CW-complexes
by Wolfgang L¨ uck and Bob Oliver
Abstract. We first construct a classifying space for defining equivariantK-theory for proper actions of discrete groups. This is then applied to construct equivariant Chern characters with values in Bredon cohomology with coefficients in the representation ring functorR(−) (tensored by the rationals). And this in turn is applied to prove some versions of the Atiyah-Segal completion theorem for real and complex K-theory in this setting.
Key words:K-theory, proper actions, vector bundles, Γ-spaces
1991 mathematics subject classification: primary 55N91, secondary 19L47
In an earlier paper [8], we showed that for any discrete groupG, equivariantK-theory for finite proper G-CW-complexes can be defined using equivariant vector bundles. This was then used to prove a version of the Atiyah-Segal completion theorem in this situation. In this paper, we continue to restrict attention to actions of discrete groups, and begin by constructing an appropriate classifying space which allows us to define KG∗(X) for an arbitrary properG-complexX. We then construct rational-valued equivariant Chern characters for such spaces, and use them to prove some more general versions of completion theorems.
In fact, we construct two different types of equivariant Chern character, both of which involve Bredon cohomology with coefficients in the system G/H7→R(H)
. The first, ch∗X :KG∗(X)−−−−−−→HG∗(X;Q⊗R(−)),
is defined for arbitrary properG-complexes. The second, a refinement of the first, is a homomorphism che∗X :KG∗(X)−−−−−−→Q⊗HG∗(X;R(−)),
but defined only for finite dimensional proper G-complexes for which the isotropy subgroups on X have bounded order. WhenX is afiniteproperG-complex (i.e.,X/Gis a finite CW-complex), thenHG∗(X;R(−)) is finitely generated, and these two target groups are isomorphic. The second Chern character is important when proving the completion theorems. The idea for defining equivariant Chern characters with values in Bredon cohomologyHG∗(X;Q⊗R(−)) was first due to S lomi´nska [12]. A complex-valued Chern character was constructed earlier by Baum and Connes [4], using very different methods.
The completion theorem of [8] is generalized in two ways. First, we prove it for real as well as complex K-theory. In addition, we prove it for families of subgroups in the sense of Jackowski [7]. This means that for each finite properG-complexX and each familyF of subgroups ofG, KG∗(EF(G)×X) is shown to be isomorphic to a certain completion of KG∗(X). In particular, when F ={1}, then EF(G) =EG, and this becomes the usual completion theorem.
The classifying spaces for equivariantK-theory are constructed here using Segal’s Γ-spaces. This seems to be the most convenient form of topological group completion in our situation. However, although Γ-spaces do produce spectra, as described in [10], the spectra they produce are connective, and hence not what is needed to define equivariant K-theory directly. So instead, we define KG−n(−) andKO−Gn(−) for all n≥0 using classifyingspaces constructed from a Γ-space, then prove Bott periodicity, and use that to define the groups in positive degrees. One could, of course, construct an equivariant spectrum (or anOr(G)-spectrum
The second author is partly supported by UMR 7539 of the CNRS.
in the sense of [6]) by combining our classifying spaceKGwith the Bott map Σ2KG→KG; but the approach we use here seems the simplest way to do it.
By comparison, in [6], equivariantK-homology groupsK∗G(X) were defined by using certain covariant functorsKtopfrom the orbit categoryOr(G) to spectra. This construction played an important role in [6] in reformulating the Baum-Connes conjecture. In general, one expects an equivariant homology theory to be classified by a covariant functor from the orbit category to spaces or spectra, and an equivariant cohomology theory to be classified by a contravariant functor. But in fact, when defining equivariantK-theory here, it turned out to be simplest to do so via a classifying G-space, rather than a classifying functor fromOr(G) to spaces.
We would like in particular to thank Chuck Weibel for suggesting Segal’s paper and the use of Γ-spaces, as a way to avoid certain problems we encountered when first trying to define the multiplicative structure onKG(X).
The paper is organized as follows. The classifying spaces forKG−n(−) andKO−Gn(−) are constructed in Section 1; and the connection withG-vector bundles is described. Products are then constructed in Section 2, and are used to define Bott homomorphisms and ring structures on KG∗(X); and thus to complete the construction of equivariantK-theory as a multiplicative equivariant cohomology theory. Homomorphisms in equivariantK-theory involving changes of groups are then constructed in Section 3. Finally, the equivariant Chern characters are constructed in Section 5, and the completion theorems are formulated and proved in Section 6. Section 4 contains some technical results about rational characters.
1. A classifying space for equivariant K-theory
Our classifying space for equivariant K-theory for proper actions of an infinite discrete group is con- structed using Γ-spaces in the sense of Segal. So we begin by summarizing the basic definitions in [10].
Let Γ be the category whose objects are finite sets, and where a morphism θ : S → T sends each elements∈S to a subsetθ(s)⊆T such that s6=s0 impliesθ(s)∩θ(s0) =∅. Equivalently, ifP(S) denotes the set of subsets ofS, one can regard a morphism in Γ as a mapP(S)→ P(T) which sends disjoint unions to disjoint unions. For alln≥0,ndenotes the object {1, . . . , n}. (In particular,0is the empty set.) There is an obvious functor from the simplicial category ∆ to Γ, which sends each object [n] ={0,1, . . . , n}in ∆ to n, and where a morphism in ∆ — an order preserving map ϕ: [m] → [n] — is sent to the morphism θϕ:m→nin Γ which sendsito{j|ϕ(i−1)< j≤ϕ(i)}.
A Γ-space is a functorA: Γop→Spaceswhich satisfies the following two conditions:
(i)A(0) is a point; and
(ii) for eachn >1, the mapA(n)−−→ Qn
i=1A(1), induced by the inclusionsκi:1→n(κi(1) ={i}), is a homotopy equivalence.
(In fact, Segal only requires thatA(0) be contractible; but for our purposes it is simpler to assume it is always a point.) Note that eachA(S) has a basepoint: the image ofA(0) induced by the unique morphismS →0.
We writeA=A(1), thought of as the “underlying space” of the Γ-spaceA. A Γ-spaceA : Γop →Spaces can be regarded as a simplicial space via restriction to ∆, and|A|denotes its topological realization (nerve) as a simplicial space.
If A is a Γ-space, then BA denotes the Γ-space BA(S) = |A(S× −)|; and this is iterated to define BnAfor all n. Thus,BnA=BnA(1) is the realization of then-simplicial space which sends (S1, . . . , Sn) to A(S1× · · · ×Sn). SinceA(0) is a point, we can identify ΣA (= Σ(A(1))) as a subspace ofBA ∼=|A|; and this induces by adjointness a map A→ΩBA. Upon iterating this, we get maps Σ(BnA)→Bn+1Afor all
n; and these make the sequenceA, BA, B2A, . . . into a spectrum. This is “almost” an Ω-spectrum, in that BnA'ΩBn+1A for alln≥1 [10, Proposition 1.4].
Note that for any Γ-spaceA, the underlying spaceA=A(1) is anH-space: multiplication is defined to be the composite of a homotopy inverse of the equivalenceA(2)−−→' A(1)×A(1) with the mapA(2)→A(1) induced bym2 :1→2(m2(1) ={1,2}). ThenA'ΩBAifπ0(A) is a group; and ΩBA is the topological group completion ofAotherwise. All of this is shown in [10,§1].
We work here with equivariant Γ-spaces; i.e., with functors A: Γop→G-Spacesfor which A(0) is a point, and for which A(n)H
→Qn
i=1 A(1)H
is a homotopy equivalence for allH ⊆G. In other words, restriction to fixed point sets of anyH ⊆Gdefines a Γ-spaceAH; and the properties of equivariant Γ-spaces follow immediately from those of nonequivariant ones. For example, Segal’s [10, Proposition 1.4] implies immediately that for any equivariant Γ-spaceA,BnA→ΩBn+1Ais a weak equivalence for alln≥1 in the sense that it restricts to an equivalence (BnA)H'(ΩBn+1A)H for allH ⊆G. This motivates the following definitions.
IfFis any family of subgroups ofG, then aweakF-equivalenceofG-spaces is aG-map whose restriction to fixed point sets of any subgroup inF is a weak homotopy equivalence in the usual sense. The following lemma about maps to weak equivalences is well known; we note it here for later reference.
Lemma 1.1. Fix a familyF of subgroups ofG, and letf :Y →Y0 be any weakF-equivalence. Then for any G-complexX all of whose isotropy subgroups are inF, the map
f∗: [X, Y]G
∼
−−−−−→= [X, Y0]G
is a bijection. More generally, if A⊆X is any G-invariant subcomplex, and all isotropy subgroups of XrA are in F, then for any commutative diagram
A −−−−→α0 Y
y f
y X −−−−→α Y0
ofG-maps, there is an extension ofα0to aG-mapαe:X →Y such thatf◦αe'α(equivariantly homotopic), andαe is unique up to equivariant homotopy.
Proof. The idea is the following. Fix a G-orbit of cells G/H×Dn → X
in X whose boundary is inA.
Then, sinceYH→(Y0)H is a weak homotopy equivalence, the mapeH×Dn→XH→(Y0)H can be lifted toYH (up to homotopy), and this extends equivariantly to aG-mapG/H×Dn →Y. Upon continuing this procedure, we obtain a lifting ofαto aG-mapαe:X→Y which extendsα0. This proves the existence of a lifting in the above square (and the surjectivity of f∗ in the special case); and the uniqueness of the lifting follows upon applying the same procedure to the pairX×I⊇(X×{0,1})∪(A×I).
Now fix a discrete groupG. Let E(G) be the category whose objects are the elements of G, and with exactly one morphism between each pair of objects. Let B(G) be the category with one object, and one morphism for each element of G. (Note that |E(G)| =EGand |B(G)| =BG; hence the notation.) When necessary to be precise, ga will denote the morphism a → ga in E(G). We let G act on E(G) via right multiplication:x∈Gacts on objects by sendingatoax and on morphisms by sendingga togax. Thus, for anyH ⊆G, the orbit categoryE(G)/H is the groupoid whose objects are the cosets inG/H, and with one morphismgaH :aH →gaH for eachg∈G: a category which is equivalent toB(H). Note in particular that B(G)∼=E(G)/G.
In order to deal simultaneously with real and complexK-theory, we letF denote one of the fieldsCor R. SetF∞=S∞
n=1Fn: the space of all infinite sequences inF with finitely many nonzero terms. LetF-mod be the category whose objects are the finite dimensional vector subspaces of F∞, and whose morphisms are F-linear isomorphisms. The set of objects of F-mod is given the discrete topology, and the space of morphisms between any two objects has the usual topology.
For any finite setS, anS-partitioned vector space is an objectV ofF-mod, together with a direct sum decomposition V =L
s∈SVs. LetFhSi-moddenote the category ofS-partitioned vector spaces inF-mod, where morphisms are isomorphisms which respect the decomposition. In particular,Fh0i-modhas just one object 0 ⊆ F∞ and one morphism. A morphism θ : S → T induces a functor Fhθi from FhTi-mod to FhSi-mod, by sendingV =L
t∈TVtto W =L
s∈SWs whereWs=L
t∈θ(s)Vt. Let VecFG be the Γ-space defined by setting
VecFG(S)def=
func(E(G), FhSi-mod)
for each finite setS. Here, func(C,D) denotes the category of functors fromCtoD. We give this theG-action induced by the action onE(G) described above. This is made into a functor on Γ via composition with the functorsFhθi.
By definition, VecFG(0) is a point. To see that VecFG is an equivariant Γ-space, it remains to show for eachnandH that the map VecFG(n)H
→Qn
i=1 VecFG(1)H
is a homotopy equivalence. The target is the nerve of the category of functors from E(G)/H to n-tuples of objects inF-mod, while the source can be thought of as the nerve of the full subcategory of functors from E(G)/H to n-tuples of vector subspaces which are independant inF∞. And these two categories are equivalent, since every object in the larger one is isomorphic to an object in the smaller (and the set of objects is discrete).
For all finiteH⊆G, VecFGH
is the disjoint union, taken over isomorphism classes of finite dimensional H-representations, of the classifying spaces of their automorphism groups. We will see later that VecFG classifies G-vector bundles over properG-complexes. So it is natural to define equivariant K-theory using the its group completionKFG
def= ΩBVecFG, regarded as a pointedG-space.
In the following definition, [−,−]G and [−,−]
·
G denote sets of homotopy classes of G-maps, and of pointedG-maps, respectively.Definition 1.2. For each properG-complexX, set
KG(X) = [X, KCG]G and KOG(X) = [X, KRG]G. For each proper G-CW-pair (X, A)and each n≥0, set
KG−n(X, A) = [Σn(X/A), KCG]
·
G and KO−Gn(X, A) = [Σn(X/A), KRG]·
G.The usual cohomological properties of the KFG−n(−) follow directly from the definition. Homotopy invariance and excision are immediate; and the exact sequence of a pair and the Mayer-Vietoris sequence of a pushout square are shown using Puppe sequences to hold in degrees≤0. Note in particular the relations
KFG−n(X)∼= Ker
KFG(Sn×X)−−→KFG(X) KFG−n(X, A)∼= Ker
KFG−n(X∪AX)−−→KFG−n(X)
, (1.3)
for any properG-CW-pair (X, A) and anyn≥0.
The following lemma will be needed in the next section. It is a special case of the fact that VecFG and KFG (at least up to homotopy) are independent of our choice of category of F-vector spaces.
Lemma 1.4. For any monomorphism α : F∞ → F∞, the induced map α∗ : VecFG → VecFG, defined by composition withF-mod−−−→α(−) F-mod, isG-homotopic to the identity. In particular,α∗induces the identity onKG(X).
Proof. The functor V 7→α(V)
is naturally isomorphic to the identity.
In [8], we defined KG(X), for any proper G-complexX, to be the Grothendieck group of the monoid of vector bundles over X. We next construct natural homomorphisms KG(X) → KG(X), for all proper
G-complexesX, which are isomorphisms if X/Gis a finite complex (this is the situation where the K∗G(X) form an equivariant cohomology theory).
For each n ≥0, let Fn-mod⊆ F-mod be the full subcategory of n-dimensional vector subspaces in F∞. LetFn-framedenote the category whose objects are the pairs (V, b), whereV is an object ofFn-mod and b is an ordered basis of V; and whose morphisms are the isomorphisms which send ordered basis to ordered basis. The set of objects is given the topology of a disjoint union of copies ofGLn(F) (one for each V inFn-mod). Note that there is a unique morphism between any pair of objects inFn-frame. Set
VecF,nG =
func(E(G), Fn-mod)
and VecgF,nG =
func(E(G), Fn-frame) ,
with the action of G×GLn(F) on VecgF,nG induced by the G-action on E(G) and the GLn(F)-action on the set of ordered bases of eachn-dimensional V. Letτn :VecgF,nG →VecF,nG be the G-map induced by the forgetful functorFn-frame→Fn-mod. ThenGLn(F) acts freely and properly onVecgF,nG . And τn induces aG-homeomorphismVecgF,nG /GLn(F)∼= VecF,nG , since for anyϕ:V →V0 inF-mod, a lifting of V orV0 to Fn-framedetermines a unique lifting of the morphism.
LetH ⊆G×GLn(F) be any subgroup. IfH∩(1×GLn(F))6= 1, then (gVecF,nG )H =∅, sinceGLn(F) acts freely on VecgF,nG . So assume H ∩(1×GLn(F)) = 1. Then H is the graph of some homomorphism ϕ : H0 → GLn(F) (H0 ⊆ G), and (gVecF,nG )H is the nerve of the (nonempty) category of ϕ-equivariant functorsE(G)→Fn-frame, with a unique morphism between any pair of objects (since there is a unique morphism between any pair of objects inFn-frame). In particular, this shows that (VecgF,nG )H is contractible.
Thus,VecgF,nG is a universal space for those (G×GLn(F))-complexes upon whichGLn(F) acts freely (cf.
[8,§2]). The frame bundle of anyn-dimensionalG-F-vector bundle over aG-complexX is such a complex, and hencen-dimensional G-F-vector bundles overX are classified by maps to VecF,nG =gVecF,nG /GLn(F). It follows that
EVecF,nG =VecgF,nG ×GLn(F)Fn −−−−−−−→VecF,nG
is a universal n-dimensional G-F-vector bundle. And [X,VecF,nG ]G ∼=VectF,nG (X): the set of isomorphism classes ofn-dimensionalG-F-vector bundles overX.
If E is any G-F-vector bundle over X, we let [[E]]∈ KFG(X) = [X, KFG]G be the composite of the classifying mapX →VecFG forE with the group completion map VecFG→ΩBVecFG=KFG. Any pairE, E0 of vector bundles overX is induced by aG-map
X −−−−−→VecFG×VecFG=|func(E(G), F-mod×F-mod)| ' |func(E(G), Fh2i-mod)|;
and upon composing with the forgetful functorFh2i-mod→F-modwe get the classifying map forE⊕E0. The direct sum operation onVectFG(X) is thus induced by the H-space structure on VecFG, and [[E⊕E0]] = [[E]] + [[E0]] for allE, E0.
Proposition 1.5. The assignment [E]7→[[E]]
defines a homomorphism γX:KFG(X)−−−−−→KFG(X),
for any properG-complexX. This extends to natural homomorphismsγX,A−n :KFG−n(X, A)→KFG−n(X, A), for all proper G-CW-pairs(X, A) and alln≥0; which are isomorphisms when restricted to the category of finite proper G-CW-pairs.
Proof. By the above remarks, [E]7→[[E]]
defines a homomorphism of monoids fromVectFG(X) toKFG(X), and hence a homomorphism of groups
γX:KFG(X)−−−−−−→KFG(X).
HomomorphismsγX,A−n (for all proper G-CW-pairs (X, A)) are then constructed via the definitions KFG−n(X)def= Ker[KFG(Sn×X)→KFG(X)]
and KFG−n(X, A) def= Ker[KFG−n(X∪AX) → KFG−n(X)] used in [8], together with the analogous relations (1.3) forKG∗(−). These homomorphisms clearly commute with boundary maps.
It remains to check thatγX−nis an isomorphism wheneverXis a finite properG-complex. SinceKFG(−) andKFG(−) are both cohomology theories in this situation, it suffices, using the Mayer-Vietoris sequences for pushout squares
G/H×Sm−1 −−−−→ G/H×Dm
ϕ
y
y
X −−−−→ (G/H×Dm)∪ϕX,
to do this whenX =G/H×Smfor finiteH⊆Gand anym≥0. Using (1.3) again, it suffices to show that γX =γX0 is an isomorphism whenever X =G/H×Y for any finite complexY with trivial G-action. By definition,
KFG(G/H×Y) =
G/H×Y, KFG
G∼= [Y,(KFG)H];
whileKFG(G/H×Y) is the Grothendieck group of the monoid VectFG(G/H×Y)∼=
G/H×Y,VecFG
G∼=
Y,(VecFG)H .
Sinceπ0((VecFG)H) is a free abelian monoid (the monoid of isomorphism classes ofH-representations), [10, Proposition 4.1] applies to show that [−,(KFG)H] is universal among representable functors from compact spaces to abelian groups which extend VectFG(G/H×−) ∼=VectFH(−). And since KH is representable as a functor on compact spaces with trivial action (H is finite), it is the universal functor, and so [Y,(KFG)H]∼= KH(Y)∼=KG(G/H×Y).
2. Products and Bott periodicity
We now want to construct Bott periodicity isomorphisms, and define the multiplicative structures on KG∗(X) andKO∗G(X). Both of these require defining pairings of classifying spaces; thus pairings of Γ-spaces.
A general procedure for doing this was described by Segal [10,§5], but a simpler construction is possible in our situation.
Fix an isomorphism µ : F∞⊗F∞ → F∞ (F = C or R), induced by some bijection between the canonical bases. This induces a functor
µ∗:FhSi-mod×FhTi-mod−−−−−→FhS×Ti-mod, and hence (for any discrete groupsH andG)
µ∗: VecFH(S)∧VecFG(T)−−−−−→VecFH×G(S×T). (2.1) This is an (H×G)-equivariant map of bi-Γ-spaces, and after taking their nerves (and loop spaces) we get maps
ΩBVecFH∧ΩBVecFG
=KFH∧KFG
−−−−→Ω2 BVecFH∧BVecFG Ω2|µ∗|
−−−−−→Ω2B2VecFH×G'ΩBVecFH×G
=KFH×G
. (2.2) By Lemma 1.4, these maps are all independent (up to homotopy) of the choice ofµ:F∞⊗F∞→F∞. Lemma 2.3. For any discrete groups H and G, any H-space X, and any G-space Y, the following square commutes:
KFH(X)⊗KFG(Y) −−−−−→γX×γY KFH(X)⊗KFG(Y)
⊗
y
µ∗
y KFH×G(X×Y) −−−−−→γX×Y KFH×G(X×Y) whereµ∗ is the homomorphism induced by (2.2).
Proof. The pullback of the universal bundle EVecFH×G, via the pairing VecFH∧VecFG →VecFH×G of (2.1), is isomorphic to the tensor product of the universal bundles EVecFH and EVecFG. This is clear if we identify EVecFG ∼=
func(E(G), F-Bdl)
(and similarly for the other two bundles), whereF-Bdlis the category of pairs (V, x) forV in F-modandx∈V.
We now consider case where H = 1, and hence whereKFH =Z×BU or Z×BO. The product map (2.2), after composition with the Bott elements inπ2(BU) orπ8(BO), induces Bott maps
β∗C: Σ2KG−−−−−→KG and βR∗ : Σ8KOG−−−−−→KOG. (2.4) Proposition 2.5. For any proper CW-pair(X, A), the Bott homomorphisms
bCX,A:KG−n(X, A)−−−−−→KG−n−2(X, A) and bRX,A:KOG−n(X, A)−−−−−→KO−Gn−8(X, A) are isomorphisms; and commute with the homomorphisms
γX,A−n :KFG−n(X, A)→KFG−n(X, A).
Proof. The last statement follows immediately from Lemma 2.3.
By Lemma 1.1, it suffices to prove that the adjoint maps
KG−−−−→Ω2KG and KOG −−−−→Ω8KOG
to the pairings in (2.4) are weak homotopy equivalences after restricting to fixed point sets of finite subgroups of G. In other words, it suffices to prove that bCX : KG(X) → KG−2(X) and bRX : KOG(X) → KO−G8(X) are isomorphisms when X =G/H×Sn for any n≥ 0 and any finite H ⊆G. And this follows since the Bott maps for KG and KOG are isomorphisms [8, Theorems 3.12 & 3.15], since KFG−n(X) ∼= KFG−n(X) (Proposition 1.5), and since these isomorphisms commute with the Bott maps.
TheKG−n(X) andKOG−n(X) can now be extended to (additive) equivariant cohomology theories in the usual way. But before stating this explicitly, we first consider the ring structure onKG(X). This is defined to be the composite
[X, KFG]G×[X, KFG]G−−−−−→[X, KFG×G]G−−−−−→∆∗ [X, KFG]G,
where the first map is induced by the pairing in (2.2), and the second by restriction to the diagonal subcat- egoryE(G)⊆ E(G×G).
Before we can prove the ring properties of this multiplication, we must look more closely at the homotopy equivalence ΩBVecFG −→' Ω2B2VecFG which appears in the definition of the product. In fact, there is more than one natural map from ΩnBnVecFG to Ωn+1Bn+1VecFG. For each n ≥ 0 and each k = 0, . . . , n, let ιkn : ΩnBnVecFG → Ωn+1Bn+1VecFG denote the map induced as Ωn(f), where f is adjoint to the map ΣBnVecFG→Bn+1VecFG, induced by identifyingBnVecFG(S1, . . . , Sn) withBn+1VecFG(. . . , Sk−1,1, Sk, . . .).
By a weakG-equivalencef :X →Y is meant a map ofG-spaces which restricts to a weak equivalence fH:XH →YHfor allH ⊆G; i.e., a weakF-equivalence in the notation of Lemma 1.1 whenF is the family of all subgroups ofG. Since we are interested equivariant Γ-spaces only as targets of maps fromG-complexes, it suffices by Lemma 1.1 to work in a category where weakG-equivalences are inverted.
Lemma 2.6. LetAbe any G-equivariant Γ-space. Then for anyn≥1, the mapsιkn: ΩnBnA→Ωn+1Bn+1A (for0≤k≤n) are all equal in the homotopy category ofG-spaces where weak G-equivalences are inverted.
Proof. For anyσ∈Σn, letσ∗: ΩnBnA→ΩnBnAbe the map induced by permuting the coordinates ofBnA as ann-simplicial set, and by switching the order of looping. Then any two of theιkn−1differ by composition with some appropriateσ∗, and so it suffices to show that theσ∗ are all homotopic to the identity.
Consider the following commutative diagram
ΩBA −−−−→ϕ Ωn+1Bn+1A ι
n
←−−−−n ΩnBnA
Id
y (1×σ)∗
y σ∗
y
ΩBA −−−−→ϕ Ωn+1Bn+1A ι
n
←−−−−n ΩnBnA,
for any σ ∈ Σn ⊆ Σn+1, where ϕ = ι0n◦· · ·◦ι01 is induced by identifying A(S) with A(S,1, . . . ,1). The diagram commutes, and all maps in it are weakG-equivalences by [10, Proposition 1.4]. So (1×σ)∗ andσ∗ are both homotopic to the identity after inverting weakG-equivalences.
We are now ready to show:
Theorem 2.7. For any discrete group and any proper G-complex X, the pairings µX define a structure of graded ring on KG∗(X) and on KOG∗(X), which make KG∗(−)and KO∗G(−) into multiplicative cohomology theories. The Bott isomorphisms
bCX :KG−n(X)→KG−n−2(X) and bRX:KO−Gn(X)→KOG−n−8(X) areKG(X)- orKOG(X)-linear. And the canonical homomorphisms
γCX:K∗G(X)→KG∗(X) and γXR :KO∗G(X)→KO∗G(X) are homomorphisms of rings.
Proof. As usual, setF =C or R. We first check thatµX makes KFG(X) into a commutative ring — i.e., that it is associative and commutative and has a unit.
To see that there is a unit, let [F1]∈VecFG denote the vertex for the constant functorE(G)7→F1 ∈ Fh1i-mod, and set [F1]Ω=ι00([F1])∈ΩBVecFG. The following diagram commutes:
ΩBVecFG [F
1]∧−
−−−−−−−→ VecFG∧ΩBVecFG −−−−−−−→µ∗ ΩBVecFG
Id
y ι
0 0∧Id
y' ι
0 1
y' ΩBVecFG [F
1]Ω∧−
−−−−−−−→ ΩBVecFG∧ΩBVecFG −−−−−−−→µ∗ Ω2B2VecFG;
and the composite in the top row is homotopic to the identity by Lemma 1.4. So the element 1∈KFG(X), represented by the constant mapX 7→[F1]Ω∈KFG, is an identity for multiplication inKFG(X).
The commutativity ofKFG(X) follows from Lemma 2.6 (the uniqueness of the map ΩBA→Ω2B2A after inverting weakG-equivalences); together with the fact that the pairing
µ∗:BVecFG∧BVecFG −−−−−→B2VecFG
commutes up to homotopy using Lemma 1.4. And associativity follows since the triple products are induced by maps
ΩBVecFG∧3
−−−−−→Ω3 (BVecFG)∧3 Ω
3|µ∗◦(µ∗∧Id)|
−−−−−−−−−−→
−−−−−−−−−−→
Ω3|µ∗◦(Id∧µ∗)|
Ω3B3VecFG←−−−−−' ΩBVecFG;
where the two maps in the middle are homotopic by Lemma 1.4, and the last could be any of the three possible maps by Lemma 2.6.
The extension of the product to negative gradings is straightforward, via the identifications of (1.3).
For anyn, m≥0, the composite
KFG(Sn×X)⊗KFG(Sm×X) proj
∗
−−−−−→KFG(Sn×Sm×X)⊗KFG(Sn×Sm×X)
−−−−−→µ KFG(Sn×Sm×X)
restricts to a product mapKFG−n(X)⊗KFG−m(X)→KFG−n−m(X). To see that the product has image in KFG−n−m(X), just note that
KFG−n−m(X)∼= Ker
KFG(Sn+m×X)−−→KFG(X)
= Ker
KFG(Sn×Sm×X)−−→KFG(Sn×X)⊕KFG(Sm×X) .
This product is clearly associative, and graded commutative (where the change in sign comes from the degree of the switching mapSn+m→Sm+n).
We next check that this product commutes with the Bott maps in the obvious way, so that it can be extended toKGi(X) for alli. This means showing that the two maps
KF(Sn)⊗KFG(X)⊗KFG(X) −−−−−−−→−−−−−−−→ KFG(Sn×X)
induced by the products constructed above are equal. And this follows from the same argument as that used to prove associativity of the internal product onKFG(X).
Finally,γ:KFG∗(X)→KFG∗(X) is a ring homomorphism by Lemma 2.3.
3. Induction, restriction, and inflation
In this section we explain how the natural maps defined on KG(X) and KOG(X) by induction and restriction carry over toKG(X) andKOG(X). Namely, we want to construct for any pairH ⊆Gof discrete groups, anyF=CorR, anyG-complexX, and any H-complexY, natural induction and restriction maps
IndGH :KFH∗(Y)−−−−→∼= KFG∗(G×HY) and ResGH:KFG∗(X)−−−−→KFH∗(X|H).
Furthermore, whenH CGis a normal subgroup, we construct an inflation homomorphism InflGG/H :KFG/H∗ (X/H)−−−−−−→KFG∗(X),
which is an isomorphism wheneverH acts freely onX. These maps correspond under the natural homomor- phismKFG∗(X)→KFG∗(X) to the obvious homomorphisms induced by induction, restriction, and pullback of vector bundles. They are all induced using the following maps between classifying spaces for equivariant K-theory.
Lemma 3.1. Let f :G0 →Gbe any homomorphism of discrete groups. Then composition with the induced functor E(f) : E(G0) → E(G) induces an G0-equivariant map f∗ : VecFG → VecFG0 of Γ-spaces, and hence a G0-equivariant map f∗ : KFG → KFG0 of classifying spaces. And for any subgroup L ⊆ G0 such that L∩Ker(f) = 1,f∗ restricts to a homotopy equivalence(KFG)f(L)'(KFG0)L.
Proof. This is immediate, except for the last statement. And if L ⊆ G0 is such that L∩Ker(f) = 1, then L∼=f(L), the categories E(G0)/L andE(G)/f(L) are both equivalent to the category B(L) with one object and endomorphism groupL; and thus (VecFG)f(L)(S) =|func(E(G)/f(L), FhSi-mod)| is homotopy equivalent to (VecFG0)L(S) =|func(E(G0)/L, FhSi-mod)|for eachS in Γ.
We first consider the restriction and induction homomorphisms.
Proposition 3.2. Fix F =Cor R, and let H ⊆Gbe any pair of discrete groups. Let i∗ :KFG →KFH be the map of Lemma 3.1.
(a) For any proper G-CW-pair(X, A),i∗ induces a homomorphism of rings ResGH:KFG∗(X, A)−−−−−−→KFH∗(X, A).
(b) For any proper H-CW-pair (Y, B),i∗ induces an isomorphism
IndGH:KFH∗(Y, B)−−−−−−→∼= KFG∗(G×HY, G×HB),
which is natural in (Y, B), and also natural with respect to inclusions of subgroups.
The restriction and induction maps both commute with the maps betweenKFG(−)andKFH(−) induced by induction and restriction of equivariant vector bundles.
Proof. It suffices to prove this whenA=∅=B and∗= 0. The fact thati∗:KFG →KFH commutes with the Bott homomorphisms and the products follows directly from the definitions. So part (a) is clear.
The inverse of the homomorphism in (b) is defined to be the composite [G×HY, KFG]G∼= [Y, KFG]H
i∗◦−
−−−−−−−→[Y, KFH]H.
And sincei∗ restricts to a homotopy equivalence (KFG)L →(KFH)L for each finiteL⊆H (Lemma 3.1), this map is an isomorphism by Lemma 1.1.
The last statement is clear from the construction and the definition ofγ:KFG(−)→KFG(−).
We next consider the inflation homomorphism.
Proposition 3.3. Fix F =CorR. Let Gbe any discrete group, and letN CGbe a normal subgroup. Then for each proper G-CW-pair(X, A), there is an inflation map
InflGG/N :KFG/N∗ (X/N, A/N)−−−−−−→KFG∗(X, A),
which is natural in (X, A), which is a homomorphism of rings (if A = ∅), and which commutes with the homomorphism KFG/N(X/N, A/N) → KFG(X, A) induced considering G/N-vector bundles as G-vector bundles. And ifN acts freely on X, then InflGG/N is an isomorphism.
Proof. Letf :G→G/N denote the natural homomorphism, and let f∗: KFG/N →KFG be the induced map of Lemma 3.1. Define InflGG/N to be the composite
[X/N, KFG/N]G/N ∼= [X, KFG/N]G f
∗◦−
−−−−−→[X, KFG]G.
IfN acts freely onX, then for each isotropy subgroupLofX,L∩N= 1, so (f∗)L: (KFG/N)L→(KFG)L is a homotopy equivalence by Lemma 3.1, and the inflation map is an isomorphism by Lemma 1.1. The other statements are clear.
Another type of natural map will be needed when constructing the equivariant Chern character. Fix a discrete groupG and a finite normal subgroupN CG, and let Irr(N) be the set of isomorphism classes of irreducible complex N-representations. Let X be any proper G/N-complex. For any V ∈ Irr(N) and any G-vector bundle E → X, let HomN(V, E) denote the vector bundle over X whose fiber over x ∈ X is HomN(V, Ex) (each fiber of E is anN-representation). If H ⊆G is any subgroup which centralizes N, then we can regard HomN(V, E) as anH-vector bundle by setting (hf)(x) =h·f(x) for anyh∈H and any f ∈HomN(V, E). We thus get a homomorphism
Ψ :KG(X)−−−−−−→KH(X)⊗R(N), where Ψ([E]) =P
V∈Irr(N)[HomN(V, E)]⊗[V]. We need a similar homomorphism defined on KG∗(X).
Proposition 3.4. LetGbe a discrete group, let NCGbe any finite normal subgroup, and letH ⊆Gbe any subgroup such that [H, N] = 1. Then for any properG/N-complexX, there is a homomorphism of rings
Ψ = ΨXG;N,H :KG∗(X)−−−−−−−→KH∗(X)⊗R(N),
which is natural in X and natural with respect to the degree-shifting maps KG∗(X) → KG∗+n(Sn×X), and which has the following properties:
(a) For any (complex)G-vector bundleE→X, Ψ([[E]]) = X
V∈Irr(N)
[[HomN(V, E)]]⊗[V].
(b) For any G0 ⊆G,N0⊆N∩G0, andH0⊆H∩G0, the following diagram commutes:
KG∗(X) Ψ
X G;N,H
−−−−−−−−→ KH∗(X)⊗R(N)
ResGG0
y Res
H H0
y⊗Res
N N0
KG∗0(X) Ψ
X G0;N0,H0
−−−−−−−−→ KH∗0(X)⊗R(N0).
Proof. Fix G, H, and N. For any irreducible N-representation V and any surjective homomorphism p : C[N]−V, composition withpdefines a monomorphism
HomN(V, W)−−−−−→−◦p HomN(C[N], W) =W
for anyN-representationW; and thus allows us to identify HomN(V, W) as a subspace ofW. In particular, there is a functor
p∗: func(Or(G)/N,ChSi-mod)−−−−−→ func(Or(H),ChSi-mod)
which sends anyαto the functorh7→HomN(V, α(hN))⊆α(hN). Ifp0 :C[N]−V0is another surjection of N-representations, whereV ∼=V0, then any isomorphismV −→∼= V0 defines a natural isomorphism between p∗and (p0)∗. We thus get a map of Γ-spaces
ψp : VecCG−−−−−−→VecCH
which is unique (independant of the projection p) up to H-equivariant homotopy. So this induces homo- morphismsψV :KG−n(X)→ KH−n(X), for all proper G/N-complexX (and all n≥0), which depend only on V and not on p. The ψV clearly commute with the Bott maps, and thus extend to homomorphisms ψV :KG∗(X)→KH∗(X). So we can define Ψ by setting Ψ(x) =P
V∈Irr(N)ψV(x)⊗[V]. Point (a) is imme- diate; as is naturality in X and naturality for restriction toG0 ⊆Gor H0 ⊆H. Naturality with respect to the degree-shifting maps holds by construction.
We next show that Ψ is natural inN; i.e., that point (b) holds whenG0=GandH0=H. LetψV be the homomorphisms defined above, for each irreducibleN-representationV; and letψW0 :KG∗(X)→KH∗(X) be the corresponding homomorphism for each irreducible N0-representation W. For eachV ∈ Irr(N) and eachW ∈Irr(N0), set
nVW = dimC HomN0(W, V)
= dimC HomN(IndNN0(W), V) .
Thus, nVW is the multiplicity of W in the decomposition of V|N0, as well as the multiplicity of V in the decomposition of IndNN0(W). So for anyx∈KG∗(X),
(Id⊗ResNN0)(ΨG;N,H(x)) = X
V∈Irr(N)
ψV(x)⊗[V|N0] = X
W∈Irr(N0)
X
V∈Irr(N)
nVW·ψV(x)
⊗[W];
and we will be done upon showing that ψW0 = P
V nVW·ψV for each W ∈ Irr(N0). Fix a surjection p0 : C[N0]−W, and a decomposition IndNN0(W) = Pk
i=1Vi (where the Vi are irreducible and k=P
V nVW).
For each 1≤i≤k, letpi :C[N]−Vi be the composite of IndNN0(p0) followed by projection toVi. Then ψp0 =
k
M
i=1
ψpi : VecCGN
−−−−−−→VecCH as maps of Γ-spaces, and soψW 'Pk
i=1ψVi as mapsKG∗(X)→KH∗(X).
It remains to show that Ψ is a homomorphism of rings. Since it is natural in N, and since R(N) is detected by characters, it suffices to prove this whenN is cyclic. For anyx, y∈KG(X),
Ψ(x)·Ψ(y) = X
V,W∈Irr(N)
ψV(x)·ψW(y)
⊗[V ⊗W] and Ψ(xy) = X
U∈Irr(N)
ψU(xy)⊗[U].
And thus Ψ(x)·Ψ(y) = Ψ(xy) since ψU◦µ∗= M
V,W∈Irr(G) V⊗W∼=U
µ∗◦(ψV ∧ψW) : VecCGN
∧ VecCGN
−−−−−−→VecCH,
as maps of Γ-spaces, for eachU ∈Irr(N).
4. Characters and class functions
Throughout this section, G will be a finite group. We prove here some results showing that certain class functions are characters; results which will be needed in the next two sections.
For any field K of characteristic zero, a K-character of G means a class function G → K which is the character of some (virtual)K-representation ofG. Two elementsg, h∈Gare calledK-conjugate ifg is conjugate toha for someaprime ton=|g|=|h|such that (ζ7→ζa)∈Gal(Kζ/K), whereζ= exp(2πi/n).
For example,gandhareQ-conjugate ifhgiandhhiare conjugate as subgroups, and areR-conjugate ifg is conjugate tohorh−1.
Proposition 4.1. Fix a finite extensionK of Q, and letA⊆K be its ring of integers. Letf :G→Abe any function which is constant onK-conjugacy classes. Then |G|·f is an A-linear combination of K-characters of G.
Proof. Setn=|G|, for short. Let V1, . . . , Vk be the distinct irreducible K[G]-representations, letχi be the character of Vi, set Di = EndK[G](Vi) (a division algebra over K), and set di = dimK(Di). Then by [11, Theorem 25, Cor. 2],
|G|·f =
k
X
i=1
riχi where ri= 1 di
X
g∈G
f(g)χi(g−1);
and we must show that ri ∈ A for alli. This means showing, for each i= 1, . . . , k, and each g ∈ G with K-conjugacy class conjK(g), that |conjK(g)|·χi(g)∈diA.
Fixi andg; and set C=hgi,m=|g|=|C|, andζ = exp(2πi/m). Then Gal(K(ζ)/K) acts freely on the set conjK(g): the element (ζ7→ζa) acts by sending hto ha. So [K(ζ):K]
|conjK(g)|.
Let Vi|C = W1a1 ⊕ · · · ⊕Wtat be the decomposition as a sum of irreducible K[C]-modules. For each j, Kj def= EndK[C](Wj) is the field generated by K and the r-th roots of unity for some r|m (m = |C|), and dimKj(Wj) = 1. So dimK(Wj)
[K(ζ):K]. Also, di
dimK(Wjaj), since Wjaj is a Di-module; and thus di
aj·|conjK(g)|. So if we setξj=χWj(g)∈A, then
|conjK(g)|·χi(g) =|conjK(g)|·
t
X
j=1
ajξj ∈diA,
and this finishes the proof.
For each prime pand each element g∈G, there are unique elements gr of order prime topandgu of p-power order, such thatg=grgu=gugr. As in [11,§10.1], we refer togras thep0-component of g. We say that a class functionf :G→Cisp-constant iff(g) =f(gr) for eachg∈G. Equivalently,f isp-constant if and only iff(g) =f(g0) for all g, g0∈Gsuch that [g, g0] = 1 andg−1g0 hasp-power order.
Lemma 4.2. Fix a finite group G, a prime p, and a field K of characteristic zero. Then a p-constant class functionϕ:G→K is aK-character ofGif and only ifϕ|H is aK-character ofH for all subgroupsH ⊆G of order prime to p.
Proof. Recall first thatGis called K-elementary if for some primeq,G=CmoQ, where Cm is cyclic of order m, q|-|m, Q is a q-group, and the conjugation action of Qon K[Cm] leaves invariant each of its field components. By [11, §12.6, Prop. 36], a K-valued class function of G is a K-character if and only if its restriction to anyK-elementary subgroup ofGis aK-character. Thus, it suffices to prove the lemma when GisK-elementary.
Assume first thatGisq-K-elementary for some prime q6=p. Fix a subgroup H ⊆Gofp-power index and order prime top, and letα:GH be the surjection withα|H = Id. Setpa=|Ker(α)|. Then
Aut(Ker(α))∼= (Z/pa)∗∼= (1 +pZ/pa)×(Z/p)∗,
where the first factor is ap-group. Hence for anyg∈H andx∈Ker(α), either [g, x] = 1 and henceg= (gx)r; or gxg−1=xi for somei6≡1 (mod p) and hence g is conjugate togx. In either case,ϕ(gx) =ϕ(g). Thus, ϕ= (ϕ|H)◦α, and this is aK-character ofGsinceϕ|H is by assumption aK-character ofH.
Now assumeGisp-K-elementary. WriteG=CmoP, wherep|-|mandP is ap-group. LetS be the set of primes which dividem. For eachI⊆S, letCI ⊆Cmbe the product of the Sylowp-subgroups forp∈I, setGI =CIoP, and letαI :GGI be the homomorphism which is the identity onGI.
For each I ⊆ S, we can consider K[CI] as a G-representation via the conjugation action of P; and eachCI-irreducible summand ofK[CI] isP-invariant and henceG-invariant. Thus, each irreducibleK[CI]- representation can be extended to aK[GI]-representation upon whichP∩CG(CI) acts trivially. Hence, since ϕ|CI is aK-character ofCI; there is aK-characterχI ofGI such thatχI(gx) =χI(x) =ϕ(x) for allx∈CI andg∈P such that [g, CI] = 1.
Now set
χ= X
J⊆I⊆S
(−1)|IrJ|(χI◦αJ),
a K-character ofG. We claim thatϕ=χ. Since both are class functions, it suffices to show that ϕ(gx) = χ(gx) for all commutingg∈P andx∈Cm=CS. Fix suchgandx, and letX⊆S be the set of all primes p
|x|. Then [g, CX] = 1, and so χ(gx) = X
J⊆I⊆S
(−1)|IrJ|χI(αJ(gx)) = X
J⊆I⊆S
(−1)|IrJ|χI(g·αJ(x))
= X
J⊆I⊆X
(−1)|IrJ|ϕ(αJ(x)) + X
J⊆I6⊆X
(−1)|IrJ|χI(g·αJ(x))
=ϕ(x) =ϕ(gx).
Note, in the second line, that all terms in the second sum cancel sinceαJ(x) =αJ0(x) ifJ =J0∩X, and all terms in the first sum cancel except that whereJ=I=X.
WhenA=ZandK=Q, Proposition 4.1 and Lemma 4.2 combine to give:
Corollary 4.3. Fix a finite group Gand a primep. Let f :G→Zbe any function which is p-constant, and constant onQ-conjugacy classes inG. Set|G|=m·pr wherep|-|m. Thenm·f is aQ-character of G.
5. The equivariant Chern character
We construct here two different equivariant Chern characters, both defined on the equivariant complex K-theory of proper G-complexes. The first is defined for arbitrary X (with proper G-action), and sends
