of our other constructions; the following results will prove G → G essential for computations in the ﬁnal chapter.

Proposition 6.5.12. Let f : Y X be a proper G-map and let α H∗ (X; C). Then [[α]] [f]= → ∈ G ◦ [f] [[f ∗α]] HH ∗ (X, Y ). ◦ ∈ G Proof. Let : Id Rf f ∗ be the unit of adjunction for the adjoint pair (f ∗, Rf ). Because is → ∗ ∗ a natural transformation one has a commutative diagram

α CX / CX

∗ ∗ Rf∗f α ∗ Rf∗f CX / Rf∗f CX

97 for every morphism α HomD+(X;C)(I, I). Since f is proper applying RΓc converts this to ∈ G

RΓc(α) RΓc(CX ) / RΓc(CX )

∗ ∗ RΓc(f ) RΓc(f ∗ RΓc(f α) RΓc(CY ) / RΓc(C).

But the fact that this diagram commutes is exactly the statement that [[α]] [f] = [f] [[f ∗α]]. ◦ ◦ Proposition 6.5.13. Let X be a G-space, Z a closed G-invariant set and U = X Z its comple- \ ment. Then for any α H∗ (X) one has ∈ G (a) [i] [[α ]] = [[α]] [i] HH ∗ (Z,U), ◦ |Z ◦ ∈ G (b) [j] [[α]] = [[α ]] [j] HH ∗ (Z,U), and ◦ |U ◦ ∈ G (c) δ [[α ]] = [[α ]] δ HH ∗ (Z,U) ◦ |U |Z ◦ ∈ G where [i] HH 0 (X,Z) represents the restriction to Z, [j] HH 0 (U,X) the inclusion of U, and ∈ G ∈ G δ HH 1 (Z,U) the boundary class. ∈ G Proof. Let I be an injective resolution of C by G-sheaves, and let α : I I be an element of X → ∗ HG(X). Then one has a corresponding map of short exact sequences of G-sheaves

0 / IU / I / IZ / 0

αU α αZ 0 / IU / I / IZ / 0.

Applying RΓc converts this to a map of exact triangles

δ RΓc(CU ) / RΓc(CX ) / RΓc(CZ ) / RΓc(CU )[1]

αU α αZ αU [1]

δ RΓc(CU ) / RΓc(CX ) / RΓc(CZ ) / RΓc(CU )[1].

But the desired results are just the statement that the various squares commute.

98 Part III

An Equivariant Bivariant Chern Character

99 Chapter 7

An Equivariant Bivariant Chern Character

7.1 The K-theory Chern Character

Our goal in this section is to prove the following proposition.

Proposition 7.1.1. For each γ G and proper cocompact G-space X one has a ‘local’ Chern ∈ tor character homomorphism

∞ chγ : K (X) Heven(Xγ; C)= H2j (Xγ; C). G G → Z(γ) Z(γ) j=0 M Let E X be a complex n-dimensional G-vector bundle on a proper (though not necessarily → cocompact) G-space X. Then because the G-action on the classifying space EG is free we can construct a complex (non-equivariant) vector bundle E = EG E X = EG X over the G ×G → G ×G corresponding Borel space XG. This allows us to deﬁne the local Chern character at the identity element e G by che E = chE , though in general one only knows that che E H2j (X; C). ∈ G G G G ∈ G Proposition 7.1.2. If X is cocompact then with the deﬁnitions above one has Q

∞ ∞ che E H2j (X; C) H2j (X; C). G ∈ G ⊆ G j=0 j=0 M Y Proof. By the results of Chapter 2 one can find a finite proper G-CW complex Z, a G-map f : X Z and a complex G-vector bundle F on Z such that E is a summand of f ∗F . Let n → denote the (locally constant) rank of E, and let Gr(F,n) Z be the Grassmann bundle of n- → dimensional subspaces in the fibres of F . Then Gr(F,n) possesses a tautological complex G-vector bundle τ Gr(F,n), and f lifts to a G-map f˜ : X Gr(F,n) such that f˜∗τ = E. It follows → → e ˜∗ e that chGE = f chGτ, and thus it suffices to prove the result for spaces of the form Gr(F,n). But one can easily show that since Z has bounded Borel G-cohomology, so does Gr(F,n).

Remark 7.1.3. In subsequent sections we shall often ﬁnd it more convenient to work with G- vector bundles over non-cocompact proper G-spaces. However in every such case the G-bundles in question can be obtained as pullbacks from proper cocompact G-spaces, and hence their Chern characters will also lie in the direct sum, rather than direct product, of the equivariant cohomology groups.

Now suppose that in addition to the vector bundle E we are given a torsion element γ G . ∈ tor Then the restriction E Xγ is an complex Z(γ)-vector bundle on the proper cocompact Z(γ)- γ | γ space X . Moreover, since γ acts trivially on X each ﬁber of E γ is representation of the |X 100 ﬁnite cyclic group γ . Let Irrep( γ )= ρ : γ U(V ) denote the set of irreducible unitary h i h i { h i → ρ } representations of γ . Then one can decompose E γ as h i |X γ E γ = Hom(X V , E γ ) V = E V , |X ∼ × ρ |X ⊗ ρ ρ ⊗ ρ ρ∈Irrep(Mhγi) ρ∈Irrep(Mhγi) γ where each Eρ is itself a complex Z(γ)-vector bundle on X .

Deﬁnition 7.1.4. The local Chern character of E at γ G is ∈ tor ∞ chγ E = ρ(γ) che E H2j (Xγ; C). G · Z(γ) ρ ∈ Z(γ) j=0 X Y Remark 7.1.5. In light of Proposition 7.1.2 one sees that if X is cocompact, or more generally γ if E is obtained as a pullback from a cocompact G-space, then chGE will actually lie in the direct sum of the cohomology groups.

γ So deﬁned one can easily verify that chG gives a morphism of semigroups

C Vect (X) Heven(Xγ ; C). G → Z(γ)

It therefore extends to the Grothendieck group KG(X) by universality. Composing this Chern character with the homomorphism H∗ (Xγ) HH ∗ (Xγ,Xγ) Z(γ) → Z(γ) constructed in Section 6.5 allows us to assign an element [[chγ E]] HH even (Xγ ,Xγ) to any G ∈ Z(γ) complex G-vector bundle E.

Example 7.1.6. Let X be a G-space and E X is an even-dimensional G-Spinc vector bundle → over X. Then in Section 2.5 we saw how to associate an element β K (M E) to the G-Spinc E ∈ G structure on E. Combined with the results from this section this leads to an element

[[chγ β ]] HH even(M Eγ ,M Eγ). G E ∈ G It’s an unfortunate fact of life that vector bundle modiﬁcation is a rather poorly behaved ∗ construction, in as much as M E⊕F = (M E)π F . To help avoid these and related issues it will 6 be useful to make a slight modiﬁcation to the previous construction. Note that the inclusion i : M , M E = S(E 1) of M as the ‘south poles’ of M E results in a short exact sequence − → ⊕ i∗ 0 HH even(M, E) HH even(M,M E) − HH even(M,M) 0. → G → G −→ G → Let p : M E M denote the natural projection and suppose α H∗ (M E) is such that i∗ α = 0. E → ∈ G − Then it follows from Proposition 6.5.13 that

[p ] [[α]] [i ] = [p ] [i ] [[i∗ α]]=0, E ◦ ◦ − E ◦ − ◦ − ∗ ∗ E and thus there is a unique class α HH G(M, E) which maps to [pE] [[α]] HH G(M,M ). ∗ ∈ ◦ ∈ Note in particular since i−βE = 0 this applies to the local Chern characters of Bott elements. The usefulness of this construction is highlighted by the following proposition, the proof of which shall consume the remainder of the section.

101 Proposition 7.1.7. Let M be a G-space and E, F M two G-Spinc vector bundles over M. → Let π : E M be the natural projection. Then E → γ γ γ even γ γ ch βE⊕F = ch βE ch βπ∗ F HH (M , (E F ) ). G G ◦ G E ∈ Z(γ) ⊕ E E p∗ F Note that if we let p denote the natural projection M M then (M ) E can be identified E → with the fibre product M E M F . Let M E M F M E M F denote the bundle of ×M ∨M ⊂ ×M bouquets of spheres constructed using the south poles of M E and M F . Then the complement of M E M F in M E M F naturally identifies with the space E F . ∨M ×M ⊕ Lemma 7.1.8. The open inclusion of E F , M E M F results in a short exact sequence ⊕ → ×M 0 HH n (M γ, (E F )γ ) HH n (M γ, (M E M F )γ ) → Z(γ) ⊕ → Z(γ) ×M → HH n (M γ , (M E M F )γ ) 0. Z(γ) ∨M → Proof. The result would follow from the associated long exact sequence if one knew that the map induced by the open inclusion were an injection. Note that the open inclusion E F , M E ⊕ → ×M M F factors through the open inclusions E F , E M F and E M F , M E M F . ⊕ → ×M ×M → ×M But the long exact sequences corresponding these inclusions both collapse into split short exact sequences, and thus the maps

HH n (M γ , (E F )γ) HH n (M γ, (E M F )γ ) Z(γ) ⊕ → Z(γ) ×M and HH n (M γ, (E M F )γ) HH n (M γ, (M E M F )γ ) Z(γ) ×M → Z(γ) ×M are injections. It follows that their composition is also an injection.

Let βE βF denote the product of the pullbacks of βE, βF to the ﬁbre product. If we let p : M E M F M represent the projection then we can form the class [pγ] [[chγ β ×M → ◦ G E β ]] HH n (M γ, (M E M F )γ ). Note that since the restriction of β β to M E M F F ∈ Z(γ) ×M E F ∨M vanishes it follows from the previous lemma that the class [pγ] [[chγ β β ]] is the image of ◦ G E F n γ γ γ a unique element in HH Z(γ)(M , (E F ) ). We shall now show that both chGβE⊕F and γ γ ⊕ ch βE ch βπ∗ F satisfy this condition, and as a result they must be equal. G ◦ G E Lemma 7.1.9. Under the open inclusion E F , M E M F the class chγ β is ⊕ → ×M G E⊕F sent to [pγ] [[chγ β β ]]. ◦ G E F Proof. The map c : M E M F M E⊕F given by collapsing M E M F ﬁbrewise results in ×M → ∨M the commutative diagram HH n (M γ , (E F )γ ) / HH n (M γ , (M E⊕F )γ ) Z(γ) ⊕ Z(γ)

Id c∗ HH n (M γ , (E F )γ ) / HH n (M γ , (M E M F )γ ). Z(γ) ⊕ Z(γ) ×M ∗ But the result now follows from the fact that c βE⊕F ∼= βE βF .

102 Lemma 7.1.10. Under the open inclusion E F , M E M F the class chγ β ⊕ → ×M G E ◦ γ γ γ ch βπ∗ F is sent to [p ] [[ch βE βF ]]. G E ◦ G Proof. The result would follow from the commutativity of the diagram

∗ γ γ HH Z(γ)(M , (E F ) ) i4 ⊕ iiii iiii iiii iiii ∗ γ γ HH Z(γ)(M , E ) kk5 UUU kkk UUUU kkk UUUU kkk UUUU kkk UU* ∗ γ γ ∗ γ F γ HH Z(γ)(M ,M ) HH (M , (E M M ) ) S Z(γ) × SSS SSS SSS SSS S) ∗ γ Eγ HH Z(γ)(M ,M ) UUU UUUU UUUU UUUU UU* HH ∗ (M γ, (M E M F )γ ) Z(γ) ×M where the vertical maps are induced by the obvious open inclusions, while those diagonally up are given by composition with classes and those diagonally down are given by compositions of · an obvious projection with the Chern character of the appropriate Bott element. This is because the composition

HH ∗ (M γ,M γ) HH ∗ (M γ, Eγ ) HH ∗ (M γ, (E F )γ ) Z(γ) → Z(γ) → Z(γ) ⊕ γ γ sends 1M γ to ch βE ch βπ∗ F , while the composition G ◦ G E HH ∗ (M γ ,M γ) HH ∗ (M γ ,M Eγ) HH ∗ (M γ , (M E M F )γ ) Z(γ) → Z(γ) → Z(γ) ×M

γ γ sends 1 γ to [p ] [[ch β β ]]. M ◦ G E F Now, as for the commutativity of the diagram; ﬁrst note that the triangles in the diagram all commute by the deﬁnition of the classes. That leaves only the commutativity of the · parallelogram. But this amounts to the fact that

γ Eγ γ F γ E F γ [E M ] [[ch βπ∗ F ]] [(E M M ) , (M M M ) ] ← ◦ G E ◦ × → × γ Eγ F γ E F γ γ = [E M ] [(E M M ) , (M M M ) ] [[ch βp∗ F ]] ← ◦ × → × ◦ G E γ Eγ Eγ E F γ γ = [E , M ] [M (M M M ) ] [[ch βp∗ F ]]. → ◦ ← × ◦ G E

7.2 The Chern Character of a G-Spinc Manifold

Deﬁnition 7.2.1. A smooth proper G-manifold M is said to be G-orientable if it possesses a G-invariant volume form. A G-orientation for M consists of a choice of G-invariant volume form.

103 Remark 7.2.2. For non-proper actions this notion of G-orientability is far too restrictive; instead one should merely require that volume form be invariant up to multiplication by positive functions. However if the action is proper an averaging argument allows one produce a G-invariant volume form from any such volume form.

Lemma 7.2.3. Let M be a smooth G-oriented manifold (without boundary) of constant dimension n. Then one has an element

HH −n(M, pt) ∈ G ZM associated to the integration of compactly supported diﬀerential n-forms.

k Proof. Recall that the sheaves ΩM of smooth (complex-valued) differential n-forms are c-soft; it then follows from the PoincaréLemma that C Ω∗ is a c-soft resolution of C . Thus M → M M C ∗ RΓc( M ) is isomorphic to the complex Γc(ΩM ) of compactly supported differential forms on M. But according to Stoke’s theorem integration defines a chain map from the complex of compactly supported forms on M to C[ n]; since the volume form is G-invariant this chain map will be − C[G]-equivariant. As a result we have an element

HomK+ C (RΓ (C ), C[ n]), ∈ ( [G]) c M − ZM and after localization this provides an morphism in the derived category.

Remark 7.2.4. Similarly reasoning shows that for any smooth G-oriented real vector bundle E over M one can deﬁne an element HH −n(E,M) corresponding to integration over the E→M ∈ G ﬁbre. R

The following lemma is an immediate consequence of the deﬁnition of the integration class.

Lemma 7.2.5. Let M be a smooth G-oriented manifold (without boundary) of constant dimension n and let U M be an open submanifold. Then ⊂

[j] = , ◦ ZM ZU where [j] HH ∗ (U,M) represents the class of the open inclusion. ∈ G Lemma 7.2.6. Let M be a smooth G-oriented manifold of dimension n. Then

δ = , ◦ ˚ ZM Z∂M 1 ˚ where δ denotes the boundary element in HH G(∂M, M).

Proof. Let res denote the restriction map Γ (Ω∗ ) Γ (Ω∗ ). Then one has a commutative ∂M c M → c ∂M 104 diagram ∗ δ ∗ Γc(Ω ) / Γc(Ω ˚ )[1] ∂M M M MM R ˚ MMMM MMM M& Id C[ n + 1] 8 − R + R ppp ∂M pMpp ppp pp ∗ Γc(Ω∂M ) / M(res∂M ) where ∂M + M denotes the chain map R R Γ (Ω∗ ) Γ (Ω∗ )[1] C[ n + 1] c ∂M ⊕ c M → − α α α + α . ∂M ⊕ M 7→ ∂M M Z∂M ZM The result now follows by computing the eﬀect of the two maps Γ (Ω∗ ) C[ n + 1]. c ∂M → − Deﬁnition 7.2.7. Let E X be a complex vector bundle over a space X. The Todd class of E → is the multiplicative characteristic class Td(E) Heven(X) associated to the even formal power ∈ series x/(1 e−x). − Remark 7.2.8. See [HBJ92] for brief survey of the theory of multiplicative characteristic classes and their relation to formal power series.

Now suppose E X is a complex G-vector bundle over a proper cocompact G-space X. The → e local Todd class at the identity is deﬁned to be TdG(E) = Td(EG); the same arguments used for e even C the Chern character show that TdGE lies in HG (X; ). Continuing along the same vein, ﬁx an element γ G . As seen in the previous section the ∈ tor restriction E γ decomposes as a direct sum E V where each E is itself a complex Z(γ)- |X ⊕ ρ ⊗ ρ ρ vector bundle on Xγ. Let θ denote the summand corresponding to the trivial representation and ν denote its complement.

Definition 7.2.9. The local Todd class of E at γ is e γ TdZ(γ)(θ) even γ TdG(E)= γ ∗ HZ(γ)(X ) chG(Λ−1ν ) ∈ where Λ ν∗ = ( 1)iΛiν∗. −1 − γ ∗ even C Remark 7.2.10.PSee [BC98] for a proof that chG(Λ−1ν ) is invertible in HG (X; ). The following is a straightforward characteristic class computation, and hinges on the fact that the spinor bundles of E C are just the even and odd graded components of its exterior ⊗ algebra. Lemma 7.2.11. Let M be a smooth G-manifold (without boundary) and E M a real G-vector → bundle. Then γ C γ ∗ γ γ [[TdG(E )]] chGβE⊗C =1M γ HH Z(γ)(M ,M ) ⊗ ◦ ◦ C γ γ ∈ Z(E⊗ ) →M where HH ∗ ((E C)γ ,M γ) denotes integration along the fibres of (E C)γ . (E⊗C)γ →M γ ∈ Z(γ) ⊗ ⊗ R 105 Our primary interest will be with the equivariant Todd classes associated to the tangent γ bundle of a smooth G-manifold M; to simplify notation we shall simply write TdG(M) instead of Tdγ (TM C). G ⊗ One might hope (alas, na¨ıvely) that one could simply define the local Chern character of a c γ G-Spin manifold to be [[TdG(M)]] M γ . Unfortunately there is no reason to expect the fixed- c ◦ point set of G-Spin manifold to be ZR(γ)-oriented, and thus the integration class M γ need not γ γ exist. However the tangent bundle T (M ) = (TM) will always be Z(γ)-orientedR [BC98], and this allows the following definition.

Deﬁnition 7.2.12. Let M be a smooth n-dimensional G-Spinc manifold (without boundary). The local Chern character at γ G of M is deﬁned to be the element ∈ tor γ γ γ ∗∗ γ chGM = [[TdG(M)]] chGβT M \⊕n HH Z(γ)(M , pt). ◦ ◦ γ ∈ Z(T M⊕n)

Remarks 7.2.13. (a) For the definition of TM \ see 2.4.11. (b) This is more or less a variation on the equivariant index formula given by Baum & Connes in [BC98]. (c) The use of TM n, as opposed to simply TM, is necessitated by the fact that we have only ⊕ defined Bott elements for even-dimensional G-Spinc vector bundles. Furthermore, note that the fixed-point manifolds (TM + n)γ will always have dimension whose parity is the same as M, so that the resulting bivariant cohomology class will only have terms in degrees with the same parity as M.

Lemma 7.2.14. Let M be a smooth G-Spinc manifold (without boundary) of constant dimension n and let U M be an open submanifold. Then ⊂ [jγ] chγ M = chγ U, ◦ G G where [jγ] HH ∗ (U γ,M γ) represents the class of the open inclusion. ∈ G Proof. This follows as a result of Proposition 6.5.13 and Lemma 7.2.5.

Proposition 7.2.15. Let M be a smooth n-dimensional G-Spinc manifold. Then the associated boundary map ∂ : HH ∗ (M˚γ , pt) HH ∗−1 (∂M γ, pt) Z(γ) → Z(γ) γ ˚ γ sends chGM to chG∂M. Proof. Let δ HH 1 (∂M γ , M˚ γ) denote the boundary class. Then it follows from Proposition ∈ Z(γ) 6.5.13 and Lemma 7.2.6 that

γ ˚ γ ˚ γ δ chGM = δ [[TdGM]] chGβT M˚\⊕n ◦ ◦ ◦ ◦ ˚ γ Z(T M⊕n) γ γ = [[(TdGM) ∂M ]] chGβT M \|∂M⊕n | ◦ ◦ γ Z(T M|∂M⊕n)

106 C But TM ∂M = T∂M 1, and thus TM ∂M n ∼= [T∂M (n 1)] . By Proposi- | ⊕ | γ ⊕ ⊕ −γ ⊕ tion 7.1.7 this allows us to decompose chGβT M \|∂M⊕n = chGβ(T ∂M \⊕(n−1))⊕C as γ γ ch β \ ch β C . Thus G T ∂M ⊕(n−1) ◦ G [T ∂M⊕(n−1)]× γ γ γ δ ch M˚ = [[Td ∂M]] ch β \ ◦ G G ◦ G T ∂M ⊕(n−1) ◦ γ chGβ[T ∂M⊕(n−1)]×C ◦ γ C Z(T ∂M⊕(n−1)) × γ γ = [[TdG∂M]] chGβT ∂M \⊕(n−1) ◦ ◦ γ Z(T ∂M⊕(n−1)) γ = chG∂M.

Proposition 7.2.16. Let M be a smooth G-Spinc manifold (without boundary) and let p : E E → M be a smooth even-dimensional G-Spinc vector bundle. Then

chγ M = [π] [[chγ β ]] chγ M E, G ◦ G E ◦ G where π : M E M is the natural projection. → Proof. Let [jγ] HH 0 (Eγ ,M Eγ) represent the class of the open inclusion of Eγ in M Eγ as ∈ Z(γ) the complement of the south poles. Then it follows from the deﬁnition of chγ β that G E [π] [[chγ β ]] = chγ β [jγ ] ◦ G E G E ◦ and so by Lemma 7.2.14

[π] [[chγ β ]] chγ M E = chγ β [jγ ] chγ M E ◦ G E ◦ G G E ◦ ◦ G = chγ β chγ E# G E ◦ G where to avoid any possible confusion we have used E# to signify the we are viewing E as a G-Spinc manifold. Now suppose that E has rank 2k, so that E# is an (n +2k)-dimensional manifold. Then

chγ β chγ E# = G E ◦ G γ γ # γ chGβE [[TdG(E )]] chGβTE\⊕(n+2k) . ◦ ◦ ◦ γ Z(TE⊕(n+2k)) Note, however, that T E = p∗ TM p∗ E, and so ∼ E ⊕ E Tdγ (E#)= p∗ Tdγ (M) p∗ Tdγ (E C). G E G · E G ⊗ As a result one sees that

chγ β chγ E# = chγ β [[p∗ Tdγ (M)]] G E ◦ G G E ◦ E G ∗ γ C γ [[pETdG(E )]] chGβTE\⊕(n+2k) ◦ ⊗ ◦ ◦ γ Z(TE⊕(n+2k))

107 chγ β chγ E# = [[Tdγ (M)]] [[Tdγ (E C)]] G E ◦ G G ◦ G ⊗ γ γ chGβE chGβTE\⊕(n+2k) . ◦ ◦ ◦ γ Z(TE⊕(n+2k)) Furthermore, the fact that T E is obtained as the pullback of E TM from M allows us to apply ⊕ Proposition 7.1.7 to conclude that the composition

γ γ ch β ch β \ G E ◦ Z(γ) TE ⊕(n+2k) γ identiﬁes with chGβT M \⊕(E⊗C)⊕(n+2k) . But by again applying Proposition 7.1.7 this class γ γ may be decomposed as ch βT M \⊕n ch βp∗ (E⊗C)⊕2k . Thus G ◦ Z(γ) T M⊕n

chγ β chγ E# = [[Tdγ (M)]] [[Tdγ (E C)]] G E ◦ G G ◦ G ⊗ γ γ ch β \ ch β ∗ C G T M ⊕n G pT M⊕n(E⊗ )⊕2k ◦ ◦ ◦ γ Z(TE⊕(n+2k))

γ γ # γ γ ch β ch E = [[Td (M)]] ch β \ G E ◦ G G ◦ G T M ⊕n γ ∗ γ [[Td (p E C)]] ch β ∗ C . G T M⊕n G pT M⊕n(E⊗ )⊕2k ◦ ⊗ ◦ ◦ γ Z(TE⊕(n+2k)) But it follows from Lemma 7.2.11 this last expression is simply

γ γ γ [[TdG(M)]] chGβT M \⊕n = chG(M). ◦ ◦ γ Z(T M⊕n)

7.3 The Bivariant Chern Character

Let (M,ξ,f) be a topological KK G-cycle for (X, A; Y ) and let γ G be a torsion element. In ∈ tor the previous sections we have seen how to obtain elements

[f] HH even ((X A)γ , M˚ γ) ∈ Z(γ) \ [π ] HH even (M˚γ , M˚ γ Y γ ) M˚ ∈ Z(γ) × [[chγ (ξ) ]] HH even (M˚γ Y γ , M˚ γ Y γ) G |M˚×Y ∈ Z(γ) × × chγ M˚ HH ∗∗ (M˚γ , pt), G ∈ Z(γ) where π : M˚ Y M˚ is the obvious projection. M˚ × → Deﬁnition 7.3.1. The local equivariant bivariant Chern character at γ G is the semigroup ∈ tor homomorphism chγ : t G(X, A; Y ) HH ∗∗ ((X A)γ , Y γ ) G E∗ → Z(γ) \ given by sending the topological KK G-cycle (M,ξ,f) to the composition

γ γ [f] [π ] [[ch (ξ) ]] (ch M˚ 1 γ ). ◦ M˚ ◦ G |M˚×Y ◦ G × Y

108 Lemma 7.3.2. Let (W,M,ξ,F ) be a G-bordism for (X, A; Y ). Then the local bivariant Chern character of the boundary cycle vanishes for every γ G . ∈ tor Proof. As in Proposition 4.1.16 one has a commutative diagram

n+1 ˚ γ γ ∂ n γ γ n ˚ γ γ HH Z(γ)(W , Y ) / HH Z(γ)(∂W , Y ) / HH Z(γ)(M , Y )

HH n+1 (∅, Y γ ) ∂ / HH n (Xγ, Y γ) / HH n ((X A)γ , Y γ ). Z(γ) Z(γ) Z(γ) \ The commutativity of the ﬁrst square comes from the naturality of the boundary map in the long exact sequence when applied to the map of Z(γ)-pairs (W γ , ∂W γ) (Xγ,Xγ), while the → commutativity of the second square comes the map of Z(γ)-pairs F : (∂W γ , (∂W M˚)γ) \ → (Xγ, Aγ ). Now consider the element

γ γ ∗∗ γ γ π [[ch (η) ]] (ch W˚ 1 γ ) HH (W˚ , Y ). W˚ ◦ G |W˚ ×Y ◦ G × Y ∈ Z(γ) Propositions 6.5.12, 6.5.13 and 7.2.15 combine to show that the image of this class in the bottom right corner is the boundary cycle of the G-bordism. But since the diagram commutes and passes ∗ ∅ through the trivial groups HH Z(γ)( , Y ) we see that the image must be zero.

Lemma 7.3.3. Let (M,ξ,f) be a topological KK G-cycle for (X, A; Y ) and let E be an even- dimensional G-Spinc vector bundle over X. Then

γ γ E chG(M,ξ,f) = chG(M,ξ,f) .

Proof. This follows immediately from Proposition 7.2.16.

Corollary 7.3.4. The equivariant bivariant Chern character at γ G descends to a well- ∈ tor deﬁned natural transformation of Z2-graded equivariant homology theories

chγ : tKK G(X; Y ) HH ∗∗ ((X A)γ , Y γ ). G ∗ → Z(γ) \ γ Proof. The fact that chG is a well-deﬁned map follows from the previous two lemmas. Hence all that remains is to prove that it is a natural transformation of homology theories, for which the only non-obvious assertion is that the diagram

G δ G tKK n+1(X, A; Y ) / tKK n (A; Y )

γ γ chG chG HH n+1 ((X A)γ , Y γ ) δ / HH n (Aγ , Y γ) Z(γ) \ Z(γ) commutes, but this readily follows from Propositions 6.5.13 and 7.2.15.

The group G acts by conjugation on the set Gtor of torsion elements; let Gtor//G denote the quotient of Gtor by this action.

Deﬁnitions 7.3.5.

109 (a) Let X and Y be two G-spaces. The global (or delocalized) equivariant bivariant cohomology of (X, Y ) is the Z-graded abelian group

∗ ∗ γ γ HH G(X, Y )= HH Z(γ)(X , Y ). [γ]∈GMtor//G d (b) Let (X, A) be a proper cocompact G-pair and Y a compact G-space. The global (or delocalized) equivariant bivariant Chern character for (X, A; Y ) is the natural transformation

of Z2-graded equivariant homology theories

∗∗ ch = chγ : tKK G(X, A; Y ) HH (X A, Y ). G G ∗ → G \ [γ]∈GMtor//G b d

Remark 7.3.6. Note that since X is a proper and cocompact G-space only ﬁnitely many of the γ γ spaces X occurring the sum are non-empty, and thus only ﬁnitely many of the morphisms chG are non-zero.

Our ﬁnal goal is to prove that the global equivariant bivariant Chern character is an isomorphism for all ﬁnite proper G-CW pairs (X, A). Standard arguments from algebraic topology show that this is true if and only if it induces an isomorphism for the G-pairs (G/H, ∅).

Lemma 7.3.7. Let H G be a ﬁnite subgroup and ﬁx an element γ G . Let h = a−1γa ≤ ∈ tor { i i i} be a choice of representatives for those conjugacy classes of H which are conjugate to γ in G. Then

γ Z(γ)/Z −1 (γ) (G/H) aiHa → i a u Z −1 (γ) ua H aiHa 7→ i is a bijection of Z(γ)-sets.

Proof. Note that (G/H)γ = aH : a−1γa H , from which it follows that the assignment aH { ∈ } 7→ a−1γa yields unique H-conjugacy class for each Z(γ)-orbit in (G/H)γ . A simple computation then shows that the stabilizer of aH under the Z(γ)-action can be identiﬁed with ZaHa−1 (γ)= Z(γ) aHa−1. ∩ Proposition 7.3.8. Let H G be a ﬁnite subgroup and Y be a compact G-space. Then there ≤ is a natural isomorphism ∗ ∗ iG : HH (pt, Y ) HH (G/H, Y ). H H → G −1 Proof. Fix a γ Gtor and letb hi =da γai be ad choice of representatives for those conjugacy ∈ { i } classes of H which are conjugate to γ in G. We shall describe a natural isomorphism

HH ∗ (pt, Y hi ) HH ∗ ((G/H)γ , Y γ). ZH (hi) → Z(γ) i M Summing over the conjugacy classes of Gtor will then give the desired global isomorphism.

110 Recall from Proposition 6.5.11 that there is a natural isomorphism

Z(hi) ∗ hi ∗ hi i : HH (pt, Y ) HH (Z(hi)/ZH (hi), Y ). ZH (hi) ZH (hi) → Z(hi) ∗ γ Conjugating by ai identiﬁes this last group with HH (Z(γ)/Z −1 (γ), Y ), and we have Z(γ) aiHai therefore constructed an isomorphism

∗ h =∼ ∗ γ −1 iHH ZH (hi)(pt, Y ) HH Z(γ)( Z(γ)/Za Ha (γ), Y ). ⊕ −→ i i i a ∗ γ γ But it follows from the previous lemma that this last term is precisely HH Z(γ)((G/H) , Y ).

Proposition 7.3.9. Let H G be a ﬁnite subgroup, and let Y be a compact G-space. Then the ≤ following diagram commutes

chH ∗∗ H b tKK ∗ (pt, Y ) / HH H (pt, Y )

iG iG H d bH chG ∗∗ G b tKK ∗ (G/H, Y ) / HH G (G/H, Y ).

Proof. We shall only prove the even dimensional case;d the proof in the odd dimensions is similar. H Let (M, ξ) be a topological cycle for tKK ∗ (pt, Y ). In light of Proposition 4.7.6 it suﬃces to consider only cycles of the form (pt, ξ), ξ K (Y ). But then ∈ H chγ (iG (pt, ξ)) = chγ (G/H, G ξ, Id) G H G ×H γ γ γ = [π ] [[ch (G ξ)]] (ch G/H 1 γ ) G/H ◦ G ×H ◦ G × Y HH ∗∗ ((G/H)γ , Y γ), ∈ Z(γ) where here and later πX denotes the projection X Y X. Restricting this to an orbit h × → Z(γ)/ZaHa−1 (γ) in (G/H) results in

γ [π ] [[ch (Z(γ) ξ γ )]] ( 1 h ). Z(γ)/ZaHa−1 (γ) Z(γ) ZaHa−1 (γ) Y Y ◦ × | ◦ Z(γ)/Z (γ) × Z aHa−1 On the other hand one has

iZ(h) (chh (pt, ξ)) = iZ(h) ([πh ] [[chh ξ]]) ZH (h) H ZH (h) pt ◦ H Z(h) h = [πZ(h)/Z (h)] ind [[ch ξ Y h )]] µ. H ◦ ZH (h) ZH (h) | ◦ But thanks to Remark 6.5.9 one sees that

µ = 1 h × Y ZZ(h)/ZH (h) and one can easily check that

Z(h) h h ind [[ch ξ Y h )]] = [[ch (Z(h) Z (h) ξ Y h )]]. ZH (h) ZH (h) | Z(h) × H | It follows that conjugating by a identiﬁes this with the previously computed component of γ G chG(iH (pt, ξ)).

111 Corollary 7.3.10. Let H G be a ﬁnite subgroup, and let Y be a compact G-space. Then the ≤ global equivariant Chern character

∗∗ ch : tKK G(G/H; Y ) HH (G/H, Y ) G ∗ → G becomes an isomorphism afterb tensoring with C. d

Proof. It follows from earlier results that the induction homomorphisms in Proposition 7.3.9 are isomorphisms, so it suﬃces to show that the Chern chH is an isomorphism after tensoring over C. But this is just a rephrasing of the K-theoretic Chern character isomorphism for ﬁnite groups b ([AS89], [BC98]).

Combining this work with our earlier results on the relationship between topological and analytic KK G-theory results in the following theorem.

Theorem 7.3.11. For any countable discrete group G, ﬁnite proper G-CW pair (X, A) and compact G-space Y there is a natural Chern character isomorphism

∗∗ KK G(X A, Y ) C HH (X A, Y ). ∗ \ ⊗ → G \ It is now a trivial matter to drop the condition thatd Y be compact.

Corollary 7.3.12. For any countable discrete group G, ﬁnite proper G-CW pair (X, A) and G-space Y there is a natural Chern character isomorphism

∗∗ KK G(X A, Y ) C HH (X A, Y ). ∗ \ ⊗ → G \ Proof. Consider the one-point compactiﬁcation Y + dof Y . Then one has a commutative diagram

0 / KK G(X A, Y ) C / KK G(X A, Y +) C / KK G(X A, pt) C / 0 ∗ \ ⊗ ∗ \ ⊗ ∗ \ ⊗ ch ch b G b G ∗∗ ∗∗ ∗∗ 0 / HH (X A, Y ) / HH (X A, Y +) / HH (X A, pt) / 0 G \ G \ G \ where each rowd is a short exact sequenced and each vertical map isd an isomorphism. It follows that there is an isomorphism

∗∗ KK G(X A, Y ) C HH (X A, Y ). ∗ \ ⊗ → G \ Similar arguments show that this map is natural ind both variables.

7.4 Concluding Remarks

Having worked so hard to obtain the Chern character isomorphism one would like to know how much further one can push the machinery. It should be clear that the argument outlined above relies quite heavily on the fact that, so long as G is discrete, KK G(X, Y ) can be described by the triples (M,ξ,f). Unfortunately this is not the case for all totally disconnected groups, due

112 in large part to the failure of the oft-used result of Lück and Oliver to extend to this more general context. Work by J. Sauer [Sau03] indicates that the limiting arguments used by Baum & Schneider to extend their isomorphism to the profinite case should also work to extend the isomorphism constructed here to actions of prodiscrete groups1; however the same paper also shows that the appropriate versions of Lück and Oliver’s result fails to be true even for very simple totally disconnected, non-prodiscrete, groups. Hence continued pursuit of such questions requires a new approach to the problem.

1A group is prodiscrete if it a direct limit of discrete groups.

113 Appendices

114 Appendix A

Straightening the Angle

Our goal in this appendix is to prove the following lemma and its corollaries, all of which play an essential role in the equivariant bordism arguments in Chapter 4.

Lemma A.1 (Straightening the Angle). Let W n be a topological G-manifold and M n−2 W ⊂ a closed G-submanifold (smooth and without boundary). Suppose further that

W M has a differentiable structure, • \ the action of G restricted to W M is smooth, and • \ there exists a G-invariant neighborhood U of M in W and a G-equivariant homeomorphism • φ : U M R R which identifies M with M 0 0 and is a diffeomorphism on → × + × + ×{ }×{ } U M. \ Then there is a differentiable structure on W which extends that on W M and makes W a smooth \ G-manifold.

Proof. Let α : R R R R be the homeomorphism induced by the complex function + × + → × + z z2; note that α is a diﬀeomorphism away from the origin. Let α denote the corresponding 7→ M map

M R R M R R × + × + → × × + (m,s,t) (m, α(s,t)). 7→ Pulling back the differentiable structure on M R R over the composition α φ endows U × × + M ◦ with a differentiable structure. Both U and W M then have differentiable structures, and the \ structure on U has been chosen so that they agree on their common intersection in W . It follows that these structure can be glued to give a differentiable structure on their union, which is the entirety of W .

Corollary A.2. n n n−1 (a) Let W1 and W2 be two proper G-manifolds, and let M be a G-manifold lying in both their boundaries. Then the gluing W = W W can be given a G-equivariant smooth 1 ∪M 2 structure such that T W = T W T W as real G-vector bundles. ∼ 1 ∪M 2 (b) Let M1 and M2 be two smooth proper G-manifolds with boundary. Then the product M = M M can be given a G-equivariant smooth structure such that TM = π∗TM π∗TM 1 × 2 ∼ 1 1 ⊕ 2 2 as real G-vector bundles.

115 Proof.

(a) Typically when one glues two manifolds W1 and W2 together along a common boundary component M one also chooses collaring neighborhoods of M which are each diffeomorphic to M [0, 1). This results in a neighborhood of the form M ( 1, 1) for M W = × × − ⊂ W W , and this allows one to put a differentiable structure on the whole of W . However 1 ∪M 2 when M is not an entire component of the boundary, difficulties arise from the non-trivial boundary of M. In this case the standard construction still suffices to put a smooth structure on W ∂M, but ∂M itself is left with a neighborhood which identifies with ∂M \ × R R . Straightening the angle allows one to remove this singularity. Finally note that + × + the initial gluing ensures that T W = (T W T W ) ; but since W equivariantly |W˚ ∼ 1 ∪M 2 |W˚ deformation retracts into W˚ this implies that T W = T W T W . ∼ 1 ∪M 2 (b) If one of the two manifolds is without boundary then M = M M can be given the 1 × 2 product differential structure; however if both manifold have boundaries then M will have ‘corners’ at Z = ∂M ∂M . Specifically, while M Z will be a smooth manifold, Z will 1 × 2 \ have a neighborhood which identifies with Z R R . Again this deficiency can be × + × + corrected by straightening the angle. Finally the smooth structure chosen for M ensures that TM = (π∗TM π∗TM ) ; but since M equivariantly deformation retracts into |M˚ ∼ 1 1 ⊕ 2 2 |M˚ M˚ this implies TM = π∗TM π∗TM . ∼ 1 1 ⊕ 2 2

116 Appendix B

Smoothing Spinc Structures

The introductory material on G-Spinc vector bundles given in Section 2.4 assumes very little about the base space of the bundle. However in most of our applications we are actually interested in G-Spinc bundles over smooth G-manifolds, and it is often essential that the G-Spinc structure have some measure of smoothness as well.

Deﬁnition B.3. Let E be a smooth G-oriented Euclidean G-vector bundle over a smooth G- manifold M. Then the oriented frame bundle F +(E) possesses a natural smooth structure. A smooth G-Spinc structure on E consists of a G-Spinc structure (P, σ) such that P M is a smooth principal (G, Spinc)-bundle, and σ : P F +(E) is a smooth G-map. → Implicit in most of the results in Chapter 4 is the fact that any topological G-Spinc structure on a smooth Euclidean G-vector bundle can be ’smoothed’, and that this smoothing is essentially unique. Attempting to prove such a result using local techniques leads nowhere fast; while the transition functions of a principal bundle can easily be replaced by smooth approximations, doing so destroys the cocycle condition. Fortunately there is a more global approach to the problem coming from similar arguments regarding the smoothing of topological Euclidean G- vector bundles. To explain this we must ﬁrst venture into the land of holomorphically closed subalgebras.

∗ Deﬁnition B.4. Let A be a C -algebra, A0 A a subalgebra. One says that A0 is holomorphi- + ⊆ cally closed if for any a A and any function f holomorphic on a neighborhood of Spec + (a ) 0 ∈ 0 A 0 one has f(a ) A+. 0 ∈ 0

Remark B.5. If A0 is a dense holomorphically closed subalgebra of A then then same is true of the matrix algebras M (A ) M (A) [Sch92]. n 0 ⊆ n The signiﬁcance of holomorphically closed subalgebras lies in the following proposition (see [Kar78] for a proof).

∗ Proposition B.6. Let A0 be a dense holomorphically closed subalgebra of the C -algebra A. Then every projection p A is homotopic to a projection p A ; moreover any two projections ∈ 0 ∈ 0 p ,p0 A which are homotopic in A are smoothly homotopic in A . 0 0 ∈ 0 0

Note that in particular this last proposition implies that there is a unique projective A0- module associated to the projection p.

∞ o Proposition B.7. Let M be a smooth proper G-manifold. Then Cc (M) G is a holomorphically closed subalgebra of C (M) o G. Similarly, if E M is a smooth Euclidean G-vector bundle 0 r → ∞ C o C o on M then Γc ( l(E)) G is a holomorphically closed subalgebra of Γ0( l(E)) r G.

117 Proof. See [BC98] for a proof of the ﬁrst result; the second result follows from a similar argument.

Corollary B.8. Every Euclidean G-vector bundle on a smooth proper cocompact G-manifold can be smoothed.

Proof. Given a topological Euclidean G-vector bundle on a proper cocompact G-space one can use local trivializations to construct a projection in Mn(C0(M)or G) whose range consists of the continuous sections of E [Phi89]. From the last two propositions it follows that this projection ∞ o can be modiﬁed to lie in Mn(Cc (M) G). Finally, we declare a compactly supported section of E to be smooth if it lies in

p [C∞(M) o G]n p [C (M) o G]n = Γ(E). ∞ c ⊂ ∞ 0 Choosing local trivializations which consist of smooth sections then results in transition functions which are smooth.

Now to translate this argument into something that will allow us to smooth G-Spinc structures.

Deﬁnition B.9. Let E be a Euclidean G-vector bundle on a proper cocompact G-space X and let Cl(E) denote the associated bundle of Cliﬀord algebras. A pair (S,ρ) consisting of a complex G- vector bundle S and G-equivariant algebra-bundle homomorphism ρ : Cl(E) End(S) is called → an irreducible (Cl(E), G)-module if each ρ : Cl(E ) End(S ) is an irreducible representation. x x → x Proposition B.10. Let E be a G-oriented Euclidean G-vector bundle of rank 2k over a proper cocompact G-space X. Then there is a canonical bijection between the set of G-Spinc structures on E and the set of irreducible (Cl(E), G)-modules. Moreover if E is smooth then there is also a canonical bijection between the set of smooth G-Spinc structures and the set of smooth irreducible (Cl(E), G)-modules.

Proof. An proof of the non-equivariant version of the topological result can be found in [Ply86], and the same argument works in the equivariant (and equivariant smooth) case.

Proposition B.11. Every topological G-Spinc structure on a smooth G-oriented Euclidean G- vector bundle can be smoothed, and any two smoothings are smoothly isomorphic.

Proof. We shall only consider the case when the rank of E is even; the case when the rank is odd then follows by stabilizing and applying a smooth form of the two-out-of-three lemma. Let E be a smooth G-oriented Euclidean G-vector bundle of rank 2k equipped with a choice of topological G-Spinc structure. By the previous proposition this topological G-Spinc structure corresponds to an irreducible (Cl(E), G)-module (S,ρ); through local trivializations of S such a module can be realized as the range of a projection in matrices over the reduced crossed product

A =Γ0(Cl(E)) or G. In order to smooth the structure it suﬃces to replace this projection with ∞ C o one lying in matrices over the subalgebra A0 = Γc ( l(E)) G, since this will simultaneously

118 provide S with a smooth structure and replace ρ with a smooth homomorphism of algebra- bundles ρ : Cl(E) End(S). But the subalgebra A is dense and holomorphically closed in ∞ → 0 A, and therefore such a projection exists. As for uniqueness, this follows from the fact that any two smoothing projections are smoothly homotopic within A0.

119 Bibliography

[AB68] Michael Atiyah and Raoul Bott, A Lefschetz ﬁxed-point formula for elliptic complexes: Ii. applications, Ann. Math. 88 (1968), no. 3, 451–491.

[AS89] Michael Atiyah and Graeme Segal, On equivariant Euler characteristics, J. Geom. Phys. 6 (1989), no. 4, 671–677.

[Ati67] Michael Atiyah, K-theory, Lecture notes by D. W. Anderson, W. A. Benjamin Inc., New York-Amsterdam, 1967.

[Ati68] , Bott periodicity and the index of elliptic operators, Quart. J. Math. Oxford. Ser. (2) 19 (1968), 113–140.

[Ati70] , Global theory of elliptic operators, Proc. Internat. Conf. on Functional Analysis and Related Topics (Tokyo, 1969), Univ. of Tokyo Press, 1970, pp. 21–30.

[BBM85] Paul Baum, Jean-Luc Brylinski, and Robert MacPherson, Cohomologie équivariante délocalisée, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), no. 17, 605–608.

[BC98] Paul Baum and Alain Connes, Chern character for discrete groups, A fˆete of topology, Academic Press, Boston, 1998, pp. 163–232.

[BCH94] Paul Baum, Alain Connes, and Nigel Higson, Classifying space for proper actions and K-theory of group C∗-algebras, C∗-algebras : 1943 – 1993 (San Antonio, TX, 1993), Contemporary Mathematics, vol. 167, Amer. Math. Soc., 1994, pp. 240–291.

[BD82] Paul Baum and Ronald Douglas, K-homology and index theory, Operator algebras and applications, Part I (Kingston, Ont., 1980), Proc. Sympos. Pure Math, vol. 38, Amer. Math. Soc., 1982, pp. 117–173.

[BDF73] Lawrence Brown, Ronald Douglas, and Peter Fillmore, Extensions of C∗-algebras, operators with compact self-commutators, and K-homology, Bull. Amer. Math. Soc. 79 (1973), 973–978.

[BG94] Jonathan Block and Ezra Getzler, Equivariant cyclic homology and equivariant diﬀer- ential forms, Ann. Sci. Ecole´ Norm. Sup. (4) 27 (1994), no. 4, 493–527.

[Bla98] Bruce Blackadar, K-theory for operator algebras, Mathematical Sciences Research In- stitute Publications, vol. 5, Cambridge University Press, Cambridge, 1998.

[Bot65] Raoul Bott, The index theorem for homogeneous diﬀerential operators, Diﬀerential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, 1965, pp. 167–186.

120 [BS02] Paul Baum and Peter Schneider, Equivariant-bivariant Chern character for proﬁnite groups, K-theory 25 (2002), no. 4, 313–353.

[Con79] Pierre Conner, Diﬀerentiable periodic maps, Lecture Notes in Mathematics, vol. 738, Springer, Berlin, 1979.

[Dav96] Kenneth Davidson, C∗-algebras by example, Fields Institute Monographs, vol. 6, Amer. Math. Soc., Providence, 1996.

[GHT00] Erik Guentner, Nigel Higson, and Jody Trout, Equivariant E-theory for C∗-algebras, Memoirs of the American Mathematical Society, vol. 148, Amer. Math. Soc., Provi- dence, 2000.

[Hat02] Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002.

[HBJ92] Friedrich Hirzebruch, Thomas Berger, and Rainer Jung, Manifolds and modular forms, Aspects of Mathematics, vol. E20, Friedr. Vieweg & Sohn, Braunschweig, 1992.

[HR00] Nigel Higson and John Roe, Analytic K-homology, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000.

[HY76] Akio Hattori and Tomoyoshi Yoshida, Lifting compact group actions in ﬁber bundles, Japan. J. Math. (N.S.) 2 (1976), no. 1, 13–25.

[Ive86] Birger Iversen, Cohomology of sheaves, Universitext, Spring-Verlag, Berlin, 1986.

[Kar78] Max Karoubi, K-theory, an introduction, Grundlehren der Math. Wissen., vol. 226, Spring-Verlag, Berlin, 1978.

[Kas80a] Gennadi Kasparov, Hilbert C∗-modules: theorems of Stinespring and Voiculescu, J. Operator Theory 4 (1980), no. 1, 133–150.

[Kas80b] , The operator K-functor and extensions of C∗-algebras, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 3, 571–636.

[KS90] Masaki Kashiwara and Pierre Shapira, Sheaves on manifolds, Fundamental Principles of Mathematical Sciences, vol. 292, Spring-Verlag, Berlin, 1990.

[LM89] H. Blaine Lawson and Marie-Louise Michelsohn, Spin geometry, Princeton Mathemat- ical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989.

[LMS83] Richard Lashof, J. Peter May, and Graeme Segal, Equivariant bundles with abelian structure group, Proceedings of the Northwestern Homotopy Theory Conference (Evanston, IL, 1982), Contemp. Math., vol. 19, Amer. Math. Soc., 1983, pp. 167–176.

[LO01] Wolfgang L¨uck and Bob Oliver, The completion theorem in K-theory for proper actions of a discrete group, Topology 40 (2001), no. 3, 585–616.

121 [L¨uc02] Wolfgang L¨uck, Chern characters for proper equivariant homology theories and applications to K- and L-theory, J. Reine Angew. Math. 543 (2002), 193–234.

[Mal65] A. I. Mal’cev, On the faithful representations of inﬁnite groups by matrices, Amer. Math. Soc. Transl. 45 (1965), 1–18.

[Nis91] Victor Nistor, A bivariant chern character for p-summable quasihomomorphisms, K- Theory 5 (1991), no. 3, 193–211.

[Phi89] Christopher Phillips, Equivariant K-theory for proper actions, Pitman Research Notes in Mathematics, vol. 178, John Wiley & Sons Inc., New York, 1989.

[Ply86] Roger Plymen, Strong morita equivalence, spinors and symplectic spinors, J Operator Theory 16 (1986), no. 2, 305–324.

[Sau03] Juliane Sauer, K-theory for proper smooth actions of totally disconnected groups, Preprintreihe SFB 478, Geometrische Strukturen in der Mathematik, 2003.

[Sch92] Larry B. Schweitzer, A short proof that Mn(A) is local if A is local and Fr´echet, Inter- nat. J. Math. 3 (1992), no. 4, 581–589.

[Seg68] Graeme Segal, Equivariant K-theory, Inst. Hautes Etudes´ Sci. Publ. Math. 34 (1968), 129–151.

[Voi03] Christian Voigt, Equivariant cyclic homology, Ph.D. thesis, University of M¨unster, 2003.

[Wei94] Charles Weibel, An introduction to homological algebra, Cambridge Studies in Ad- vanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994.

122 JEFF RAVEN 915 Southgate Drive, Apt 13 McAllister Building State College, PA 16801 Department of Mathematics [email protected] Pennsylvania State University tel. (814) 861-2080 University Park, PA 16802

ACADEMIC EDUCATION 1996 – present Pennsylvania State University, Ph.D. in Mathematics. Adviser : Paul Baum. 1991 – 1995 Pennsylvania State University, B.S. with High Distinction in Mathematics and Physics, Honors in Mathematics.

TEACHING EXPERIENCE 2001 – present Teaching Assistant, Pennsylvania State University. Taught courses in calculus, vector calculus and diﬀerential equations. Lectured, wrote and graded exams. 2000 Graduate Sequences Seminar, Pennsylvania State University. Taught preparatory seminar for the algebra, topology and analysis Ph.D. qualifying examinations. 1999 MASS Program Acting Director, Pennsylvania State University. MASS (Mathematics Advanced Studies Semester) is a program for talented under- graduates from across the United States. 1998 MASS Teaching Assistant, Pennsylvania State University. Assisted Profes- sor Ken Ono with the MASS Number Theory course. 1996 – 1997 Teaching Assistant, Pennsylvania State University. Taught introductory courses in algebra, combinatorics and calculus.

AWARDS AND HONORS Pritchard Dissertation Fellowship, 2003. Graduate Assistant Award For Outstanding Teaching, 2003. University-wide award for outstanding teaching by a graduate student. Charles H. Hoover Memorial Teaching Award, 2002. Award for outstanding teaching by a mathematics graduate assistant. ZZRQ Award, 2002. Award for promoting sense of community within the math department. Vollmer-Kleckner Scholarship in Science, 1999. Graduate Scholars Award, 1997. Phi Beta Kappa, Academic Honor Society. Braddock Scholar, Pennsylvania State University, 1991–1995. Full academic scholarship. National Merit Scholar, 1991–1995.

PROFESSIONAL ACTIVITIES American Mathematical Society JGFA Seminar, Pennsylvania State University, 1999 – present. Regular participant and speaker at this graduate student-run reading seminar in Geometric Functional Analysis. Topology/Geometry Seminar, Pennsylvania State University, 1999 – present. Regular participant and speaker. Graduate Teaching Mentor, Pennsylvania State University, 2001. Institute for Mathematics and Its Applications Summer School, University of Chicago, 1997. Month-long program in Algebraic Topology.