Chern characters for equivariant K-theory of proper G-CW-complexes

by Wolfgang Lück and Bob Oliver^{∗}

In an earlier paper [10], we showed that for any discrete group G, equivariant K-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 K_{G}^{∗}(X) for an arbitrary proper G-complex X. 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/H ↦ R(H)). The first,

ch_{X}^{∗} : K_{G}^{∗}(X) ⟶ H_{G}^{∗}(X; ℚ ⊗ R(−)),

is defined for arbitrary proper G-complexes. The second, a refinement of the first, is a homomorphism

c̃h_{X}^{∗} : K_{G}^{∗}(X) ⟶ ℚ ⊗ H_{G}^{∗}(X; R(−)),

but defined only for finite dimensional proper G-complexes for which the isotropy subgroups on X have bounded order. When X is a finite proper G-complex (i.e., X/G is a finite CW-complex), then H_{G}^{∗}(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 cohomology H_{G}^{∗}(X; ℚ ⊗ R(−)) was first due to Słomińska [14]. A complex-valued Chern character was constructed earlier by Baum and Connes [5], using very different methods.

The completion theorem of [10] 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 [9]. This means that for each finite proper G-complex X and each family F of subgroups of G, K_{G}^{∗}(E_{F}(G) × X) is shown to be isomorphic to a certain completion of K_{G}^{∗}(X). In particular, when F = {1}, then E_{F}(G) = EG, and this becomes the usual completion theorem.

The classifying spaces for equivariant K-theory are constructed here using Segal’s Γspaces. 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 [12], the spectra they produce are connective, and hence not what is needed to define equivariant K-theory directly. So instead, we define K_{G}^{−n}(−) and KO_{G}^{−n}(−) for all n ≥ 0 using classifying spaces 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 an Or(G)- spectrum in the sense of [7]) by combining our classifying space K_{G} with the Bott map Σ^{2}K_{G} → K_{G}; but the approach we use here seems the simplest way to do it.

∗Partly supported by UMR 7953 of the CNRS

By comparison, in [7], equivariant K-homology groups K_{∗}^{G}(X) were defined by using certain covariant functors K^{top} from the orbit category Or(G) to spectra. This construction played an important role in [7] 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 equivariant K-theory here, it turned out to be simplest to do so via a classifying G-space, rather than a classifying functor from Or(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 on K_{G}(X).

The paper is organized as follows. The classifying spaces for K_{G}^{−n}(−) and KO_{G}^{−n}(−) are constructed in Section 1; and the connection with G-vector bundles is described. Products are then constructed in Section 2, and are used to define Bott homomorphisms and ring structures on K_{G}^{∗}(X); and thus to complete the construction of equivariant K-theory as a multiplicative equivariant cohomology theory. Homomorphisms in equivariant K-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 constructed using Γ-spaces in the sense of Segal. So we begin by summarizing the basic definitions in [12].

Let Γ be the category whose objects are finite sets, and where a morphism θ : S → T sends each element s ∈ S to a subset θ(s) ⊆ T such that s ≠ s′ implies θ(s) ∩ θ(s′) = ∅. Equivalently, if P(S) denotes the set of subsets of S, one can regard a morphism in Γ as a map P(S) → P(T) which sends disjoint unions to disjoint unions. For all n ≥ 0, n denotes the object {1, …, n}. (In particular, 0 is the empty set.) There is an obvious functor from the simplicial category Δ to Γ, which sends each object [n] = {0, 1, …, n} in Δ to n, and where a morphism in Δ — an order preserving map φ : [m] → [n] — is sent to the morphism θ_{φ} : m → n in Γ which sends i to {j | φ(i−1) < j ≤ φ(i)}.

A Γ-space is a functor \underline{A} : Γ^{op} → Spaces which satisfies the following two conditions:

(i) \underline{A}(0) is a point; and

(ii) for each n > 1, the map \underline{A}(n) ⟶ ∏_{i=1}^{n} \underline{A}(1), induced by the inclusions κ_{i} : 1 → n (κ_{i}(1) = {i}), is a homotopy equivalence.

(In fact, Segal only requires that \underline{A}(0) be contractible; but for our purposes it is simpler to assume it is always a point.) Note that each A(S) has a basepoint: the image of A(0) induced by the unique morphism S → 0. We write A = \underline{A}(1), thought of as the “underlying space” of the Γ-space \underline{A}. A Γ-space \underline{A} : Γ^{op} → Spaces can be regarded as a simplicial space via restriction to Δ, and |\underline{A}| denotes its topological realization (nerve) as a simplicial space.

If \underline{A} is a Γ-space, then \underline{BA} denotes the Γ-space \underline{BA}(S) = |\underline{A}(S × −)|; and this is iterated to define \underline{B^{n}A} for all n. Thus, B^{n}A = \underline{B^{n}A}(1) is the realization of the n-simplicial space which sends (S_{1}, …, S_{n}) to \underline{A}(S_{1} × ⋯ × S_{n}). Since \underline{A}(0) is a point, we can identify ΣA (= Σ(\underline{A}(1))) as a subspace of BA ≅ |\underline{A}|; and this induces by adjointness a map A → ΩBA. Upon iterating this, we get maps Σ(B^{n}A) → B^{n+1}A for all n; and these make the sequence A, BA, B^{2}A, … into a spectrum. This is “almost” an Ω-spectrum, in that B^{n}A ≃ ΩB^{n+1}A for all n ≥ 1 [12, Proposition 1.4].

Note that for any Γ-space \underline{A}, the underlying space A = \underline{A}(1) is an H-space: multiplication is defined to be the composite of a homotopy inverse of the equivalence \underline{A}(2) ⟶^{≃} \underline{A}(1)×\underline{A}(1) with the map \underline{A}(2) → \underline{A}(1) induced by m_{2} : 1 → 2 (m_{2}(1) = \{1, 2\}). Then A ≃ ΩBA if π_{0}(A) is a group; and ΩBA is the topological group completion of A otherwise. All of this is shown in [12, §1].

We work here with equivariant Γ-spaces; i.e., with functors \underline{A} : Γ^{op} → G-Spaces for which \underline{A}(0) is a point, and for which (\underline{A}(n))^{H} → ∏_{i=1}^{n}(\underline{A}(1))^{H} is a homotopy equivalence for all H ⊆ G. In other words, restriction to fixed point sets of any H ⊆ G defines a Γ-space \underline{A}^{H}; and the properties of equivariant Γ-spaces follow immediately from those of nonequivariant ones. For example, Segal’s [12, Proposition 1.4] implies immediately that for any equivariant Γ-space \underline{A}, B^{n}A → ΩB^{n+1}A is a weak equivalence for all n ≥ 1 in the sense that it restricts to an equivalence (B^{n}A)^{H} ≃ (ΩB^{n+1}A)^{H} for all H ⊆ G. This motivates the following definitions.

If F is any family of subgroups of G, then a weak F-equivalence of G-spaces is a G-map whose restriction to fixed point sets of any subgroup in F 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 family F of subgroups of G, and let f : Y → Y′ be any weak Fequivalence. Then for any G-complex X all of whose isotropy subgroups are in F, the map

f_{∗} : [X, Y]_{G} ⟶^{≅} [X, Y′]_{G}

is a bijection. More generally, if A ⊆ X is any G-invariant subcomplex, and all isotropy subgroups of X∖A are in F, then for any commutative diagram

\begin{matrix}A & ⟶^{α_{0}} & Y \\ ↓ & & ^{f}↓ \\ X & ⟶^{α} & Y′ \end{matrix}

of G-maps, there is an extension of α_{0} to a G-map α̃ : X → Y such that f ∘α̃ ≃ α (equivariantly homotopic), and α̃ is unique up to equivariant homotopy.

\underline{\underline{Proof}} : The idea is the following. Fix a G-orbit of cells (G/H × D^{n} → X) in X whose boundary is in A. Then, since Y^{H} → (Y′)^{H} is a weak homotopy equivalence, the map eH × D^{n} → X^{H} → (Y′)^{H} can be lifted to Y^{H} (up to homotopy), and this extends equivariantly to a G-map G/H × D^{n} → Y. Upon continuing this procedure, we obtain a lifting of α to a G-map α̃ : X → Y which extends α_{0}. 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 pair X×I ⊇ (X×{0, 1}) ∪ (A×I).

Now fix a discrete group G. 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)| = EG and |B(G)| = BG; hence the notation.) When necessary to be precise, g_{a} will denote the morphism a → ga in E(G). We let G act on E(G) via right multiplication: x ∈ G acts on objects by sending a to ax and on morphisms by sending g_{a} to g_{ax}. Thus, for any H ⊆ G, the orbit category E(G)/H is the groupoid whose objects are the cosets in G/H, and with one morphism g_{aH} : aH → gaH for each g ∈ G: a category which is equivalent to B(H). Note in particular that B(G) ≅ E(G)/G.

In order to deal simultaneously with real and complex K-theory, we let F denote one of the fields ℂ or ℝ. Set F^{∞} = ⋃_{n=1}^{∞} F^{n}: the space of all infinite sequences in F with finitely many nonzero terms. Let F -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 set S, an S-partitioned vector space is an object V of F -mod, together with a direct sum decomposition V = ⨁_{s∈S} V_{s}. Let F ⟨S⟩-mod denote the category of Spartitioned vector spaces in F -mod, where morphisms are isomorphisms which respect the decomposition. In particular, F ⟨0⟩-mod has just one object 0 ⊆ F^{∞} and one morphism. A morphism θ : S → T induces a functor F ⟨θ⟩ from F ⟨T ⟩-mod to F ⟨S⟩-mod, by sending V = ⨁_{t∈T} V_{t} to W = ⨁_{s∈S} W_{s} where W_{s} = ⨁_{t∈θ(s)} V_{t}.

Let \underline{Vec}_{G}^{F} be the Γ-space defined by setting

\underline{Vec}_{G}^{F}(S) =^{def} |func(E(G), F ⟨S⟩-mod)|

for each finite set S. Here, func(C, D) denotes the category of functors from C to D. We give this the G-action induced by the action on E(G) described above. This is made into a functor on Γ via composition with the functors F ⟨θ⟩.

By definition, \underline{Vec}_{G}^{F}(0) is a point. To see that \underline{Vec}_{G}^{F} is an equivariant Γ-space, it remains to show for each n and H that the map (\underline{Vec}_{G}^{F}(n))^{H} → ∏_{i=1}^{n}(\underline{Vec}_{G}^{F}(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 in F -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 in F^{∞}. 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 finite H ⊆ G, (Vec_{G}^{F})^{H} 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 Vec_{G}^{F} classifies G-vector bundles over proper G-complexes. So it is natural to define equivariant K-theory using the its group completion KF_{G} =^{def} ΩBVec_{G}^{F}, regarded as a pointed G-space.

In the following definition, [−, −]_{G} and [−, −]·_{G} denote sets of homotopy classes of Gmaps, and of pointed G-maps, respectively.

Definition 1.2 For each proper G-complex X, set

K_{G}(X) = [X, Kℂ_{G}]_{G} and KO_{G}(X) = [X, Kℝ_{G}]_{G}.

For each proper G-CW-pair (X, A) and each n ≥ 0, set

K_{G}^{−n}(X, A) = [Σ^{n}(X/A), Kℂ_{G}]·_{G} and KO_{G}^{−n}(X, A) = [Σ^{n}(X/A), Kℝ_{G}]·_{G}.

The usual cohomological properties of the KF_{G}^{−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

\begin{aligned}KF_{G}^{−n}(X) & ≅ Ker[KF_{G}(S^{n} × X) ⟶ KF_{G}(X)] \\ KF_{G}^{−n}(X, A) & ≅ Ker[KF_{G}^{−n}(X ∪_{A} X) ⟶ KF_{G}^{−n}(X)], \end{aligned}  (1.3)

for any proper G-CW-pair (X, A) and any n ≥ 0.

The following lemma will be needed in the next section. It is a special case of the fact that \underline{Vec}_{G}^{F} and KF_{G} (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_{∗} : \underline{Vec}_{G}^{F} → \underline{Vec}_{G}^{F},

defined by composition with F -mod ⟶^{α(−)} F -mod, is G-homotopic to the identity. In particular, α_{∗} induces the identity on K_{G}(X).

\underline{\underline{Proof}} : The functor (V ↦ α(V)) is naturally isomorphic to the identity.

In [10], we defined 𝕂_{G}(X), for any proper G-complex X, to be the Grothendieck group of the monoid of vector bundles over X. We next construct natural homomorphisms 𝕂_{G}(X) → K_{G}(X), for all proper G-complexes X, which are isomorphisms if X/G is a finite complex (this is the situation where the 𝕂_{G}^{∗}(X) form an equivariant cohomology theory).

For each n ≥ 0, let F^{n}-mod ⊆ F -mod be the full subcategory of n-dimensional vector subspaces in F^{∞}. Let F^{n}-frame denote the category whose objects are the pairs (V, b), where V is an object of F^{n}-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 of GL_{n}(F) (one for each V in F^{n}-mod). Note that there is a unique morphism between any pair of objects in F^{n}-frame . Set

Vec_{G}^{F,n} = |func(E(G), F^{n}-mod)| and Ṽec_{G}^{F,n} = |func(E(G), F^{n}-frame )|,

with the action of G×GL_{n}(F) on Ṽec_{G}^{F,n} induced by the G-action on E(G) and the GL_{n}(F)- action on the set of ordered bases of each n-dimensional V. Let τ_{n} : Ṽec_{G}^{F,n} → Vec_{G}^{F,n} be the G-map induced by the forgetful functor F^{n}-frame → F^{n}-mod. Then GL_{n}(F) acts freely and properly on Ṽec_{G}^{F,n}. And τ_{n} induces a G-homeomorphism Ṽec_{G}^{F,n}/GL_{n}(F) ≅ Vec_{G}^{F,n}, since for any φ : V → V′ in F -mod, a lifting of V or V′ to F^{n}-frame determines a unique lifting of the morphism.

Let H ⊆ G × GL_{n}(F) be any subgroup. If H ∩ (1 × GL_{n}(F)) ≠ 1, then (Ṽec_{G}^{F,n})^{H} = ∅, since GL_{n}(F) acts freely on Ṽec_{G}^{F,n}. So assume H ∩ (1 × GL_{n}(F)) = 1. Then H is the graph of some homomorphism φ : H′ → GL_{n}(F) (H′ ⊆ G), and (Ṽec_{G}^{F,n})^{H} is the nerve of the (nonempty) category of φ-equivariant functors E(G) → F^{n}-frame , with a unique morphism between any pair of objects (since there is a unique morphism between any pair of objects in F^{n}-frame ). In particular, this shows that (Ṽec_{G}^{F,n})^{H} is contractible.

Thus, Ṽec_{G}^{F,n} is a universal space for those (G × GL_{n}(F))-complexes upon which GL_{n}(F) acts freely (cf. [10, §2]). The frame bundle of any n-dimensional G-F -vector bundle over a G-complex X is such a complex, and hence n-dimensional G-F -vector bundles over X are classified by maps to Vec_{G}^{F,n} = Ṽec_{G}^{F,n}/GL_{n}(F). It follows that

EVec_{G}^{F,n} = Ṽec_{G}^{F,n} ×_{GL_{n}(F)} F^{n} ⟶ Vec_{G}^{F,n}

is a universal n-dimensional G-F -vector bundle. And [X, Vec_{G}^{F,n}]_{G} ≅ 𝕍ect_{G}^{F,n}(X): the set of isomorphism classes of n-dimensional G-F -vector bundles over X.

If E is any G-F -vector bundle over X, we let [[E]] ∈ KF_{G}(X) = [X, KF_{G}]_{G} be the composite of the classifying map X → Vec_{G}^{F} for E with the group completion map Vec_{G}^{F} → ΩBVec_{G}^{F} = KF_{G}. Any pair E, E′ of vector bundles over X is induced by a G-map

X ⟶ Vec_{G}^{F} × Vec_{G}^{F} = | func(E(G), F -mod × F -mod)| ≃ | func(E(G), F ⟨2⟩-mod)|; and upon composing with the forgetful functor F ⟨2⟩-mod → F -mod we get the classifying map for E ⊕ E′. The direct sum operation on 𝕍ect_{G}^{F}(X) is thus induced by the H-space structure on Vec_{G}^{F}, and [[E ⊕ E′]] = [[E]] + [[E′]] for all E, E′.

Proposition 1.5 The assignment ([E] ↦ [[E]]) defines a homomorphism

γ_{X} : 𝕂F_{G}(X) ⟶ KF_{G}(X),

for any proper G-complex X. This extends to natural homomorphisms γ_{X,A}^{−n} : 𝕂F_{G}^{−n}(X, A) → KF_{G}^{−n}(X, A), for all proper G-CW-pairs (X, A) and all n ≥ 0; which are isomorphisms when restricted to the category of finite proper G-CW-pairs.

\underline{\underline{Proof}} : By the above remarks, ([E] ↦ [[E]]) defines a homomorphism of monoids from 𝕍ect_{G}^{F}(X) to KF_{G}(X), and hence a homomorphism of groups

γ_{X} : 𝕂F_{G}(X) ⟶ KF_{G}(X).

Homomorphisms γ_{X,A}^{−n} (for all proper G-CW-pairs (X, A)) are then constructed via the definitions

𝕂F_{G}^{−n}(X) =^{def} Ker[𝕂F_{G}(S^{n} × X) → 𝕂F_{G}(X)]

and 𝕂F_{G}^{−n}(X, A) =^{def} Ker[𝕂F_{G}^{−n}(X∪_{A}X) → 𝕂F_{G}^{−n}(X)] used in [10], together with the analogous relations (1.3) for K_{G}^{∗}(−). These homomorphisms clearly commute with boundary maps.

It remains to check that γ_{X}^{−n} is an isomorphism whenever X is a finite proper G-complex. Since 𝕂F_{G}(−) and KF_{G}(−) are both cohomology theories in this situation, it suffices, using the Mayer-Vietoris sequences for pushout squares

G/H × S^{m−1} ⟶  G/H × D^{m}
φ↓  ↓
X  ⟶ (G/H × D^{m}) ∪_{φ} X,

to do this when X = G/H × S^{m} for finite H ⊆ G and any m ≥ 0. Using (1.3) again, it suffices to show that γ_{X} = γ_{X}^{0} is an isomorphism whenever X = G/H × Y for any finite complex Y with trivial G-action. By definition,

KF_{G}(G/H × Y) = [G/H × Y, KF_{G}]_{G} ≅ [Y, (KF_{G})^{H}];

while 𝕂F_{G}(G/H × Y) is the Grothendieck group of the monoid

𝕍ect_{G}^{F}(G/H × Y) ≅ [G/H × Y, Vec_{G}^{F}]_{G} ≅ [Y, (Vec_{G}^{F})^{H}].

Since π_{0}((Vec_{G}^{F})^{H}) is a free abelian monoid (the monoid of isomorphism classes of H-representations), [12, Proposition 4.1] applies to show that [−, (KF_{G})^{H}] is universal among representable functors from compact spaces to abelian groups which extend 𝕍ect_{G}^{F}(G/H×−) ≅ 𝕍ect_{H}^{F}(−). And since 𝕂_{H} is representable as a functor on compact spaces with trivial action (H is finite), it is the universal functor, and so [Y, (KF_{G})^{H}] ≅ 𝕂_{H}(Y) ≅ 𝕂_{G}(G/H × Y).

2. Products and Bott periodicity

We now want to construct Bott periodicity isomorphisms, and define the multiplicative structures on K_{G}^{∗}(X) and KO_{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 [12, §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_{∗} : F ⟨S⟩-mod × F ⟨T ⟩-mod ⟶ F ⟨S×T ⟩-mod,

and hence (for any discrete groups H and G)

μ_{∗} : \underline{Vec}_{H}^{F}(S) ∧ \underline{Vec}_{G}^{F}(T) ⟶ \underline{Vec}_{H×G}^{F}(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

ΩBVec_{H}^{F} ∧ ΩBVec_{G}^{F} = KF_{H} ∧ KF_{G} ⟶ Ω^{2}(BVec_{H}^{F} ∧ BVec_{G}^{F}) ⟶^{Ω^{2}|μ_{∗}|} Ω^{2}B^{2}Vec_{H×G}^{F} ≃ {ΩBVec_{H×G}^{F} \atop = KF_{H×G}}.

(2.2)

By Lemma 1.4, these maps are all independent (up to homotopy) of the choice of μ : 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:

𝕂F_{H}(X) ⊗ 𝕂F_{G}(Y) ⟶^{γ_{X}×γ_{Y}} KF_{H}(X) ⊗ KF_{G}(Y)
⊗↓  ^{μ_{∗}}↓
𝕂F_{H×G}(X × Y) ⟶^{γ_{X×Y}} KF_{H×G}(X × Y)

where μ_{∗} is the homomorphism induced by (2.2).

\underline{\underline{Proof}} : The pullback of the universal bundle EVec_{H×G}^{F}, via the pairing Vec_{H}^{F} ∧ Vec_{G}^{F} → Vec_{H×G}^{F} of (2.1), is isomorphic to the tensor product of the universal bundles EVec_{H}^{F} and EVec_{G}^{F}. This is clear if we identify EVec_{G}^{F} ≅ |func(E(G), F -Bdl)| (and similarly for the other two bundles), where F -Bdl is the category of pairs (V, x) for V in F -mod and x ∈ V.

We now consider case where H = 1, and hence where KF_{H} = ℤ × BU or ℤ × BO. The product map (2.2), after composition with the Bott elements in π_{2}(BU) or π_{8}(BO), induces Bott maps

β_{∗}^{ℂ} : Σ^{2}K_{G} ⟶ K_{G} and β_{∗}^{ℝ} : Σ^{8}KO_{G} ⟶ KO_{G}.  (2.4)

Proposition 2.5 For any proper CW-pair (X, A), the Bott homomorphisms

b_{X,A}^{ℂ} : K_{G}^{−n}(X, A) ⟶ K_{G}^{−n−2}(X, A) and b_{X,A}^{ℝ} : KO_{G}^{−n}(X, A) ⟶ KO_{G}^{−n−8}(X, A)

are isomorphisms; and commute with the homomorphisms

γ_{X,A}^{−n} : 𝕂F_{G}^{−n}(X, A) → KF_{G}^{−n}(X, A).

\underline{\underline{Proof}} : The last statement follows immediately from Lemma 2.3.

By Lemma 1.1, it suffices to prove that the adjoint maps

K_{G} ⟶ Ω^{2}K_{G} and KO_{G} ⟶ Ω^{8}KO_{G}

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 b_{X}^{ℂ} : K_{G}(X) → K_{G}^{−2}(X) and b_{X}^{ℝ} : KO_{G}(X) → KO_{G}^{−8}(X) are isomorphisms when X = G/H × S^{n} for any n ≥ 0 and any finite H ⊆ G. And this follows since the Bott maps for 𝕂_{G} and 𝕂𝕆_{G} are isomorphisms [10, Theorems 3.12 & 3.15], since 𝕂F_{G}^{−n}(X) ≅ KF_{G}^{−n}(X) (Proposition 1.5), and since these isomorphisms commute with the Bott maps.

The K_{G}^{−n}(X) and KO_{G}^{−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 on K_{G}(X). This is defined to be the composite

[X, KF_{G}]_{G} × [X, KF_{G}]_{G} ⟶ [X, KF_{G×G}]_{G} ⟶^{Δ^{∗}} [X, KF_{G}]_{G},

where the first map is induced by the pairing in (2.2), and the second by restriction to the diagonal subcategory E(G) ⊆ E(G×G).

Before we can prove the ring properties of this multiplication, we must look more closely at the homotopy equivalence ΩBVec_{G}^{F} ⟶^{≃} Ω^{2}B^{2}Vec_{G}^{F} which appears in the definition of the product. In fact, there is more than one natural map from Ω^{n}B^{n}Vec_{G}^{F} to Ω^{n+1}B^{n+1}Vec_{G}^{F}. For each n ≥ 0 and each k = 0, …, n, let ι_{n}^{k} : Ω^{n}B^{n}Vec_{G}^{F} → Ω^{n+1}B^{n+1}Vec_{G}^{F} denote the map induced as Ω^{n}(f), where f is adjoint to the map ΣB^{n}Vec_{G}^{F} → B^{n+1}Vec_{G}^{F}, induced by identifying B^{n}Vec_{G}^{F}(S_{1}, …, S_{n}) with B^{n+1}Vec_{G}^{F}(…, S_{k−1}, 1, S_{k}, …).

By a weak G-equivalence f : X → Y is meant a map of G-spaces which restricts to a weak equivalence f^{H} : X^{H} → Y^{H} for all H ⊆ G; i.e., a weak F-equivalence in the notation of Lemma 1.1 when F is the family of all subgroups of G. Since we are interested equivariant Γ-spaces only as targets of maps from G-complexes, it suffices by Lemma 1.1 to work in a category where weak G-equivalences are inverted.

Lemma 2.6 Let \underline{A} be any G-equivariant Γ-space. Then for any n ≥ 1, the maps ι_{n}^{k} : Ω^{n}B^{n}A → Ω^{n+1}B^{n+1}A (for 0 ≤ k ≤ n) are all equal in the homotopy category of G-spaces where weak G-equivalences are inverted.

\underline{\underline{Proof}} : For any σ ∈ Σ_{n}, let σ_{∗} : Ω^{n}B^{n}A → Ω^{n}B^{n}A be the map induced by permuting the coordinates of B^{n}A as an n-simplicial set, and by switching the order of looping. Then any two of the ι_{n−1}^{k} differ by composition with some appropriate σ_{∗}, and so it suffices to show that the σ_{∗} are all homotopic to the identity.

Consider the following commutative diagram

ΩBA ⟶^{φ} Ω^{n+1}B^{n+1}A ⟵^{ι_{n}^{n}} Ω^{n}B^{n}A
Id↓  ^{(1×σ)_{∗}}↓  ^{σ_{∗}}↓
ΩBA ⟶^{φ} Ω^{n+1}B^{n+1}A ⟵^{ι_{n}^{n}} Ω^{n}B^{n}A,

for any σ ∈ Σ_{n} ⊆ Σ_{n+1}, where φ = ι_{n}^{0}∘ ⋯ ∘ι_{1}^{0} is induced by identifying \underline{A}(S) with \underline{A}(S, 1, …, 1). The diagram commutes, and all maps in it are weak G-equivalences by [12, Proposition 1.4]. So (1 × σ)_{∗} and σ_{∗} are both homotopic to the identity after inverting weak G-equivalences.

We are now ready to show:

Theorem 2.7 For any discrete group and any proper G-complex X, the pairings μ_{X} define a structure of graded ring on K_{G}^{∗}(X) and on KO_{G}^{∗}(X), which make K_{G}^{∗}(−) and KO_{G}^{∗}(−) into multiplicative cohomology theories. The Bott isomorphisms

b_{X}^{ℂ} : K_{G}^{−n}(X) → K_{G}^{−n−2}(X)  and  b_{X}^{ℝ} : KO_{G}^{−n}(X) → KO_{G}^{−n−8}(X)

are K_{G}(X)- or KO_{G}(X)-linear. And the canonical homomorphisms

γ_{X}^{ℂ} : 𝕂_{G}^{∗}(X) → K_{G}^{∗}(X)  and  γ_{X}^{ℝ} : 𝕂𝕆_{G}^{∗}(X) → KO_{G}^{∗}(X)

are homomorphisms of rings.

\underline{\underline{Proof}} : As usual, set F = ℂ or ℝ. We first check that μ_{X} makes KF_{G}(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 [F^{1}] ∈ Vec_{G}^{F} denote the vertex for the constant functor E(G) ↦ F^{1} ∈ F ⟨1⟩-mod, and set [F^{1}]_{Ω} = ι_{0}^{0}([F^{1}]) ∈ ΩBVec_{G}^{F}. The following diagram commutes:

ΩBVec_{G}^{F} ⟶^{[F^{1}]∧−} Vec_{G}^{F} ∧ ΩBVec_{G}^{F} ⟶^{μ_{∗}} ΩBVec_{G}^{F}
Id↓  ^{ι_{0}^{0}∧Id}↓^{≃}  ^{ι_{1}^{0}}↓^{≃}
ΩBVec_{G}^{F} ⟶^{[F^{1}]_{Ω}∧−} ΩBVec_{G}^{F} ∧ ΩBVec_{G}^{F} ⟶^{μ_{∗}} Ω^{2}B^{2}Vec_{G}^{F};

and the composite in the top row is homotopic to the identity by Lemma 1.4. So the element 1 ∈ KF_{G}(X), represented by the constant map X ↦ [F^{1}]_{Ω} ∈ KF_{G}, is an identity for multiplication in KF_{G}(X).

The commutativity of KF_{G}(X) follows from Lemma 2.6 (the uniqueness of the map ΩBA → Ω^{2}B^{2}A after inverting weak G-equivalences); together with the fact that the pairing

μ_{∗} : BVec_{G}^{F} ∧ BVec_{G}^{F} ⟶ B^{2}Vec_{G}^{F}

commutes up to homotopy using Lemma 1.4. And associativity follows since the triple products are induced by maps

(ΩBVec_{G}^{F})^{∧3} ⟶ Ω^{3}((BVec_{G}^{F})^{∧3}) \overrightarrow{⟶}_{Ω^{3}|μ_{∗}∘(Id ∧μ_{∗})|}^{Ω^{3}|μ_{∗}∘(μ_{∗}∧Id)|} Ω^{3}B^{3}Vec_{G}^{F} ⟵^{≃} ΩBVec_{G}^{F};

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 any n, m ≥ 0, the composite

KF_{G}(S^{n} × X) ⊗ KF_{G}(S^{m} × X) ⟶^{proj^{∗}} KF_{G}(S^{n} × S^{m} × X) ⊗ KF_{G}(S^{n} × S^{m} × X)

⟶^{μ} KF_{G}(S^{n} × S^{m} × X)

restricts to a product map KF_{G}^{−n}(X) ⊗ KF_{G}^{−m}(X) → KF_{G}^{−n−m}(X). To see that the product has image in KF_{G}^{−n−m}(X), just note that

\begin{aligned}KF_{G}^{−n−m}(X) & ≅ Ker[KF_{G}(S^{n+m} × X) ⟶ KF_{G}(X)] \\ & = Ker[KF_{G}(S^{n} × S^{m} × X) ⟶ KF_{G}(S^{n} × X) ⊕ KF_{G}(S^{m} × X)]. \end{aligned}

This product is clearly associative, and graded commutative (where the change in sign comes from the degree of the switching map S^{n+m} → S^{m+n}).

We next check that this product commutes with the Bott maps in the obvious way, so that it can be extended to K_{G}^{i}(X) for all i. This means showing that the two maps

KF (S^{n}) ⊗ KF_{G}(X) ⊗ KF_{G}(X) \overrightarrow{⟶} KF_{G}(S^{n} × 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 on KF_{G}(X).

Finally, γ : 𝕂F_{G}^{∗}(X) → KF_{G}^{∗}(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 𝕂_{G}(X) and 𝕂𝕆_{G}(X) by induction and restriction carry over to K_{G}(X) and KO_{G}(X). Namely, we want to construct for any pair H ⊆ G of discrete groups, any F = ℂ or ℝ, any G-complex X, and any H-complex Y, natural induction and restriction maps

Ind_{H}^{G} : KF_{H}^{∗}(Y) ⟶^{≅} KF_{G}^{∗}(G×_{H}Y) and Res_{H}^{G} : KF_{G}^{∗}(X) ⟶ KF_{H}^{∗}(X|_{H}).

Furthermore, when H ◁ G is a normal subgroup, we construct an inflation homomorphism

Infl_{G/H}^{G} : KF_{G/H}^{∗}(X/H) ⟶ KF_{G}^{∗}(X),

which is an isomorphism whenever H acts freely on X. These maps correspond under the natural homomorphism 𝕂F_{G}^{∗}(X) → KF_{G}^{∗}(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 : G′ → G be any homomorphism of discrete groups. Then composition with the induced functor E(f) : E(G′) → E(G) induces an G′-equivariant map f^{∗} : \underline{Vec}_{G}^{F} → \underline{Vec}_{G′}^{F} of Γ-spaces, and hence a G′-equivariant map f^{∗} : KF_{G} → KF_{G′} of classifying spaces. And for any subgroup L ⊆ G′ such that L∩Ker(f) = 1, f^{∗} restricts to a homotopy equivalence (KF_{G})^{f(L)} ≃ (KF_{G′})^{L}.

\underline{\underline{Proof}} : This is immediate, except for the last statement. And if L ⊆ G′ is such that L ∩ Ker(f) = 1, then L ≅ f(L), the categories E(G′)/L and E(G)/f(L) are both equivalent to the category B(L) with one object and endomorphism group L; and thus (\underline{Vec}_{G}^{F})^{f(L)}(S) = | func(E(G)/f(L), F ⟨S⟩-mod)| is homotopy equivalent to (\underline{Vec}_{G′}^{F})^{L}(S) = | func(E(G′)/L, F ⟨S⟩-mod)| for each S in Γ.

We first consider the restriction and induction homomorphisms.

Proposition 3.2 Fix F = ℂ or ℝ, and let H ⊆ G be any pair of discrete groups. Let i^{∗} : KF_{G} → KF_{H} be the map of Lemma 3.1.

(a) For any proper G-CW-pair (X, A), i^{∗} induces a homomorphism of rings

Res_{H}^{G} : KF_{G}^{∗}(X, A) ⟶ KF_{H}^{∗}(X, A).

(b) For any proper H-CW-pair (Y, B), i^{∗} induces an isomorphism

Ind_{H}^{G} : KF_{H}^{∗}(Y, B) ⟶^{≅} KF_{G}^{∗}(G×_{H}Y, G×_{H}B),

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 between 𝕂F_{G}(−) and 𝕂F_{H}(−) induced by induction and restriction of equivariant vector bundles.

\underline{\underline{Proof}} : It suffices to prove this when A = ∅ = B and ∗ = 0. The fact that i^{∗} : KF_{G} → KF_{H} 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 ×_{H} Y, KF_{G}]_{G} ≅ [Y, KF_{G}]_{H} ⟶^{i^{∗}∘−} [Y, KF_{H}]_{H}.

And since i^{∗} restricts to a homotopy equivalence (KF_{G})^{L} → (KF_{H})^{L} for each finite L ⊆ H (Lemma 3.1), this map is an isomorphism by Lemma 1.1.

The last statement is clear from the construction and the definition of γ : 𝕂F_{G}(−) → KF_{G}(−).

We next consider the inflation homomorphism.

Proposition 3.3 Fix F = ℂ or ℝ. Let G be any discrete group, and let N ◁ G be a normal subgroup. Then for each proper G-CW-pair (X, A), there is an inflation map

Infl_{G/N}^{G} : KF_{G/N}^{∗}(X/N, A/N) ⟶ KF_{G}^{∗}(X, A),

which is natural in (X, A), which is a homomorphism of rings (if A = ∅), and which commutes with the homomorphism 𝕂F_{G/N}(X/N, A/N) → 𝕂F_{G}(X, A) induced considering G/Nvector bundles as G-vector bundles. And if N acts freely on X, then Infl_{G/N}^{G} is an isomorphism.

\underline{\underline{Proof}} : Let f : G → G/N denote the natural homomorphism, and let f^{∗} : KF_{G/N} → KF_{G} be the induced map of Lemma 3.1. Define Infl_{G/N}^{G} to be the composite

[X/N, KF_{G/N}]_{G/N} ≅ [X, KF_{G/N}]_{G} ⟶^{f^{∗}∘−} [X, KF_{G}]_{G}.

If N acts freely on X, then for each isotropy subgroup L of X, L ∩ N = 1, so (f^{∗})^{L} : (KF_{G/N})^{L} → (KF_{G})^{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 group G and a finite normal subgroup N ◁ G, 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 Hom_{N}(V, E) denote the vector bundle over X whose fiber over x ∈ X is Hom_{N}(V, E_{x}) (each fiber of E is an N-representation). If H ⊆ G is any subgroup which centralizes N, then we can regard Hom_{N}(V, E) as an H-vector bundle by setting (hf)(x) = h·f(x) for any h ∈ H and any f ∈ Hom_{N}(V, E). We thus get a homomorphism

Ψ : 𝕂_{G}(X) ⟶ 𝕂_{H}(X) ⊗ R(N),

where Ψ([E]) = ∑_{V ∈Irr(N)}[Hom_{N}(V, E)] ⊗ [V]. We need a similar homomorphism defined on K_{G}^{∗}(X).

Proposition 3.4 Let G be a discrete group, let N ◁ G be any finite normal subgroup, and let H ⊆ G be any subgroup such that [H, N] = 1. Then for any proper G/N-complex X, there is a homomorphism of rings

Ψ = Ψ_{G;N,H}^{X} : K_{G}^{∗}(X) ⟶ K_{H}^{∗}(X) ⊗ R(N),

which is natural in X and natural with respect to the degree-shifting maps K_{G}^{∗}(X) → K_{G}^{∗+n}(S^{n}×X), and which has the following properties:

(a) For any (complex) G-vector bundle E → X,

Ψ([[E]]) = ∑_{V ∈Irr(N)}[[Hom_{N}(V, E)]] ⊗ [V].

(b) For any G′ ⊆ G, N′ ⊆ N∩G′, and H′ ⊆ H∩G′, the following diagram commutes:

K_{G}^{∗}(X) ⟶^{Ψ_{G;N,H}^{X}} K_{H}^{∗}(X) ⊗ R(N)
Res_{G′}^{G} ↓  ^{Res}_{H′}^{H} ↓^{⊗ Res_{N′}^{N}}
K_{G′}^{∗}(X) ⟶^{Ψ_{G′;N′,H′}^{X}} K_{H′}^{∗}(X) ⊗ R(N′).

\underline{\underline{Proof}} : Fix G, H, and N. For any irreducible N-representation V and any surjective homomorphism p : ℂ[N] −↠ V, composition with p defines a monomorphism

Hom_{N}(V, W) ⟶^{−∘p} Hom_{N}(ℂ[N], W) = W

for any N-representation W; and thus allows us to identify Hom_{N}(V, W) as a subspace of W. In particular, there is a functor

p^{∗} : func(Or(G)/N, ℂ⟨S⟩-mod) ⟶ func(Or(H), ℂ⟨S⟩-mod)

which sends any α to the functor h ↦ Hom_{N}(V, α(hN)) ⊆ α(hN). If p′ : ℂ[N] −↠ V′ is another surjection of N-representations, where V ≅ V′, then any isomorphism V ⟶^{≅} V′ defines a natural isomorphism between p^{∗} and (p′)^{∗}. We thus get a map of Γ-spaces

ψ_{p} : \underline{Vec}_{G}^{ℂ} ⟶ \underline{Vec}_{H}^{ℂ}

which is unique (independant of the projection p) up to H-equivariant homotopy. So this induces homomorphisms ψ_{V} : K_{G}^{−n}(X) → K_{H}^{−n}(X), for all proper G/N-complex X (and all n ≥ 0), which depend only on V and not on p. The ψ_{V} clearly commute with the Bott maps, and thus extend to homomorphisms ψ_{V} : K_{G}^{∗}(X) → K_{H}^{∗}(X). So we can define Ψ by setting Ψ(x) = ∑_{V ∈Irr(N)} ψ_{V} (x) ⊗ [V]. Point (a) is immediate; as is naturality in X and naturality for restriction to G′ ⊆ G or H′ ⊆ H. Naturality with respect to the degree-shifting maps holds by construction.

We next show that Ψ is natural in N; i.e., that point (b) holds when G′ = G and H′ = H. Let ψ_{V} be the homomorphisms defined above, for each irreducible N-representation V; and let ψ′_{W} : K_{G}^{∗}(X) → K_{H}^{∗}(X) be the corresponding homomorphism for each irreducible N′- representation W. For each V ∈ Irr(N) and each W ∈ Irr(N′), set

n_{W}^{V} = dim_{ℂ}(Hom_{N′}(W, V)) = dim_{ℂ}(Hom_{N}(Ind_{N′}^{N}(W), V)).

Thus, n_{W}^{V} is the multiplicity of W in the decomposition of V |_{N′}, as well as the multiplicity of V in the decomposition of Ind_{N′}^{N}(W). So for any x ∈ K_{G}^{∗}(X),

(Id ⊗ Res_{N′}^{N})(Ψ_{G;N,H}(x)) = ∑_{V ∈Irr(N)} ψ_{V} (x) ⊗ [V |_{N′}] = ∑_{W ∈Irr(N′)}( ∑_{V ∈Irr(N)} n_{W}^{V} ·ψ_{V} (x)) ⊗ [W];

and we will be done upon showing that ψ′_{W} = ∑_{V} n_{W}^{V} ·ψ_{V} for each W ∈ Irr(N′). Fix a surjection p_{0} : ℂ[N′] −↠ W, and a decomposition Ind_{N′}^{N}(W) = ∑_{i=1}^{k} V_{i} (where the V_{i} are irreducible and k = ∑_{V} n_{W}^{V}). For each 1 ≤ i ≤ k, let p_{i} : ℂ[N] −↠ V_{i} be the composite of Ind_{N′}^{N}(p_{0}) followed by projection to V_{i}. Then

ψ_{p_{0}} = ⨁_{i=1}^{k} ψ_{p_{i}} : (\underline{Vec}_{G}^{ℂ})^{N} ⟶ \underline{Vec}_{H}^{ℂ}

as maps of Γ-spaces, and so ψ_{W} ≃ ∑_{i=1}^{k} ψ_{V_{i}} as maps K_{G}^{∗}(X) → K_{H}^{∗}(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 when N is cyclic. For any x, y ∈ K_{G}(X),

Ψ(x)·Ψ(y) = ∑_{V,W ∈Irr(N)}(ψ_{V} (x)·ψ_{W} (y)) ⊗ [V ⊗ W] and Ψ(xy) = ∑_{U∈Irr(N)} ψ_{U}(xy) ⊗ [U].

And thus Ψ(x)·Ψ(y) = Ψ(xy) since

ψ_{U} ∘μ_{∗} = ⨁_{\begin{matrix}V,W ∈Irr(G) \\ V ⊗W ≅U \end{matrix}} μ_{∗}∘(ψ_{V} ∧ ψ_{W}) : (\underline{Vec}_{G}^{ℂ})^{N} ∧ (\underline{Vec}_{G}^{ℂ})^{N} ⟶ \underline{Vec}_{H}^{ℂ},

as maps of Γ-spaces, for each U ∈ 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 of G. Two elements g, h ∈ G are called K-conjugate if g is conjugate to h^{a} for some a prime to n = |g| = |h| such that (ζ ↦ ζ^{a}) ∈ Gal(Kζ/K), where ζ = exp(2πi/n). For example, g and h are ℚ-conjugate if ⟨g⟩ and ⟨h⟩ are conjugate as subgroups, and are ℝ-conjugate if g is conjugate to h or h^{−1}.

Proposition 4.1 Fix a finite extension K of ℚ, and let A ⊆ K be its ring of integers. Let f : G → A be any function which is constant on K-conjugacy classes. Then |G|·f is an A-linear combination of K-characters of G.

\underline{\underline{Proof}} : Set n = |G|, for short. Let V_{1}, …, V_{k} be the distinct irreducible K[G]- representations, let χ_{i} be the character of V_{i}, set D_{i} = End_{K[G]}(V_{i}) (a division algebra over K), and set d_{i} = dim_{K}(D_{i}). Then by [13, Theorem 25, Cor. 2],

|G|·f = ∑_{i=1}^{k} r_{i}χ_{i} where r_{i} = \frac{1}{d_{i}} ∑_{g∈G} f(g)χ_{i}(g^{−1});

and we must show that r_{i} ∈ A for all i. This means showing, for each i = 1, …, k, and each g ∈ G with K-conjugacy class conj_{K}(g), that |conj_{K}(g)|·χ_{i}(g) ∈ d_{i}A.

Fix i and g; and set C = ⟨g⟩, m = |g| = |C|, and ζ = exp(2πi/m). Then Gal(K(ζ)/K) acts freely on the set conj_{K}(g): the element (ζ ↦ ζ^{a}) acts by sending h to h^{a}. So [K(ζ):K]||conj_{K}(g)|.

Let V_{i}|_{C} = W_{1}^{a_{1}} ⊕ ⋯ ⊕ W_{t}^{a_{t}} be the decomposition as a sum of irreducible K[C]-modules. For each j, K_{j} =^{def} End_{K[C]}(W_{j}) is the field generated by K and the r-th roots of unity for some r|m (m = |C|), and dim_{K_{j}}(W_{j}) = 1. So dim_{K}(W_{j})|[K(ζ):K]. Also, d_{i}| dim_{K}(W_{j}^{a_{j}}), since W_{j}^{a_{j}} is a D_{i}-module; and thus d_{i}|a_{j}·|conj_{K}(g)|. So if we set ξ_{j} = χ_{W_{j}}(g) ∈ A, then

|conj_{K}(g)|·χ_{i}(g) = |conj_{K}(g)|· ∑_{j=1}^{t} a_{j}ξ_{j} ∈ d_{i}A,

and this finishes the proof.

For each prime p and each element g ∈ G, there are unique elements g_{r} of order prime to p and g_{u} of p-power order, such that g = g_{r}g_{u} = g_{u}g_{r}. As in [13, §10.1], we refer to g_{r} as the p′-component of g. We say that a class function f : G → ℂ is p-constant if f(g) = f(g_{r}) for each g ∈ G. Equivalently, f is p-constant if and only if f(g) = f(g′) for all g, g′ ∈ G such that [g, g′] = 1 and g^{−1}g′ has p-power order.

Lemma 4.2 Fix a finite group G, a prime p, and a field K of characteristic zero. Then a pconstant class function φ : G → K is a K-character of G if and only if φ|_{H} is a K-character of H for all subgroups H ⊆ G of order prime to p.

\underline{\underline{Proof}} : Recall first that G is called K-elementary if for some prime q, G = C_{m} ⋊ Q, where C_{m} is cyclic of order m, q| ; ∤ ; |m, Q is a q-group, and the conjugation action of Q on K[C_{m}] leaves invariant each of its field components. By [13, §12.6, Prop. 36], a K-valued class function of G is a K-character if and only if its restriction to any K-elementary subgroup of G is a K-character. Thus, it suffices to prove the lemma when G is K-elementary.

Assume first that G is q-K-elementary for some prime q ≠ p. Fix a subgroup H ⊆ G of p-power index and order prime to p, and let α : G ↠ H be the surjection with α|_{H} = Id. Set p^{a} = | Ker(α)|. Then

Aut(Ker(α)) ≅ (ℤ/p^{a})^{∗} ≅ (1 + pℤ/p^{a}) × (ℤ/p)^{∗},

where the first factor is a p-group. Hence for any g ∈ H and x ∈ Ker(α), either [g, x] = 1 and hence g = (gx)_{r}; or gxg^{−1} = x^{i} for some i ≢ 1 (mod p) and hence g is conjugate to gx. In either case, φ(gx) = φ(g). Thus, φ = (φ|_{H})∘α, and this is a K-character of G since φ|_{H} is by assumption a K-character of H.

Now assume G is p-K-elementary. Write G = C_{m}⋊P, where p| ; ∤ ; |m and P is a p-group. Let S be the set of primes which divide m. For each I ⊆ S, let C_{I} ⊆ C_{m} be the product of the Sylow p-subgroups for p ∈ I, set G_{I} = C_{I}⋊P, and let α_{I} : G ↠ G_{I} be the homomorphism which is the identity on G_{I}.

For each I ⊆ S, we can consider K[C_{I}] as a G-representation via the conjugation action of P; and each C_{I}-irreducible summand of K[C_{I}] is P -invariant and hence G-invariant. Thus, each irreducible K[C_{I}]-representation can be extended to a K[G_{I}]-representation upon which P ∩ C_{G}(C_{I}) acts trivially. Hence, since φ|_{C_{I}} is a K-character of C_{I}; there is a K-character χ_{I} of G_{I} such that χ_{I}(gx) = χ_{I}(x) = φ(x) for all x ∈ C_{I} and g ∈ P such that [g, C_{I}] = 1.

Now set

χ = ∑_{J⊆I⊆S}(−1)^{|I∖J|}(χ_{I} ∘α_{J}),

a K-character of G. We claim that φ = χ. Since both are class functions, it suffices to show that φ(gx) = χ(gx) for all commuting g ∈ P and x ∈ C_{m} = C_{S}. Fix such g and x, and let X ⊆ S be the set of all primes p||x|. Then [g, C_{X}] = 1, and so

χ(gx) = ∑_{J⊆I⊆S}(−1)^{|I∖J|}χ_{I}(α_{J}(gx)) = ∑_{J⊆I⊆S}(−1)^{|I∖J|}χ_{I}(g·α_{J}(x))

= ∑_{J⊆I⊆X}(−1)^{|I∖J|}φ(α_{J}(x)) + ∑_{J⊆I⊈X}(−1)^{|I∖J|}χ_{I}(g·α_{J}(x))

= φ(x) = φ(gx).

Note, in the second line, that all terms in the second sum cancel since α_{J}(x) = α_{J′}(x) if J = J′ ∩ X, and all terms in the first sum cancel except that where J = I = X.

When A = ℤ and K = ℚ, Proposition 4.1 and Lemma 4.2 combine to give:

Corollary 4.3 Fix a finite group G and a prime p. Let f : G → ℤ be any function which is p-constant, and constant on ℚ-conjugacy classes in G. Set |G| = m·p^{r} where p| ; ∤ ; |m. Then m·f is a ℚ-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 K_{G}^{∗}(X) to the Bredon cohomology group H_{G}^{∗}(X; ℚ⊗_{ℤ}R(−)). The second is defined only when X is finite dimensional and has bounded isotropy, and takes values in ℚ⊗_{ℤ}H_{G}^{∗}(X; R(−)).

We first fix our notation for dealing with Bredon cohomology [6]. Let Or(G) denote the orbit category: the category whose objects are the orbits G/H for H ⊆ G, and where Mor_{Or (G)}(G/H, G/K) is the set of G-maps. A coefficient system for Bredon cohomology is a functor F : Or(G)^{op} → Ab. For any such functor F and any G-complex X, the Bredon cohomology H_{G}^{∗}(X; F) is the cohomology of a certain cochain complex C_{G}^{∗}(X; F), where C_{G}^{n}(X; F) is the direct product over all orbits of n-cells of type G/H of the groups F (G/H). This can be expressed functorially as a group of morphisms of functors on Or(G):

C_{G}^{n}(X, F) = Hom_{Or (G)}(\underline{C}_{n}(X), F ),

where \underline{C}_{n}(X) : Or(G)^{op} → Ab is the functor \underline{C}_{n}(X)(G/H) = C_{n}(X^{H}).

Clearly, the coefficient system F need only be defined on the subcategory of orbit types which occur in the G-complex X. In particular, since we work here only with proper actions, we restrict attention to the full subcategory Or_{f}(G) of orbits G/H for finite H ⊆ G. Let R(−) denote the functor on Or_{f}(G) which sends G/H to R(H): a functor on the orbit category via the identification R(H) ≅ K_{G}^{0}(G/H). More precisely, a morphism G/H → G/K in Or_{f}(G), where gH ↦ gaK for some a ∈ G with a^{−1}Ha ⊆ K, is sent to the homomorphism R(K) → R(H) induced by restriction and conjugation by a.

Since R(−) is a functor from the orbit category to rings, there is a pairing

C_{G}^{∗}(X; R(−)) ⊗ C_{G}^{∗}(X; R(−)) ⟶ C_{G}^{∗}(X × X; R(−))

for any proper X, and hence a similar pairing in cohomology. Via restriction to the diagonal subspace X ⊆ X × X this defines a ring structure on H_{G}^{∗}(X; R(−)).

The equivariant Chern character will be constructed here by first reinterpreting H_{G}^{∗}(X; ℚ⊗R(−)) as a certain group of homomorphisms of functors, and then directly constructing a map from K_{G}(X) to such homomorphisms. This will be done with the help of another category, Sub_{f}(G), which is closely related to Or_{f}(G). The objects of Sub_{f}(G) are the finite subgroups of G, and

Mor_{Sub _{f} (G)}(H, K) ⊆ Hom(H, K)/ Inn(K)

is the subset consisting of those monomorphisms induced by conjugation and inclusion in G. There is a functor Or_{f}(G) → Sub_{f}(G) which sends an orbit G/H to the subgroup H, and which sends a morphism (xH ↦ xaK) in Or_{f}(G) to the homomorphism (x ↦ a^{−1}xa) from H to K. Via this functor, we can think of Sub_{f}(G) as a quotient category of Or_{f}(G).

Let \underline{C}_{∗}^{qt}(X), \underline{H}_{∗}^{qt}(X) : Sub_{f}(G)^{op} ⟶ Ab be the functors

\underline{C}_{∗}^{qt}(X)(H) = C_{∗}(X^{H}/C_{G}(H)) and \underline{H}_{∗}^{qt}(X)(H) = H_{∗}(X^{H}/C_{G}(H)).  (5.1)

For any functor F : Sub_{f}(G)^{op} → Ab, regarded also as a functor on Or_{f}(G)^{op},

Hom_{Or_{f} (G)}(\underline{C}_{∗}(X), F) ≅ Hom_{Sub _{f} (G)}(\underline{C}_{∗}^{qt}(X), F),  (5.2)

since

Hom_{C_{G}(H)}(C_{∗}(X^{H}), F (H)) ≅ Hom(C_{∗}(X^{H}/C_{G}(H)), F (H))

for each H (and C_{G}(H) is the group of automorphisms of G/H in Or_{f}(G) sent to the identity in Sub_{f}(G)). In particular, (5.2) will be applied when F = R(−), regarded as a functor on Sub_{f}(G) as well as on Or_{f}(G).

As noted above, for any coefficient system F, the cochain complex C_{G}^{∗}(X; F) can be identified as a group of homomorphisms of functors on Or(G). The following lemma says that the Bredon cohomology groups H_{G}^{∗}(X; ℚ ⊗ R(−)) have a similar description, but using functors on Sub_{f}(G)^{op}.

Lemma 5.3 Fix a discrete group G and a proper G-complex X. Then (5.2) induces an isomorphism of rings

Φ_{X} : H_{G}^{∗}(X; ℚ ⊗ R(−)) ⟶^{≅} Hom_{Sub _{f} (G)}(\underline{H}_{∗}^{qt}(X), ℚ⊗R(−)).

\underline{\underline{Proof}} : Since

C_{G}^{∗}(X; ℚ⊗R(−)) ≅ Hom_{Or_{f} (G)}(\underline{C}_{∗}(X), ℚ⊗R(−)) ≅ Hom_{Sub _{f} (G)}(\underline{C}_{∗}^{qt}(X), ℚ⊗R(−)),

this will follow immediately once we show that ℚ⊗R(−) is injective as a functor Sub_{f}(G)^{op} → Ab. It suffices to prove this after tensoring with ℂ; i.e., it suffices to prove that Cl(−) (complex valued class functions) is injective. And this holds since for any F : Sub_{f}(G)^{op} → Ab,

Hom_{Sub _{f} (G)}(F, Cl(−)) ≅ ∏_{g} Hom_{Sub _{f} (G)}(F, Cl_{g}(−)) ≅ ∏_{g} Hom(F (⟨g⟩), ℂ);

where both products are taken over any set of conjugacy class representatives for elements of finite order in G, and where Cl_{g}(H) denotes the space of class functions on H which vanish on all elements not G-conjugate to g.

We are now ready to define the Chern character

ch_{X}^{∗} : K_{G}^{∗}(X) ⟶ H_{G}^{∗}(X; ℚ⊗R(−))

for any proper G-complex X. Here and in the following theorem, we regard K_{G}^{∗}(−) as being ℤ/2-graded; so that ch_{X}^{∗} sends K_{G}^{0}(X) to H_{G}^{ev}(X; ℚ ⊗ R(−)) and sends K_{G}^{1}(X) to H_{G}^{odd}(X; ℚ ⊗ R(−)). By Lemma 5.3, it suffices to define homomorphisms

ch_{X}^{H} : K_{G}^{∗}(X) ⟶ Hom(H_{∗}(X^{H}/C_{G}(H)) , ℚ⊗R(H)),

for each finite subgroup H ⊆ G, which are natural in H in the obvious way. We define ch_{X}^{H} to be the following composite:

K_{G}^{∗}(X) ⟶^{Res} K_{N_{G}(H)}^{∗}(X^{H}) ⟶^{Ψ} K_{C_{G}(H)}^{∗}(X^{H}) ⊗ R(H) ⟶^{(proj)^{∗}} K_{C_{G}(H)}^{∗}(EG×X^{H}) ⊗ R(H)

⟶_{≅}^{Infl^{−1}} K^{∗}(EG×_{C_{G}(H)}X^{H}) ⊗ R(H) ⟶^{ch ⊗ Id} H^{∗}(EG×_{C_{G}(H)}X^{H}; ℚ) ⊗ R(H)

⟵_{≅}^{(proj)^{∗}} H^{∗}(X^{H}/C_{G}(H); ℚ) ⊗ R(H) ≅ Hom(H_{∗}(X^{H}/C_{G}(H)), ℚ ⊗ R(H)). (5.4)

Here, Ψ is the homomorphism defined in Proposition 3.4, ch denotes the ordinary Chern character, and (proj)^{∗} in the bottom line is an isomorphism since all fibers of the projection from EG ×_{C_{G}(H)} X^{H} to X^{H}/C_{G}(H) are ℚ-acyclic (classifying spaces of finite groups). By the naturality properties of Ψ shown in Proposition 3.4, ∏_{H} ch_{X}^{H} takes values in Hom_{Sub _{f} (G)}(\underline{H}_{∗}^{qt}(X), ℚ⊗R(−)), and hence (via Lemma 5.3) defines an equivariant Chern character

ch_{X}^{∗} : K_{G}^{∗}(X) ⟶ H_{G}^{∗}(X; ℚ ⊗ R(−)) ≅ Hom_{Sub _{f} (G)}(\underline{H}_{∗}^{qt}(X), ℚ⊗R(−)).

All of the maps in (5.4) are homomorphisms of rings, and hence ch_{X}^{∗} is also a homomorphism of rings. Also, the ch_{X}^{∗} commute with degree-changing maps K_{G}^{∗}(X) → K^{∗+m}(S^{m}×X) (i.e., product with the fundamental class of S^{m}) and similarly in cohomology, since all maps in (5.4) do so. They are thus natural with respect to boundary maps in Mayer-Vietoris sequences.

Theorem 5.5 For any finite proper G-complex X, the Chern character ch_{X}^{∗} extends to an isomorphism of rings

ℚ ⊗ ch_{X}^{∗} : ℚ ⊗ K_{G}^{∗}(X) ⟶^{≅} H_{G}^{∗}(X; ℚ ⊗ R(−)).

\underline{\underline{Proof}} : For any finite subgroup H ⊆ G,

K_{G}^{0}(G/H) ≅ R(H) ≅ H_{G}^{0}(G/H; R(−)), and K_{G}^{1}(G/H) = 0 = H_{G}^{≠0}(G/H; R(−)).

From the definition in (5.4) (and since the non-equivariant Chern character K(pt) → H^{0}(pt) is the identity map), we see that ℚ⊗ch_{G/H}^{∗} is the identity map under the above identifications. The Chern characters for G/H×D^{n} and G/H×S^{n−1}) are thus isomorphisms for all n. The theorem now follows by induction on the number of orbits of cells in X, together with the Mayer-Vietoris sequences for pushouts X = X′ ∪_{φ} (G/H×D^{n}) (and the 5-lemma).

Theorem 5.5 means that the ℚ-localization of the classifying space K_{G} splits as a product of equivariant Eilenberg-Maclane spaces. Hence for any proper G-complex X, there is an isomorphism K_{G}^{∗}(X; ℚ) ⟶^{≅} H_{G}^{∗}(X; ℚ⊗R(−)), where the first group is defined via the localized spectrum (and is not in general isomorphic to ℚ ⊗ K_{G}^{∗}(X, A)).

The coefficient system ℚ ⊗ R(−), and hence its cohomology, splits in a natural way as a product indexed over cyclic subgroups of G of finite order. For any cyclic group S of order n < ∞, we let ℤ[ζ_{S}] ⊆ ℚ(ζ_{S}) denote the cyclotomic ring and field generated by the n-th roots of unity; but regarded as quotient rings of the group rings ℤ[S^{∗}] ⊆ ℚ[S^{∗}] (S^{∗} = Hom(S, ℂ^{∗})). In other words, we fix an identification of the n-th roots of unity in ℚ(ζ_{S}) with the irreducible characters of S. The kernel of the homomorphism R(S) ≅ ℤ[S^{∗}] ↠ ℤ[ζ_{S}] is precisely the ideal of elements whose characters vanish on all generators of S.

Lemma 5.6 Fix a discrete group G, and let S(G) be a set of conjugacy class representatives for the cyclic subgroups S ⊆ G of finite order. Then for any proper G-complex X, there is an isomorphism of rings

H_{G}^{∗}(X; ℚ ⊗ R(−)) ≅ ∏_{S∈S(G)}(H^{∗}(X^{S}/C_{G}(S); ℚ(ζ_{S})))^{N(S)},

where N(S) acts via the conjugation action on ℚ(ζ_{S}) and translation on X^{S}/C_{G}(S). If, furthermore, the isotropy subgroups on X have bounded order, then the homomorphism of rings

H_{G}^{∗}(X; R(−)) ⟶ ∏_{S∈S(G)} H((C^{∗}(X^{S}/C_{G}(S); ℤ[ζ_{S}]))^{N(S)})

⟶ ∏_{S∈S(G)}(H^{∗}(X^{S}/C_{G}(S); ℤ[ζ_{S}]))^{N(S)}, (1)

induced by restriction to cyclic subgroups and by the projections R(S) −↠ ℤ[ζ_{S}], has kernel and cokernel of finite exponent.

\underline{\underline{Proof}} : By (5.2),

C_{G}^{∗}(X; R(−)) ≅ Hom_{Or_{f} (G)}(\underline{C}_{∗}(X), R(−)) ≅ Hom_{Sub _{f} (G)}(\underline{C}_{∗}^{qt}(X), R(−)).

For each S ∈ S(G), let χ_{S} ∈ Cl(G) be the idempotent class function: χ_{S}(g) = 1 if ⟨g⟩ is conjugate to S, and χ_{S}(g) = 0 otherwise. By Proposition 4.1, for each finite subgroup H ⊆ G, (χ_{S})|_{H} is the character of an idempotent e_{S}^{H} ∈ ℚ⊗R(H). Set ℚR_{S}(H) = e_{S}^{H}·(ℚ⊗R(H)), and let R_{S}(H) ⊆ ℚR_{S}(H) be the image of R(H) under the projection. This defines a splitting ℚ ⊗ R(−) = ∏_{S∈S(G)} ℚR_{S}(−) of the coefficient system. For each S and H,

ℚR_{S}(S) = ℚ(ζ_{S}) and so ℚR_{S}(H) ≅ map_{N(S)}(Mor_{Sub _{f} (G)}(S, H) , ℚ(ζ_{S})).

It follows that

C_{G}^{∗}(X; ℚR_{S}(−)) ≅ Hom_{Sub _{f} (G)}(\underline{C}_{∗}^{qt}(X), ℚR_{S}(−)) ≅ Hom_{ℚ[N(S)]}(C_{∗}(X^{S}/C_{G}(S)), ℚ(ζ_{S}));

and hence H_{G}^{∗}(X; ℚR_{S}(−)) ≅ (H^{∗}(X^{S}/C_{G}(S)); ℚ(ζ_{S}))^{N(S)}.

Now assume there is a bound on the orders of isotropy subgroups on X, and let m be the least common multiple of the |G_{x}|. By Proposition 4.1 again, me_{S}^{H} ∈ R(H) for each S ∈ S(G) and each isotropy subgroup H. So there are homomorphisms of functors

R(−) \overrightarrow{⟵}_{j}^{i} ∏_{S∈S(G)} R_{S}(−),

where i is induced by the projections R(H) ↠ R_{S}(H) and j by the homomorphisms

R_{S}(H) ⟶^{me_{S}^{H}·} R(H) (regarding R_{S}(H) as a quotient of R(H)); and i∘j and j∘i are both multiplication by m. For each S, the monomorphism

C_{G}^{∗}(X; R_{S}(−)) ≅ Hom_{ℤ[N(S)]}(C_{∗}(X^{S}/C_{G}(S)), ℤ[ζ_{S}]) ⟶ C^{∗}(X^{S}/C_{G}(S); ℤ[ζ_{S}])

is split by the norm map for the action of N(S)/C_{G}(S), and hence the kernel and cokernel of the induced homomorphism

H_{G}^{∗}(X; R_{S}(−)) ⟶ (H^{∗}(X^{S}/C_{G}(S); ℤ[ζ_{S}]))^{N(S)}

have exponent dividing φ(m) (since |N(S)/C_{G}(S)||| Aut(S)||φ(m)). The composite in (1) thus has kernel and cokernel of exponent m·φ(m).

By the first part of Proposition 5.6, the equivariant Chern character can be regarded as a homomorphism

ch_{X}^{∗} : K_{G}^{∗}(X) ⟶ ∏_{S∈S(G)}(H^{∗}(X^{S}/C_{G}(S); ℚ(ζ_{S})))^{N(S)},

where S(G) is as above. This is by construction a product of ring homomorphisms.

We now apply the splitting of Lemma 5.6 to construct a second version of the equivariant rational Chern character: one which takes values in ℚ ⊗ H_{G}^{∗}(X; R(−)) rather than in H_{G}^{∗}(X; ℚ⊗R(−)). The following lemma handles the nonequivariant case.

Lemma 5.7 There is a homomorphism n!ch : K^{∗}(X) → H^{≤2n}(X; ℤ), natural on the category of CW-complexes, whose composite to H^{∗}(X; ℚ) is n! times the usual Chern character truncated in degrees greater than 2n. Furthermore, n!ch is natural with respect to suspension isomorphisms K^{∗}(X) ≅ K̃^{∗+m}(Σ^{m}(X_{+})), and is multiplicative in the sense that (n!ch(x))·(n!ch(y)) = n!·(n!ch(xy)) for all x, y ∈ K(X) (in both cases after restricting to the appropriate degrees).

\underline{\underline{Proof}} : Define n!ch : K^{0}(X) → H^{ev,≤2n}(X; ℤ) to be the following polynomial in the Chern classes:

n!· ∑_{i=1}^{n}(1 + x_{i} + \frac{x_{i}^{2}}{2!} + ⋯ + \frac{x_{i}^{n}}{n!} ) ∈ ℤ[c_{1}, …, c_{n}] = ℤ[x_{1}, x_{2}, …, x_{n}]^{Σ_{n}}.

Here, as usual, c_{k} is the k-th elementary symmetric polynomial in the x_{i}. This is extended to K^{−1}(X) ≅ K̃(Σ(X_{+})) in the obvious way. The relations all follow from the usual relations between Chern classes in the rings H^{∗}(BU(m)).

We are now ready to construct the integral Chern character. What this really means is that under certain restrictions on X, some multiple of the rational Chern character ch_{X}^{∗} of Theorem 5.5 can be lifted to the integral Bredon cohomology group H_{G}^{∗}(X; R(−)).

Proposition 5.8 Let G be a discrete group, and let X be a finite dimensional proper Gcomplex whose isotropy subgroups have bounded order. Then there is a homomorphism

c̃h_{X}^{∗} : K_{G}^{∗}(X) ⟶ ℚ ⊗ H_{G}^{∗}(X; R(−)),

natural in such X, whose composite to H_{G}^{∗}(X; ℚ ⊗ R(−)) is the map ch_{X}^{∗} of Theorem 5.5. Furthermore, c̃h_{X}^{∗} induces an isomorphism of rings ℚ ⊗ K_{G}^{∗}(X) ⟶^{≅} ℚ ⊗ H_{G}^{∗}(X; R(−)). And for any finite subgroup K ⊆ G, c̃h_{G/K}^{0} is the identity map under the identifications K_{G}(G/K) ≅ R(G/K) ≅ H_{G}^{0}(G/K; R(−)).

\underline{\underline{Proof}} : Fix X, and choose any integer n ≥ dim(X)/2. Set m = lcm\{|G_{x}| | x ∈ X\} and N = n!·m^{4n}. For each S ∈ G of finite order, let c̃h_{X}^{S} be the following composite:

K_{G}^{∗}(X) ⟶^{Res} K_{N_{G}(S)}^{∗}(X^{S}) ⟶^{Ψ} K_{C_{G}(S)}^{∗}(X^{S}) ⊗ R(S) ⟶^{(proj)^{∗}} K_{C_{G}(S)}^{∗}(EG × X^{S}) ⊗ R(S)

⟶_{≅}^{Infl^{−1}} K^{∗}(EG×_{C_{G}(S)}X^{S}) ⊗ R(S) ⟶^{n!ch} H^{≤2n}(EG×_{C_{G}(S)}X^{S}) ⊗ R(S)

⟶^{m^{4n}(proj^{∗})^{−1}} H^{∗}(X^{S}/C_{G}(S)) ⊗ R(S) ⟶ H^{∗}(X^{S}/C_{G}(S); ℤ[ζ_{S}]).

Here, Ψ is the homomorphism of Lemma 3.4, and Infl is the inflation isomorphism of Proposition 3.3. The first map in the bottom row is well defined since H^{∗}(X^{S}/C(S); ℤ[ζ_{S}]) ⟶^{proj^{∗}} H^{∗}(EG×_{C(S)} X^{S}; ℤ[ζ_{S}]) has kernel and cokernel of exponent m^{2n} (this follows from the spectral sequence for the projection, all of whose fibers are of the form BG_{x} for x ∈ X). The last map is induced by the projection R(S) −↠ ℤ[ζ_{S}]. All of these maps are homomorphisms of rings (up to the obvious integer multiples).

Now let S(G) be any set of conjugacy class representatives for cyclic subgroups S ⊆ G of finite order. Define c̃h_{X}^{∗} to be the composite

c̃h_{X}^{∗} : K_{G}^{∗}(X) \underrightarrow{\frac{1}{N} ∏ c̃h_{X}^{S}} ℚ ⊗ ( ∏_{S∈S(G)}(H^{∗}(X^{S}/C_{G}(S); ℤ[ζ_{S}]))^{N(S)}) ≅ ℚ ⊗ H_{G}^{∗}(X; R(−)),

where the isomorphism is that of Lemma 5.6. The naturality of c̃h_{X}^{∗}, its independence of the choice of n, and its relation with ch_{X}^{∗}, are immediate from the construction. Also, c̃h_{X}^{∗} is natural with respect to the degree-changing maps K^{∗}(X) → K^{∗+m}(S^{m}×X) (and similarly in cohomology). In particular, this means that it commutes with all maps in Mayer-Vietoris sequences.

It remains to prove that c̃h_{X}^{∗} induces an isomorphism on ℚ ⊗ K_{G}^{∗}(X). This is done by induction on dim(X), using the obvious Mayer-Vietoris sequences. So it suffices to show it for (possibly infinite) disjoint unions ∐_{i∈I} G/H_{i} of orbits. Both groups are zero in odd degrees. And in even degrees,

ℚ ⊗ K_{G}(∐_{i∈I} G/H_{i}) ; ≅ ℚ ⊗ (∏_{i} R(H_{i})) ⟶^{c̃h_{X}^{0}} ℚ ⊗ H_{G}^{ev}(∐_{i∈I} G/H_{i}; R(−)) ; ≅ ℚ ⊗ (∏_{i} R(H_{i}))

is the identity map under these identifications.

6. Completion theorems

Throughout this section, G is a discrete group. We want to prove completion theorems for finite proper G-complexes: theorems which show that K_{G}^{∗}(E × X), when E is a “universal space” of a certain type, is isomorphic to a certain completion of K_{G}^{∗}(X). The key step will be to construct elements of K_{G}^{∗}(X) whose restrictions to orbits in X are sufficiently “interesting”. And this requires a better understanding of the “edge homomorphism” for K_{G}^{∗}(X).

For any finite dimensional proper G-complex X, the skeletal filtration of K_{G}^{∗}(X) induces a spectral sequence

E_{2}^{p,2∗} ≅ H_{G}^{p}(X; R(−)) ⟹ K_{G}^{∗}(X).

If X also has bounded isotropy, the Chern character c̃h_{X} of Proposition 5.8 is an isomorphism (after tensoring with ℚ) from the limit of this spectral sequence to its E_{2}-term. It follows that the spectral sequence collapses rationally; i.e., that the images of all differentials in the spectral sequence consist of torsion elements.

Of particular interest is the edge homomorphism of the spectral sequence. This is a homomorphism

ϵ_{X} : K_{G}^{∗}(X) ⟶ H_{G}^{0}(X; R(−)),

which is induced by restriction to the 0-skeleton of X under the identification

H_{G}^{0}(X; R(−)) = Ker[K_{G}(X^{(0)}) ⟶ K_{G}^{1}(X^{(1)}, X^{(0)})] = Im[K_{G}(X^{(1)}) ⟶ K_{G}(X^{(0)})].

Alternatively, H_{G}^{0}(X; R(−)) can be thought of as the inverse limit, taken over all isotropy subgroups H of X and all connected components of X^{H}, of the representation rings R(H); and the edge homomorphism sends an element of K_{G}^{∗}(X) to the collection of its restrictions to elements of K_{G}^{∗}(Gx) ≅ R(G_{x}) at all points x ∈ X.

As an application of the integral Chern character of Proposition 5.8, we get:

Proposition 6.1 Let X be any finite dimensional proper G-complex whose isotropy subgroups have bounded order. Then for any ξ ∈ H_{G}^{0}(X; R(−)), there is k > 0 such that k·ξ and ξ^{k} lie in the image of the edge homomorphism

ϵ_{X} : K_{G}(X) ⟶ H_{G}^{0}(X; R(−)).

Similarly, for any ξ ∈ H_{G}^{0}(X; RO(−)), there is k > 0 such that k·ξ and ξ^{k} lie in the image of the edge homomorphism

ϵ_{X} : KO_{G}(X) ⟶ H_{G}^{0}(X; RO(−)).

\underline{\underline{Proof}} : The usual homomorphisms between R(−) and RO(−), and between K_{G}^{∗}(−) and KO_{G}^{∗}(−), induced by (ℂ⊗_{ℝ}) and by forgetting the complex structure, show that up to 2torsion, KO_{G}^{∗}(X) and H_{G}^{0}(X; RO(−)) are the fixed point sets under complex conjugation of the groups K_{G}^{∗}(X) and H_{G}^{0}(X; R(−)), respectively. So the edge homomorphism in the orthogonal case is also surjective modulo torsion. The rest of the argument is identical in the real and complex cases; we restrict to the complex case for simplicity.

By Proposition 5.8, the integral Chern character for X^{(0)} is the identity under the usual identifications K_{G}(G/K) ≅ R(K) ≅ H_{G}^{0}(G/K; R(−)) for an orbit G/K (K finite). So by the naturality of c̃h_{X}, the composite

ℚ ⊗ K_{G}(X) ⟶_{≅}^{c̃h_{X}^{0}} ℚ ⊗ H_{G}^{ev}(X; R(−)) −↠ ℚ ⊗ H_{G}^{0}(X; R(−)) ⊆ ℚ ⊗ K_{G}(X^{(0)}) (1)

is just the map induced by restriction to X^{(0)}. So rationally, the edge homomorphism is just the projection of the integral Chern character c̃h onto H_{G}^{0}(X; R(−)), and is in particular surjective. And hence, for any ξ ∈ H_{G}^{0}(X; R(−)), there is some k > 0 such that k·ξ ∈ ϵ_{X}(K_{G}(X)).

It remains to show that ξ^{k} ∈ Im(ϵ_{X}) for some k. If we knew that the Atiyah-Hirzebruch spectral sequence

E_{2}^{p,2∗} ≅ H_{G}^{p}(X; R(−)) ⟹ K_{G}^{∗}(X)

were multiplicative (i.e., that the differentials were derivations), then the result would follow directly. As we have seen, all differentials in the spectral sequence have finite order. Hence, for each r ≥ 2 and each η ∈ E_{r}^{0,2∗}, there is some k > 0 such that

k·d_{r}(η) = 0, and hence d_{r}(η^{k}) = k·d_{r}(η)η^{k−1} = 0.

Upon iteration, this shows that for any ξ ∈ H_{G}^{0}(X; R(−)) = E_{2}^{0,0}, there exists k > 0 such that k·ξ and ξ^{k} both survive to E_{∞}^{0,0}; and hence lie in the image of the edge homomorphism.

Rather than prove the multiplicativity of the spectral sequence, we give the following more direct argument. Identify

ξ ∈ H_{G}^{0}(X; R(−)) = Im[K_{G}(X^{(1)}) ⟶ K_{G}(X^{(0)})].

Assume, for some r ≥ 2, that ξ lies in the image of K_{G}(X^{(r−1)}); we prove that some power of ξ lies in the image of K_{G}(X^{(r)}).

Fix ξ̃∈ K_{G}(X^{(r−1)}) such that res_{X^{(0)}}(ξ̃) = ξ. Since r ≥ 2,

Im[H_{G}^{0}(X^{(r−1)}; R(−)) ⟶ H_{G}^{0}(X^{(0)}; R(−))] = Im[H_{G}^{0}(X^{(r)}; R(−)) ⟶ H_{G}^{0}(X^{(0)}; R(−))].

Hence, since the Chern character is rationally an isomorphism, there exists k such that k·ξ lies in the image of K_{G}(X^{(r)}), or equivalently such that

k·ξ̃∈ Ker[K_{G}(X^{(r−1)}) ⟶ K_{G}(X^{(0)}) ⟶^{d} K_{G}^{1}(X^{(r)}, X^{(0)})]
= Ker[K_{G}(X^{(r−1)}) ⟶^{d} K_{G}^{1}(X^{(r)}, X^{(r−1)}) ⟶ K_{G}^{1}(X^{(r)}, X^{(0)})].  (1)

In Lemma 6.2 below, we will show that there is a K_{G}(X^{(r−1)})-module structure on the relative group K_{G}^{1}(X^{(r)}, X^{(r−1)}) which makes the boundary map d : K_{G}(X^{(r−1)}) → K_{G}^{1}(X^{(r)}, X^{(r−1)}) into a derivation. Then d(ξ̃^{k}) = k·ξ̃^{k−1}·d(ξ̃), so ξ̃^{k} lies in the kernel in (1), and hence ξ^{k} = res_{X^{(0)}}(ξ̃^{k}) lies in the image of K_{G}(X^{(r)}).

It remains to prove:

Lemma 6.2 Let X be any proper G-complex. Then, for any r ≥ 2, one can put a K_{G}(X^{(r−1)})-module structure on K_{G}^{1}(X^{(r)}, X^{(r−1)}) in such a way that for any α, β ∈ K_{G}(X^{(r−1)}),

d(αβ) = α·dβ + β·dα ∈ K_{G}^{1}(X^{(r)}, X^{(r−1)}).

\underline{\underline{Proof}} : We can assume X = X^{(r)}. Write Y = X^{(r−1)}, for short. Fix a map Δ : X →

Z =^{def} X × Y ∪ Y × X which is homotopic to the diagonal, and such that Δ|_{Y} is equal to the diagonal map. Since Z contains the r + 1-skeleton of X × X, Δ is unique up to homotopy (rel Y). In particular, if T : Z → Z is the map which switches coordinates, then T ∘Δ ≃ Δ (rel Y).

Now, for α ∈ K_{G}(Y) and x ∈ K_{G}^{1}(X, Y), let α·x ∈ K_{G}^{1}(X, Y) be the image of α×x under the following composite

α × x ∈ K_{G}^{1}(Y × X, Y × Y) ≅ K_{G}^{1}(Z, X × Y) ⟶^{incl^{∗}} K_{G}^{1}(Z, Y × Y) ⟶^{Δ^{∗}} K_{G}^{1}(X, Y).

Here, the external product α × x is induced by the pairing K_{G} ∧ K_{G} → K_{G×G} → K_{G} of (2.2); or equivalently is defined to be the internal product of proj_{1}^{∗}(α) ∈ K_{G}(Y × X) and proj_{2}^{∗}(x) ∈ K_{G}^{1}(Y × X, Y × Y). We can thus consider K_{G}(X, Y) as a K_{G}(Y)-module. In particular, the relation (αβ)·x = α·(β·x) follows since the two composites (Δ × Id_{X})∘Δ and (Id_{X} ×Δ)∘Δ are homotopic as maps from X to

(X×Y ×Y) ∪ (Y ×X×Y) ∪ (Y ×Y ×X).

Now consider the following commutative diagram:

K_{G}(Y × Y) ⟶^{d} K_{G}^{1}(Z, Y × Y) ⟶^{≅} K_{G}^{1}((X, Y) × Y ) ⊕ K_{G}^{1}(Y × (X, Y))
Δ^{∗}↓  ^{Δ^{∗}}↓  ≅↑
K_{G}(Y) ⟶^{d} K_{G}^{1}(X, Y) ⟵ K_{G}^{1}(Z, Y × X) ⊕ K_{G}^{1}(Z, X × Y )

where the isomorphisms hold by excision. For any α, β ∈ K_{G}(Y), the external product α × β ∈ K_{G}(Y × Y) is sent, by the maps in the top row, to the pair (dα × β, α × dβ). This follows from the linearity of the differential (which holds in any multiplicative cohomology theory). And since T ∘Δ ≃ Δ, as noted above, we have

d(αβ) = Δ^{∗}(d(α × β)) = β·dα + α·dβ.

As an immediate consequence of Proposition 6.1, we now get:

Corollary 6.3 Assume that G is discrete. Fix any family F of finite subgroups of G of bounded order, and let

V = (V_{H}) ∈ \underleftarrow{lim}_{H∈F}R(H)  or  V′ = (V′_{H}) ∈ \underleftarrow{lim}_{H∈F}RO(H)

be any system of compatible (virtual) representations. Then for any finite dimensional proper G-complex X all of whose isotropy subgroups lie in F, there is an integer k > 0, and elements α, β ∈ K_{G}(X) (or α′, β′ ∈ KO_{G}(X)), such that α|_{x} = k·V_{G_{x}} and β|_{x} = (V_{G_{x}})^{k} (or α′|_{x} = k·V′_{G_{x}} and β′|_{x} = (V′_{G_{x}})^{k}) for all x ∈ X.

\underline{\underline{Proof}} : Let ξ be the image of V under the ring homomorphism

\underleftarrow{lim}_{H∈F}R(H) ⟶ H_{G}^{0}(X; R(−))

(or similarly in the orthogonal case); and apply Proposition 6.1.

Corollary 6.3 can be thought of as a generalization of [10, Theorem 2.7]. It was that result which was the key to proving the completion theorem in [10], and Corollary 6.3 plays a similar role in proving the more general completion theorems here.

In what follows, a family of subgroups of a discrete group G will always mean a set of subgroups closed under conjugation and closed under taking subgroups.

Lemma 6.4 Let X be a proper n-dimensional G-complex. Set

I = Ker[K_{G}^{∗}(X) ⟶^{res} K_{G}^{∗}(X^{(0)})].

Then I^{n+1} = 0.

\underline{\underline{Proof}} : Fix any elements x ∈ I^{n} and y ∈ I. By induction, we can assume that x vanishes in K_{G}^{∗}(X^{n−1}), and hence that it lifts to an element x′ ∈ K_{G}^{∗}(X, X^{(n−1)}). Recall that K_{G}^{∗}(X, X^{(n−1)}) is a K_{G}^{∗}(X)-module, and the map K_{G}^{∗}(X, X^{(n−1)}) → K_{G}^{∗}(X) is K_{G}^{∗}(X)-linear. But I·K_{G}^{∗}(X, X^{(n−1)}) = 0, since I vanishes on orbits; so yx′ = 0, and hence yx = 0 in K_{G}^{∗}(X).

As in earlier sections, in order to handle the complex and real cases simultaneously, we set F = ℂ or ℝ, and write KF_{G}^{∗}(−) and RF (−) for the equivariant K-theory and representation rings over F.

Fix any finite proper G-complex X, and let f : X → L be any map to a finite dimensional proper G-complex L whose isotropy subgroups have bounded order. Let F be any family of finite subgroups of G. Regard KF_{G}^{∗}(X) as a module over the ring KF_{G}(L). Set

I = I_{F,L} = Ker[KF_{G}(L) ⟶^{res} ∏_{H∈F} KF_{H}(L^{(0)})].

For any n ≥ 0, the composite

I^{n}·KF_{G}^{∗}(X) ⊆ KF_{G}^{∗}(X) ⟶^{proj^{∗}} KF_{G}^{∗}(E_{F}(G) × X) ⟶^{res} KF_{G}^{∗}((E_{F}(G) × X)^{(n−1)})

is zero, since the image is contained in IKF_{G}^{∗}((E_{F}(G)×_{G}X)^{(n−1)})^{n} = 0 which vanishes by Lemma 6.4. This thus defines a homomorphism of pro-groups

λ_{F}^{X,f} : {KF_{G}^{∗}(X)/I^{n}·K_{G}^{∗}(X)}_{n≥1} ⟶ {KF_{G}^{∗}((E_{F}(G) × X)^{(n−1)})}_{n≥1}.

Theorem 6.5 Fix F = ℂ or ℝ. Let G be a discrete group, and let F be a family of subgroups of G closed under conjugation and under subgroups. Fix a finite proper G-complex X, a finite dimensional proper G-complex Z whose isotropy subgroups have bounded order, and a G-map f : X → Z. Regard KF_{G}^{∗}(X) as a module over KF_{G}(Z), and set

I = I_{F,Z}^{F} = Ker[KF_{G}(Z) ⟶^{res} ∏_{H∈F} KF_{H}(Z^{(0)})].

Then

λ_{F}^{X,f} : {KF_{G}^{∗}(X)/I^{n}·KF_{G}^{∗}(X)}_{n≥1} ⟶ {KF_{G}^{∗}((E_{F}(G) × X)^{(n−1)})}_{n≥1}

is an isomorphism of pro-groups. Also, the inverse system {KF_{G}^{∗}((E_{F}(G) × X)^{(n)})}_{n≥1} satisfies the Mittag-Leffler condition. In particular,

\underleftarrow{lim}^{1}KF_{G}^{∗}((E_{F}(G) × X)^{(n)}) = 0;

and λ_{F}^{X,f} induces an isomorphism

KF_{G}^{∗}(X)_{I} ⟶^{≅} KF_{G}^{∗}(E_{F}(G) × X) ≅ \underleftarrow{lim}KF_{G}^{∗}((E_{F}(G) × X)^{(n)}).

\underline{\underline{Proof}} : Assume that λ_{F}^{X,f} is an isomorphism. Then the system \{KF_{G}^{∗}((E_{F}(G)×X)^{(n)})\}_{n≥1} satisfies the Mittag-Leffler condition because {KF_{G}^{∗}(X)/I^{n}} does. In particular,

\underleftarrow{lim}^{1}KF_{G}^{∗}((E_{F}(G)×X)^{(n)}) = 0, and so KF_{G}^{∗}(E_{F}(G)×X) ≅ \underleftarrow{lim}KF_{G}^{∗}((E_{F}(G)×X)^{(n)})

(cf. [4, Proposition 4.1]).

It remains to show that λ_{F}^{X,f} is an isomorphism.

Step 1 Assume first that X = G/H, for some finite subgroup H ⊆ G. Let F|H be the family of subgroups of H contained in F, and consider the following commutative diagram:

KF_{G}(Z) ⟶^{f^{∗}} KF_{G}^{∗}(G/H) ⟶^{pr_{1}} KF_{G}^{∗}(E_{F}(G)×G/H)
ev^{eH} ↓≅  ^{q}↓≅
RF (H) ⟶_{≅} KF_{H}^{∗}(pt) ⟶^{pr_{2}} KF_{H}^{∗}(E_{F|H}(H)).

Here, pr_{2} induces an isomorphism of pro-groups

{KF_{H}^{∗}(∗)/I_{F}(H)^{n}·KF_{H}^{∗}(∗)}_{n≥1} ⟶ {KF^{∗}((BH)^{(n−1)})}_{n≥1}

by the theorem of Jackowski [9, Theorem 5.1], where

I_{F}(H) = Ker[RF (H) ⟶ ∏_{L∈F|H} RF (L)] ⊇ I′ =^{def} ev_{f(eH)}(I).

(The theorem in [9] is stated only for complex K-theory, but as noted afterwards, the proof applies equally well to the real case.) We want to show that pr_{1} induces an isomorphism of pro-groups

{KF_{G}^{∗}(G/H)/I^{n}·KF_{G}^{∗}(G/H)}_{n≥1} ⟶ {KF_{G}^{∗}((E_{F}(G) × G/H)^{(n−1)})}_{n≥1}.

So we must show that for some k, I_{F}(H)^{k} ⊆ I′.

This means showing that the ideal I_{F}(H)/I′ is nilpotent; or equivalently (since R(H) is noetherian) that it is contained in all prime ideals of R(H)/I′ (cf. [3, Proposition 1.8]). In other words, we must show that every prime ideal of R(H) which contains I′ also contains I_{F}(H). Fix any prime ideal 𝔓 ⊆ R(H) which does not contain I_{F}(H). Set ζ = exp(2πi/|H|) and A = ℤ[ζ]. By a theorem of Atiyah [1, Lemma 6.2], there is a prime ideal 𝔭 ⊆ A and an element s ∈ H such that

𝔓 = {v ∈ R(G) | χ_{v}(s) ∈ 𝔭}.

(This is stated in [1] only in the complex case, but the same arguement applies to prime ideals in the real representation ring.) Also, s is not an element of any L ∈ F, since 𝔓 ⊉ I_{F}(H). Set p = char(A/𝔭) (possibly p = 0).

For any g ∈ G of finite order, we let g_{r} represent its p-regular component: the unique g_{r} ∈ ⟨g⟩ such that p| ; ∤ ; ||g_{r}| and |(g_{r})^{−1}g| is a power of p (g_{r} = g if p = 0). By [1, Lemma 6.3], we can replace s by s_{r} without changing the ideal 𝔓; and can thus assume that p| ; ∤ ; ||s|.

Let m′ be the least common multiple of the orders of isotropy subgroups in Z, and let m be the largest divisor of m′ prime to p (m = m′ if p = 0). Define φ : tors(G) → ℤ by setting φ(g) = 0 if g_{r} ∈ L for some L ∈ F, and φ(g) = m otherwise. By Corollary 4.3, φ|_{L} is a rational character of L for each L ∈ Isotr(Z). So by Corollary 6.3, there is k > 0 and an element ξ ∈ K_{G}(Z) whose restriction to any orbit has character the restriction of φ^{k}. In other words, ξ ∈ I = I_{F,Z}^{F}, and so φ^{k}|_{H} is the character of an element v ∈ I′. But then χ_{v}(s) = φ(s)^{k} ∉ 𝔭, so v ∉ 𝔓, and thus 𝔓 ⊉ I′.

Step 2 By Step 1, the theorem holds when dim(X) = 0. So we now assume that dim(X) = m > 0. Assume X = Y ∪_{φ} (G/H×D^{m}), for some attaching map φ : G/H×S^{m−1} → Y. We can assume inductively that the theorem holds for Y, G/H×S^{m−1}, and G/H×D^{m} ≃ G/H.

All terms in the Mayer-Vietoris sequence

⟶ KF_{G}^{∗}(X) ⟶ KF_{G}^{∗}(Y) ⊕ KF_{G}^{∗}(G/H×D^{m}) ⟶ KF_{G}^{∗}(G/H×S^{m−1}) ⟶

are KF_{G}(X)-modules and all homomorphisms are KF_{G}(X)-linear; and the KF_{G}(Z)-module structure on each term is induced from the KF_{G}(X)-module structure. So if we let I′ ⊆ KF_{G}(X) be the ideal generated by the image of I; then dividing out by (I′)^{n} is the same as dividing out by I^{n} for all terms. In addition, KF_{G}(X) is noetherian (in fact, a finitely generated abelian group), and so this Mayer-Vietoris sequence induces an exact sequence of pro-groups

⟶ {KF_{G}^{∗}(X)/I^{n}}_{n≥1} ⟶ {KF_{G}^{∗}(Y)/I^{n} ⊕ KF_{G}^{∗}(G/H×D^{m})/I^{n}}_{n≥1}

⟶ {KF_{G}^{∗}(G/H×S^{m−1})/I^{n}}_{n≥1} ⟶

by [10, Lemma 4.1]. There is a similar Mayer-Vietoris exact sequence of the pro-groups {KF_{G}^{∗}((E_{F}(G) × −)^{(n−1)})}_{n≥1}; and the theorem now follows from the 5-lemma for progroups together with the induction assumptions.

References

[1] M. Atiyah, Characters and cohomology of finite groups, Publ. Math. I.H.E.S. \underline{9} (1961), 23–64

[2] M. Atiyah, Bott periodicity and the index of elliptic operators, Quart. J. Math. Oxford \underline{19} (1968), 113–140

[3] M. Atiyah & I. Macdonald, Introduction to commutative algebra, Addison-Wesley (1969)

[4] M. Atiyah & G. Segal, Equivariant K-theory and completion, J. Diff. Geometry \underline{3} (1969), 1–18

[5] P. Baum and A. Connes: Chern character for discrete groups, in: Matsumoto, Miyutami, and Morita (eds.), A fête of topology; dedicated to Tamura, 163–232, Academic Press (1988)

[6] G. Bredon, Equivariant cohomology theories, Lecture notes in mathematics \underline{34}, Springer-Verlag (1967)

[7] J. Davis & W. Lück, Spaces over a category and assembly maps in isomorphism conjectures in K-and L-Theory, MPI-preprint (1996), to appear in K-theory

[8] D. Grayson, Higher algebraic K-theory: II, Algebraic K-theory, Evanston, 1976, Springer Lecture Notes in Math. 551 (1976).

[9] S. Jackowski, Families of subgroups and completion, J. Pure Appl. Algebra \underline{37} (1985), 167–179

[10] W. Lück & B. Oliver: The completion theorem in K-theory for proper actions of a discrete group, preprint (1997)

[11] S. Mac Lane, Categories for the working mathematician, Springer-Verlag (1971)

[12] G. Segal, Categories and cohomology theories, Topology \underline{13} (1974), 293–312 [13] J.-P. Serre, Linear representations of finite groups, Springer-Verlag (1977)

[14] J. Słomińska, On the equivariant Chern homomorphism, Bull. Acad. Pol. Sci. \underline{24} (1976), 909–913

Addresses:

Wolfgang Lück Bob Oliver Institut für Mathematik und Informatik Laboratoire de Mathématiques Westfälische Wilhelms-Universtität Université Paris Nord Einsteinstr. 62 Avenue J.-B. Clément 48149 Münster 93430 Villetaneuse Germany France lueck@math.uni-muenster.de bob@math.univ-paris13.fr http://wwwmath.uni-muenster.de/math/u/lueck http://zeus.math.univ-paris13.fr/∼bob