The completion theorem in K-theory for proper actions of a discrete group

by Wolfgang Lück and Bob Oliver^{∗}

Abstract

We prove a version of the Atiyah-Segal completion theorem for proper actions of an infinite discrete group G. More precisely, for any finite proper G-CW-complex X, K^{∗}(EG×_{G}X) is the completion of K_{G}^{∗}(X) with respect to a certain ideal. We also show, for such G and X, that K_{G}(X) can be defined as the Grothendieck group of the monoid of G-vector bundles over X. Key words: K-theory, proper actions, vector bundles 1991 mathematics subject classification: primary 55N91, secondary 19L47

Let G be any discrete group. For such G, a G-CW-complex is a CW-complex with Gaction which permutes the cells, such that an element g ∈ G sends a cell to itself only by the identity map. A G-CW-complex X is proper if all of its isotropy subgroups have finite order, and is finite if it is made up of finitely many orbits of cells. A G-CW-pair is a pair of G-spaces (X, A), where X is a G-CW-complex and A is a G-invariant subcomplex.

The main results of this paper are Theorems 3.2 and 4.3 below. The first says that equivariant K-theory K_{G}^{∗}(−) can be defined on the category of finite proper G-CW-pairs using (finite dimensional) G-vector bundles, in the sense that this does define an equivariant cohomology theory. In particular, for any X, K_{G}(X) is just the Grothendieck group of the monoid of G-vector bundles over X.

The second theorem is an extension of the Atiyah-Segal completion theorem to this situation. It says that for any finite proper G-CW-complex X, K^{∗}(EG×_{G}X) is the completion of K_{G}^{∗}(X) with respect to a certain ideal. In particular, when the universal proper G-space E_{FIN} (G) (= \underline{E}G in the notation of Baum and Connes [7]) has the homotopy type of a finite G-CW-complex, then this completion is taken with respect to the augmentation ideal of K_{G}(E_{FIN} (G)). For example, when X = E_{FIN} (G), Theorem 4.3 implies that K^{∗}(BG) is the completion of K_{G}^{∗}(E_{FIN} (G)) with respect to the augmentation ideal in K_{G}(E_{FIN} (G)).

There are two ways in which the proofs of these theorems, when G is infinite and discrete, diverge from the usual proofs for finite group actions. First, since the category of spaces with proper G-action doesn’t contain cones or suspensions (fixed points are not allowed), we need to find other ways to define K_{G}(X, A) and K_{G}^{−n}(X). This is easily handled. A more crucial difference is that special constructions are needed, carried out in Section 2, to get around the lack of “sufficiently many” product bundles. This second difficulty is illustrated by the fact that both of these theorems fail in general when G is a positive dimensional noncompact Lie group. This is discussed in detail, with examples, in Section 5. Examples which show that K_{G}(−) defined using G-vector bundles is not an equivariant cohomology theory in this situation were originally due to Phillips [15], who instead defined K_{G}(−) using infinite dimensional G-vector bundles with Hilbert space fibers (see also [17]).

∗URA CNRS 742

In a separate paper, we will construct an equivariant cohomology theory K_{G}^{∗}(−) for arbitrary (not necessarily proper) G-CW-complexes using spectra. More precisely, this will be done using Or(G)-spectra: contravariant functors from the orbit category of G to spectra. We will also construct an equivariant Chern character for proper G-CW -complexes which takes values in equivariant Bredon cohomology, and which is rationally an isomorphism for finite proper G-CW -complexes.

Let 𝕂_{G}^{∗}(X) be the Grothendieck group of (finite-dimensional) G-vector bundles over X. There is a natural transformation φ_{G} : 𝕂_{G}^{∗}(X) → K_{G}^{∗}(X), which is an isomorphism for finite proper G-CW-complexes. In the nonequivariant case, this is well known to be an isomorphism for any finite dimensional CW-complex X (since any map X → BU factors through some BU(n)). But even for finite G ≠ 1, Example 3.11 below shows that 𝕂_{G}^{∗} is not a cohomology theory on the category of all finite dimensional proper G-CW-pairs.

The paper is organized as follows:

1. G-vector bundles over proper G-CW-complexes
2. Constructions of G-vector bundles
3. Equivariant K-theory for finite proper G-CW-complexes
4. The completion theorem
5. Proper actions of Lie groups

References

1. G-vector bundles over proper G-CW-complexes

Throughout this section G is a Lie group. We collect here some basic facts about G-vector bundles over proper G-CW-complexes.

A G-CW-complex X is a space with G-action, which is filtered by its “skeleta” X^{(n)}, such that X has the weak topology as the union of the X^{(n)}, and such that each X^{(n)} is obtained from X^{(n−1)} by attaching orbits of cells G/H_{i}×D^{n} via attaching maps G/H_{i}×S^{n−1} → X^{(n−1)}. (Here X^{(−1)} = ∅.) When G is discrete, a G-CW-complex can be thought of as a CW-complex with G-action which permutes the cells, such that an element g ∈ G sends a cell to itself only by the identity map. Note that the orbit space of a G-CW-complex inherits the structure of an (ordinary) CW-complex. For more details about G-CW-complexes, see, e.g., [9, §§II.1–2] or [13, §§I.1–2].

A G-CW-complex X is finite if it is made up of finitely many orbits of cells G/H×D^{n}, or equivalently if X/G is a finite CW-complex. A G-CW-complex X will be called proper if all of its isotropy subgroups are compact. (For G-CW-complexes, this is equivalent to the various definitions of proper actions which have been given in more general situations.)

A G-vector bundle over a G-CW-complex X consists of a (complex) vector bundle p : E → X, together with a G-action on E such that p is G-equivariant and each g ∈ G acts on E and X via a bundle isomorphism. We let E|_{x} denote the fiber over a point x ∈ X. A map of G-vector bundles from p : E → X to p′ : E′ → X′ is just a map (f, f) of vector bundles, such that f : E → E′ and f : X → X′ are G-equivariant. Here, we assume only that f restricts to a linear map E|_{x} → E′|_{f(x)} for each x ∈ X. We call (f, f) a strong map if f restricts to a linear isomorphism E|_{x} ⟶^{≅} E′|_{f(x)} for each x ∈ X. This is clearly equivalent to the condition that p : E → X is isomorphic to the pullback of p′ : E′ → X′ over f.

Most of the properties of G-vector bundles over G-CW-complexes we need will be easy consequences of the following elementary lemma.

Lemma 1.1 (a) Any G-vector bundle over an orbit of cells G/H × D^{n} is isomorphic to G×_{H}(V × D^{n}) for some H-representation V.

(b) For any G-CW-complex X, both X and X/G are paracompact.

(c) Fix a G-vector bundle p : E → X over a G-CW-complex X. Let X^{(n)} be the n-skeleton of X, and set E_{n} = p^{−1}(X^{(n)}). Then the squares

\begin{matrix}∐_{j∈J} G/H_{j} × S^{n−1} & \underrightarrow{∐_{j∈J} q_{j}} & X^{(n−1)} \\ ↓ & & ↓ \\ ∐_{j∈J} G/H_{j} × D^{n} & \underrightarrow{∐_{j∈J} Q_{j}} & X^{(n)} \end{matrix} and \begin{matrix}∐_{j∈J} G×_{H_{j}}(V_{j} × S^{n−1}) & \underrightarrow{∐_{j∈J} q_{j}} & E_{n−1} \\ ↓ & & ↓ \\ ∐_{j∈J} G×_{H_{j}}(V_{j} × D^{n}) & \underrightarrow{∐_{j∈J} Q_{j}} & E_{n} \end{matrix}

are pushout squares for each n. Also, X and E have the weak topology with respect to the subspaces X^{(n)} and E_{n}, respectively. More generally, if {X_{i}}_{i∈I} is any set of subcomplexes which cover X, then X and E have the weak topology with respect to the subspaces X_{i} and p^{−1}(X_{i}), respectively.

(d) For any G-CW-pair (X, A), there is a neighborhood W of A in X, which can be chosen to be closed or open, such that A is an equivariant strong deformation retract of W.

\underline{\underline{Proof}} : (a) Note that for any G-map p : X ⟶ G/H, the canonical map G×_{H} p^{−1}(eH) ⟶ X is a G-homeomorphism. (This will be used frequently throughout the paper.) It thus suffices to show that any H-vector bundle over D^{n} is isomorphic to the product bundle V × D^{n} for some H-representation V, and this follows from [3, Proposition 1.6.2].

(c) The pushout square for X^{(n)}, and the fact that X has the weak topology with respect to its skeleta, follow from the definition of a G-CW-complex. In particular, a function X → Y (for any space Y) is continuous if and only if its composite with each equivariant cell G/H × D^{n} → X is continuous; and from this one sees immediately that X has the weak topology with respect to any covering set of subcomplexes. The G-pushout property for E_{n} follows the pushout property for X^{(n)}, together with (a) and [13, Lemma 1.26].

We now claim for any X, any vector bundle p : E → X, and any covering X = ∪_{i∈I}X_{i} by closed subspaces, that E has the weak topology with respect to its subsets p^{−1}(X_{i}) if X has the weak topology with respect to the X_{i}. Upon restricting to a neighborhood of any given x ∈ X, this is reduced to the case where E is a product bundle; and the result then follows easily since the fibers are locally compact.

(b) Given an open covering U of X or of X/G, a partition of unity subordinate to U can be constructed by applying Zorn’s lemma to the set of such partitions of unity over subcomplexes of X (and using (c) above). For more details (in the case of a nonequivariant CW-complex), see [14, Theorem II.4.2].

(d) For each n, one easily constructs a collar neighborhood V_{n} of X^{(n)} in X^{(n+1)} (open or closed), together with an equivariant deformation retraction ρ_{n} : V_{n} → X^{(n)}, which restricts to a deformation retraction of ρ_{n}^{−1}(B) to B for any B ⊆ X^{(n)}. Now set

W_{−1} = A, W_{n} = A ∪ ρ_{n}^{−1}(W_{n−1} ∩ X^{(n)}) (all n ≥ 0), and W = ∪_{n=0}^{∞}W_{n},

let r_{n} : W_{n} → W_{n−1} be the identity on A and ρ_{n} on W_{n}∖A, and let r : W → A be the composite of the r_{n}.

The next three results, which list some of the standard properties of G-vector bundles, are easy consequences of Lemma 1.1. We begin with homotopy invariance. As usual, I denotes the unit interval [0, 1].

Theorem 1.2 Let X be a proper G-CW-complex, let p : E → X×I be a G-vector bundle, and set E_{0} = E|_{X×0}, regarded as a G-vector bundle over X. Then there is an isomorphism ρ : E ⟶^{≅} E_{0} × I of G-vector bundles, which is the identity on E_{0} and covers the identity on X × I. If, in addition, A ⊆ X is any G-invariant subcomplex, then ρ can be chosen to extend any given isomorphism ρ_{A} : E|_{A×I} ⟶^{≅} E_{0}|_{A} × I.

\underline{\underline{Proof}} : Using Lemma 1.1(c), this is quickly reduced to the case where (X, A) = (G/H × D^{n}, G/H × S^{n−1}). By Lemma 1.1(a), E = G×_{H}(V × D^{n} × I). Thus, ρ_{A} is equivalent to a map ρ′ : S^{n−1} ×I → Aut_{H}(V) which sends S^{n−1} ×0 to the identity, and this can be extended to D^{n} × I since S^{n−1} → D^{n} is a cofibration.

The proof of the next two lemmas is similiar to that of Theorem 1.2.

Lemma 1.3 Let (X, A) be a proper G-CW-pair, and let E and E′ be G-vector bundles over X. Then any map f : E′|_{A} ⟶ E|_{A} of G-vector bundles over A extends to a map f : E′ ⟶ E of G-vector bundles over X.

\underline{\underline{Proof}} : Via Lemma 1.1, it suffices to prove this when (X, A) = (G/H × D^{n}, G/H × S^{n−1}), E = G×_{H}(V × D^{n}), and E′ = G×_{H}(V′ × D^{n}). A map E′ → E of G-vector bundles thus corresponds to an H-map D^{n} → Hom_{H}(V′, V), and any map over A extends to a map over X since Hom_{H}(V′, V) is contractible.

Lemma 1.4 Let (X, A) be a proper G-CW-pair, and let E be a G-vector bundle over X. Then any G-invariant Hermitian metric of E|_{A} extends to a G-invariant Hermitian metric on E.

\underline{\underline{Proof}} : Again, it suffices to prove this when (X, A) = (G/H × D^{n}, G/H × S^{n−1}), and E = G×_{H}(V ×D^{n}). A Hermitian metric over X then corresponds to a map D^{n} → Herm_{H}(V) (the space of H-invariant Hermitian metrics over V); and any such map on S^{n−1} can be extended to one on D^{n} since Herm_{H}(V) is convex (and hence contractible).

To finish the section, we check that a pushout of G-vector bundles is a G-vector bundle over the pushout of the base spaces. This will, of course, be used to prove excision in Section 3.

Lemma 1.5 Let φ : (X_{1}, X_{0}) → (X, X_{2}) be a map of G-CW-pairs, set φ_{0} = φ|_{X_{0}}, and assume that X ≅ X_{2}∪_{φ_{0}}X_{1}. Let p_{1} : E_{1} → X_{1} and p_{2} : E_{2} → X_{2} be G-vector bundles, let φ_{0} : E_{1}|_{X_{0}} → E_{2} be a strong map covering φ_{0}, and set E = E_{2}∪_{φ_{0}}E_{1}. Then p = p_{1} ∪ p_{2} : E → X is a G-vector bundle over X.

\underline{\underline{Proof}} : The only problem is to show that p : E ⟶ X is locally trivial (in a nonequivariant sense). Since E_{1} is locally trivial, so is E|_{X∖X_{2}} ≅ E_{1}|_{X_{1}∖X_{0}}. So it remains to find a neighborhood of X_{2} over which E is locally trivial. Choose a closed neighborhood W_{1} of X_{0} in X_{1} for which there is a strong deformation retraction r : W_{1} → X_{0} (Lemma 1.1(d)). By the homotopy invariance for nonequivariant vector bundles over paracompact spaces (cf. [10, Corollary 3.4.5]), r is covered by a strong map of vector bundles r : E_{1}|_{W_{1}} ⟶ E_{0} which extends i_{1}. Set W = X_{2}∪_{φ_{0}}W_{1}. Then r extends, via the pushout, to a strong map of vector bundles E|_{W} → E_{2} which extends i_{2}, and hence E|_{W} is locally trivial.

Let p : E ⟶ B be a G-vector bundle over a proper G-CW-complex. Each orbit Gx ⊆ X has a G-invariant neighborhood U_{x} such that Gx is an equivariant retract of U_{x}, and the neighborhood can in fact be chosen such that the retraction is covered by a strong map E|_{U_{x}} → E|_{Gx}. There is thus a G-covering U of X such that each E|_{U} (for U ∈ U) is “trivial” in the sense that it is the pullback of a bundle over a (proper) orbit. Also, since X/G is paracompact by Lemma 1.1(b), U is G-numerable in the sense that there is a locally finite partition of unity {t_{U} | U ∈ U} by G-invariant functions t_{U} with supp(t_{U}) ⊆ U. Hence our notion of G-vector bundles agrees with that of tom Dieck in [9, I.9]. For these same reasons, the results of this section can easily be extended to proper G-spaces which have paracompact quotients (any such space has tubes, i.e., equivariant neighborhood retracts of orbits).

2. Constructions of G-vector bundles

The main result in this section is Theorem 2.7. Given a discrete group G, a G-CW-complex X, and a family {V_{H}} of representations of the isotropy subgroups in X, we would like to be able to construct a G-vector bundle E → X whose fiber over any x ∈ X is isomorphic to V_{G_{x}}. This is in general not possible, even for finite G, for reasons discussed at the end of the section. What we show here is that we can do this, assuming certain conditions on X and the V_{H}, but only after replacing the V_{H} by some iterated direct sum (V_{H})^{k}, or by some iterated tensor product (V_{H})^{⊗k}. These bundles are the crucial ingredients in the proof that G-vector bundles define an equivariant cohomology theory (Theorem 3.2), and the proof of the completion theorem (Theorem 4.3).

Throughout the first part of the section, G and Γ will denote arbitrary Lie groups. A family F of subgroups of G is a set of (closed) subgroups of G which is closed under conjugation. We will need to work with some classifying spaces and universal spaces: first for (proper) G-actions and then for bundles.

Definition 2.1 For any family F of subgroups of G, let E_{F}(G) denote the topological category whose objects are the pairs (G/H, gH) for H ∈ F and g ∈ G, and where Mor((G/H, gH), (G/K, g′K)) is the set of G-maps G/H → G/K which send gH to g′K (a set of cardinality at most one). Let E_{F}(G) be the realization of the nerve of E_{F}(G), considered as a G-CW-complex:

E_{F}(G) = (∐_{n=0}^{∞} ∐_{G/H_{0}→···→G/H_{n}} G/H_{0} × Δ^{n})/∼,

where the identifications are those induced by the obvious face and degeneracy maps.

As usual, we are assuming that E_{F}(G) has the weak topology with respect to its cellular structure.

Lemma 2.2 Fix a Lie group G and a family F of subgroups of G.

(a) For any K ∈ F, (E_{F}(G))^{K} is contractible.

(b) Let (X, A) be any G-CW-pair such that G_{x} ∈ F for all x ∈ X. Then any G-map f_{A} : A → E_{F}(G), extends to a G-map f_{X} : X → E_{F}(G), and any two such extensions are G-homotopic relative A.

\underline{\underline{Proof}} : (a) For any K ⊆ G, (E_{F}(G))^{K} is the nerve of the full subcategory of E_{F}(G) with objects those (G/H, gH) such that K ⊆ gHg^{−1}. And if K ∈ F, then this category has the initial object (G/K, eK).

(b) This follows immediately from point (a) (see [13, Proposition 2.3 on page 35]).

A G-equivariant Γ-bundle (or (G, Γ)-bundle for short) consists of a Γ-principal bundle p : E → X, together with left G-actions on E and X, such that p is G-equivariant, and such that the left G-action and the right Γ-action on E commute. We let Bdl_{G,Γ}(X) denote the set of isomorphism classes of (G, Γ)-bundles over the G-space X.

One natural example of this is the case Γ = U(n). A (G, U(n))-bundle E → X is just the principal bundle associated with the G-vector bundle E×_{U(n)}ℂ^{n} → X. Similarly, a (G, Σ_{n})- bundle is the principal bundle associated with a G-equivariant n-sheeted covering space. In the constructions below, we will have to consider (G, Γ)-bundles for certain finite subgroups Γ ⊆ U(n).

Now fix a family F of compact subgroups of G. For each H ∈ F, set Rep_{Γ}(H) = Hom(H, Γ)/ Inn(Γ); i.e., the set of conjugacy classes of homomorphisms from H to Γ. For example, Rep_{U(n)}(H) is the set of isomorphism classes of n-dimensional complex representations of H, and Rep_{Σ_{n}}(H) is the set of isomorphism classes of H-sets of order n. Note that for any H and Γ, there are natural bijections

Rep_{Γ}(H) ≅ Bdl_{H,Γ}(pt) ≅ Bdl_{G,Γ}(G/H).  (2.3)

We need a way to specify the isomorphism types of the fibers of a (G, Γ)-bundle. Suppose we are given an element

A = (α_{H}) ∈ \underleftarrow{lim}_{H∈F} Rep_{Γ}(H) ⊆ ∏_{H∈F} Rep_{Γ}(H),

where the limit is taken with respect to all homomorphisms induced by inclusions and conjugation in G. This is equivalent to an element in \underleftarrow{lim}_{Or_{F} (G)} Bdl_{G,Γ}(−), where Bdl_{G,Γ}(−) is considered as a contravariant functor (via pullback) on the orbit category Or_{F}(G). If X is a G-space all of whose isotropy subgroups lie in F, then we define a (G, A)-bundle over X to be a (G, Γ)-bundle such that the fiber over any point x ∈ X is isomorphic to (Γ, α_{G_{x}}), regarded as a (G_{x}, Γ)-bundle over a point (see (2.3)). When Γ = U(n), this corresponds to those G-vector bundles whose fibers are isomorphic to certain given representations of the isotropy subgroups.

We want to define classifying spaces for (G, Γ)-bundles and for (G, A)-bundles. In fact, these are just the universal (G × Γ)-CW-complexes with respect to appropriate families.

Definition 2.4 Let F be a family of compact subgroups of G. Define

E_{F}(G, Γ) = E_{F_{Γ}}(G × Γ)  and  B_{F}(G, Γ) = E_{F}(G, Γ)/Γ,

where

F_{Γ} = {H ⊆ G × Γ | pr_{1}(H) ∈ F, H ∩ (1 × Γ) = 1}.

For any element

A = (α_{H}) ∈ \underleftarrow{lim}_{H∈F} Rep_{Γ}(H) ⊆ ∏_{H∈F} Rep_{Γ}(H),

define

F_{A} = {H ⊆ G × Γ | H = graph(α : K → Γ), some K ∈ F, some α conjugate to α_{K}},

and set

E_{F}(G, A) = E_{F_{A}}(G × Γ)  and  B_{F}(G, A) = E_{F}(G, A)/Γ.

In the above situation, if E ⟶^{p} X is any (G, Γ)-bundle, where X is a proper G-CW-complex all of whose isotropy subgroups lie in F, then E is a proper (G × Γ)-CW-complex all of whose isotropy subgroups lie in F_{Γ}. Conversely, if E is any proper (G×Γ)-CW-complex all of whose isotropy subgroups lie in F_{Γ}, then E/Γ is a proper G-CW-complex all of whose isotropy subgroups lie in F, and the projection E → E/Γ is a (G, Γ)-bundle. Similarly, for any A, there is a correspondence between (G, A)-bundles and (G × Γ)-CW-complexes all of whose isotropy subgroups lie in F_{A}. This leads to the following:

Lemma 2.5 Fix a family F of compact subgroups of G, and an element

A = (α_{H}) ∈ \underleftarrow{lim}_{H∈F} Rep_{Γ}(H),

where the limit is taken with respect to inclusions and conjugation in G. Then the following hold:

(a) For each H ∈ F, let C_{Γ}(α_{H}) denote the centralizer of the image of α_{H} : H → Γ (well defined up to conjugacy). Then there is a homotopy equivalence

(B_{F}(G, A))^{H} ≃ BC_{Γ}(α_{H}),

which is natural with respect to maps induced by homomorphisms Γ → Γ′.

(b) The (G, A)-bundle

E_{F}(G, A) ⟶ B_{F}(G, A)

is the universal (G, A)-bundle in that it defines, via pullbacks, a bijection

[X, B_{F}(G, A)]_{G} ⟶ Bdl_{G,A}(X),

for any proper G-CW-complex X all of whose isotropy subgroups are in F.

\underline{\underline{Proof}} : (a) Fix H, and write C = C_{Γ}(α_{H}) for short. Consider the (G, A)-bundle

G ×_{H} (EC×_{C}Γ) ⟶ G/H × BC,

where H acts on EC×_{C}Γ via h(x, γ) = (x, α_{H}(h)γ). The classifying map for this bundle restricts to a map

BC ⟶ (B_{F}(G, A))^{H}.

Similarly, the restriction of the universal (G, A)-bundle over (B_{F}(G, A))^{H} is an (H, α_{H})- bundle over a space with trivial H-action, and hence has structure group C = C_{Γ}(α_{H}). It is thus classified by a map

(B_{F}(G, A))^{H} ⟶ BC,

and the above two maps are homotopy inverses by the universal properties of the spaces.

(b) This follows immediately from Lemma 2.2(b).

We now assume, throughout the rest of the section, that G is discrete. We need to construct maps to the classifying spaces defined in Definition 2.4. The obstructions to doing so lie in certain Bredon cohomology groups.

Let (X, A) be any G-CW-pair such that the isotropy group of each point in X∖A lies in F. For each n ≥ 0, let

\underline{C}_{n}(X, A) : Or_{F}(G) ⟶ Ab

denote the contravariant functor which sends G/H to C_{n}(X^{H}, A^{H}). Here, C_{n}(X^{H}, A^{H}) is the free abelian group with one generator for each n-cell in X^{H}∖A^{H}. For any contravariant functor M : Or_{F}(G) → Ab, Hom_{Or_{F} (G)}(\underline{C}_{n}(X, A), M) is the direct sum of one copy of M(G/H) for each orbit G/H×D^{n} of n-cells in X∖A. In particular, \underline{C}_{n}(X, A) is projective in the category Or_{F}(G)-mod of contravariant functors Or_{F}(G) → Ab. The Bredon cohomology groups H_{G}^{∗}(X, A; M) are thus the homology groups of the cochain complex

0 ⟶ Hom_{Or_{F} (G)}(\underline{C}_{0}(X, A), M) ⟶^{δ} Hom_{Or_{F} (G)}(\underline{C}_{1}(X, A), M)

⟶^{δ} Hom_{Or_{F} (G)}(\underline{C}_{2}(X, A), M) ⟶^{δ} ….

Lemma 2.6 Assume that G is discrete. Fix a family F of finite subgroups of G, a finite group Γ, and a system of representations

A = (α_{H})_{H∈F} ∈ \underleftarrow{lim}_{H∈F} Rep_{Γ}(H).

Set B = B_{F}(G, A), and let

β_{A} : B ⟶ E_{F}(G)

be any G-map. (This exists and is unique up to G-homotopy by Lemma 2.2(b), since all isotropy subgroups for B lie in F.) Let Z denote the mapping cylinder of β_{A}. Let M : Or_{F}(G) → Ab be any contravariant functor. Then for each n ≥ 0,

|Γ|^{n} · H_{G}^{n}(Z, B; M) = 0.

\underline{\underline{Proof}} : There is a cohomology spectral sequence

E_{2}^{p,q} = Ext_{Or_{F} (G)}^{p}(\underline{H}_{q}(Z, B), M) ⟹ H_{G}^{p+q}(Z, B; M),

where \underline{H}_{p}(Z, B) denotes the functor Or_{F}(G) → Ab which assigns to G/H the abelian group H_{p}(Z^{H}, B^{H}). It is induced by the double complex Hom_{Or_{F} (G)}(\underline{C}_{q}(Z, B), I_{p}), where \{I_{p}\} is any injective resolution in Or_{F}(G)-mod of M. We have just seen that the \underline{C}_{q}(Z, B) are all projective in Or_{F}(G)-mod. This category does have enough injectives by, e.g., [19, Example 2.3.13].

Since Z^{H} ≃ (E_{F}(G))^{H} is contractible by Lemma 2.2(a), we conclude from Lemma 2.5(a) that

H_{q}(Z^{H}, B^{H}) ≅ H̃_{q−1}(B^{H}) ≅ H̃_{q−1}(C_{Γ}(α_{H})).

In particular, since C_{Γ}(α_{H}) ⊆ Γ, this shows that

|Γ| · H_{∗}(Z^{H}, B^{H}) = 0.

So |Γ| annihilates all terms in the above spectral sequence, and hence (since E_{2}^{p,0} = 0) |Γ|^{n} annihilates H_{G}^{p}(Z, B; M).

Given any A = (α_{H}) ∈ \underleftarrow{lim}_{H∈F} Rep_{Γ}(H), and any homomorphism ρ : Γ → U(n), there is a natural map

ρ_{∗} : B_{F}(G, A) ⟶ B_{F}(G, ρ∘A), where ρ∘A = (ρ∘α_{H}) ∈ \underleftarrow{lim}_{H∈F} Rep_{U(n)}(H);

and ρ_{∗} commutes with the maps β_{A} and β_{ρ∘A} to E_{F}(G).

Theorem 2.7 Assume that G is discrete. Fix any family F of finite subgroups of G, and let

V = (V_{H}) ∈ \underleftarrow{lim}_{H∈F} Rep_{U(n)}(H)

be any system of compatible n-dimensional representations. Assume that there is a finite group Γ, a system

A = (α_{H}) ∈ \underleftarrow{lim}_{H∈F} Rep_{Γ}(H),

and a homomorphism ρ : Γ → U(n) such that V = ρ∘A. Then for any d > 0 there is an integer k = k(d) > 0, such that for any d-dimensional G-CW-complex X all of whose isotropy subgroups lie in F, there are G-vector bundles E, E′ → X such that the fibers E|_{x} and E′|_{x} over each point x ∈ X are isomorphic as G_{x}-representations to (V_{G_{x}})^{k} and (V_{G_{x}})^{⊗k}, respectively.

\underline{\underline{Proof}} : We only treat the case of direct sums here; the tensor product case is analogous. By the universal property of E_{F}(G) (Lemma 2.2(b)), it suffices to prove this when X = E_{F}(G)^{(d)} (the d-skeleton).

Write B = B_{F}(G, A) and B′_{k} = B_{F}(G, V^{k}) for short (any k ≥ 1), and let Z be the mapping cylinder of β_{A} : B → E_{F}(G). We must construct, for some k, a map Z^{(d)} → B′_{k}; and we will do so by extending the map ρ_{∗}^{k} : B → B′_{k}. By Lemma 2.5(a), (B′_{k})^{H} ≃ B Aut_{H}(V_{H}^{k}) for each H ∈ F, and is in particular a product of BU(m)’s and hence simply connected. So there is no obstruction to extending ρ_{∗}^{1} : B → B′_{1} to a map f_{2} : B ∪ Z^{(2)} → B′_{1}.

Assume inductively that f_{d−1} : B ∪ Z^{(d−1)} → B′_{r} has been constructed, (where r = k(d − 1)). We now apply standard equivariant obstruction theory. For each H ∈ F, let

c^{d}(f_{d−1})(G/H) : C_{d}(Z^{H}, B^{H}) ⟶ π_{d−1}((B′_{r})^{H})

be the map which sends each generator, corresponding to a d-cell σ in Z^{H}∖B^{H}, to the element f_{d−1}(∂σ). This is well defined independently of the basepoint, since (B′_{r})^{H} is simply connected. By naturality, this defines an element

c^{d}(f_{d−1}) ∈ C_{G}^{d}(Z, B; \underline{π}_{d−1}(B′_{r})) = Hom_{Or_{F} (G)}(\underline{C}_{d}(Z, B), \underline{π}_{d−1}(B′_{r})),

and f_{d−1} can be extended to a map B ∪ Z^{(d)} → B′_{r} if and only if c^{d}(f_{d−1}) = 0. Furthermore, c^{d}(f_{d−1}) is a cocycle by [20, Theorem V.5.6], and hence defines an element

o^{d}(f_{d−1}) ∈ H_{G}^{d}(Z, B; \underline{π}_{d−1}(B′_{r})).

Finally, for any c̃ ∈ C_{G}^{d−1}(Z, B; \underline{π}_{d−1}(B′_{r})), there is a map f′ : B ∪ Z^{(d−1)} → B′_{r} such that f′ agrees with f_{d−1} on B ∪ Z^{(d−2)}, and such that c^{d}(f′) − c^{d}(f_{d−1}) = δ(c̃) (as in [20, Theorem V.5.6′]). So if o^{d}(f_{d−1}) = 0, then f_{d−1}|B ∪ Z^{(d−2)} can be extended to B ∪ Z^{(d)}. For more details, see [20, §V.5], and also [8, §II.1] (where equivariant obstruction theory is developed for actions of a finite group).

By Lemma 2.6, H_{G}^{d}(Z, B; \underline{π}_{d−1}(B′_{r})) has exponent |Γ|^{d}. Furthermore, by Lemma 2.5(a) again, \underline{π}_{d−1}(B′_{r}) is the functor G/H ↦ π_{d−1}(B Aut_{H}(V_{H}^{r})). Write m = |Γ|^{d} for short, and consider the homomorphisms

π_{d−1}(B Aut_{H}(V_{H}^{r})) \overrightarrow{⟶}_{diag}^{i_{1},...,i_{m}} (π_{d−1}(B Aut_{H}(V_{H}^{r})))^{m} ⟶^{B⊕_{∗}} π_{d−1}(B Aut_{H}(V_{H}^{mr})).

These are all homomorphisms of functors on Or_{F}(G). Also, B⊕_{∗}∘i_{s} = B⊕_{∗}∘i_{1} for all s, since the corresponding maps between spaces differ by conjugation by an element of Aut_{H}(V_{H}^{mr}) (and the automorphism group is connected). Since diag = ∑_{s=1}^{m} i_{s}, it follows that (B⊕_{∗})∘diag factors through multiplication by m, and hence that the induced map

H_{G}^{d}(Z, B; \underline{π}_{d−1}(B′_{r})) ⟶^{Δ_{∗}^{m}} H_{G}^{d}(Z, B; \underline{π}_{d−1}(B′_{mr}))

is zero. Here, Δ^{m} : U(rn) → U(mrn) denotes the diagonal inclusion. We can thus extend Δ_{∗}^{m}∘f_{d−1}|B ∪ Z^{(d−2)} to a map

f_{d} : B ∪ Z^{(d)} ⟶ B′_{mr} = B_{F}(G, V^{mr}).

Set k = k(d) = mr; the pullback to E_{F}(G)^{(d)} ⊆ Z^{(d)} of the (G, V^{k})-vector bundle

E_{F}(G, V^{k}) ×_{U(nk)} ℂ^{nk} ⟶ B_{F}(G, V^{k})

now has the desired properties.

As a first consequence of Theorem 2.7, we show the following result, which will be needed when proving excision for equivariant K-theory defined via G-vector bundles.

Corollary 2.8 Assume that G is discrete, and let X be any finite dimensional proper G-CW-complex whose isotropy subgroups have bounded order. Then there is a G-vector bundle E → X such that for each x ∈ X, the fiber E|_{x} is a multiple of the regular representation of G_{x}.

\underline{\underline{Proof}} : Let F be the family of isotropy subgroups in X, and let n be the least common multiple of their orders. For each H ∈ F, let V_{H} be the free complex H-representation of dimension n, and let α_{H} : H → Σ_{n} be a homomorphism corresponding to a free H-set of order n. Then

V = (V_{H}) ∈ \underleftarrow{lim}_{H∈F} Rep_{U(n)}(H) and A = (α_{H}) ∈ \underleftarrow{lim}_{H∈F} Rep_{Σ_{n}}(H),

and these satisfy the hypotheses of Theorem 2.7 (with Γ = Σ_{n}). So by the theorem, there is some k, and a G-vector bundle E → X, such that for each x ∈ X, E|_{x} ≅ V_{G_{x}}^{k} is a multiple of the regular representation of G_{x}.

An H-representation V will be called p′-free if for any subgroup K ⊆ H of order prime to p, V |K is a multiple of the regular representation of K. This is equivalent to the condition that the character of any element h ∈ H not of p-power order is zero. The next result, a second consequence of Theorem 2.7, is the main technical ingredient in our extension of the completion theorem of Atiyah and Segal from finite groups and compact spaces to arbitrary discrete groups and finite proper G-CW-complexes.

Corollary 2.9 Assume that G is discrete, and let X be any finite dimensional proper G-CW-complex whose isotropy subgroups have bounded order. Then for any prime p, there is a G-vector bundle E → X of dimension prime to p, such that for each x ∈ X, E|_{x} is p′-free as a G_{x}-representation.

\underline{\underline{Proof}} : Let F be the family of isotropy subgroups in X, and let m be the least common multiple of the |H| for H ∈ F. For each H ∈ F, let α_{H} : H → Σ_{m} be the homomorphism corresponding to any free action of H on {1, …, m}. The α_{H} clearly form an element

A = (α_{H}) ∈ \underleftarrow{lim}_{H∈F} Rep_{Σ_{m}}(H).

Set n = |Σ_{m}/Syl_{p}(Σ_{m})|, let ρ : Σ_{m} → U(n) be the corresponding permutation representation, and (for each H) let V_{H} be the n-dimensional representation defined by ρ∘α_{H}.

By Theorem 2.7, there is k > 0 and a G-bundle E → X, such that the fiber E|_{x} over any point x ∈ X is isomorphic to (V_{G_{x}})^{⊗k}. This bundle has dimension n^{k}, which is prime to p. Furthermore, for each H ∈ F, and each subgroup K ⊆ H of order prime to p, (V_{H})|_{K} is a free ℂ[K]-module by construction, and so the same holds for (V_{H}^{⊗k})|_{K}. In other words, E|_{x} is p′-free as a G_{x}-representation for each x, and so E has all of the required properties. This finishes the proof of Corollary 2.9.

In view of Theorem 2.7, the following question arises. Let X be a G-CW-complex. Given a compatible family {V_{H}} of representations of the isotropy subgroups of X, is there a G-vector bundle E → X such that E|_{x} ≅ V_{G_{x}} as G_{x}-representations for each x ∈ X? Here, “compatible” means that if α(K) ⊆ H, where α is an inner automorphism of G, then (α^{∗}V_{H})|_{K} ≅ V_{K}. This question can also be posed more generally, requiring different representations on different components of fixed point sets.

It is in fact easy to find counterexamples to this question, even in the case where G is finite. Fix a finite group G and a normal subgroup H ◁ G, and set I_{H} = Ker[R(G) ⟶^{res} R(H)]. By a theorem of Jackowski [11, Theorem 5.1 & Example 5.5], the pro-rings {K_{G}(E(G/H)^{(n)})}_{n≥1} and {R(G)/(I_{H})^{n}}_{n≥1} are isomorphic. In particular, for n sufficiently large, the fibers of any G-vector bundle over E(G/H)^{(n)}, considered as Hrepresentations, can always be extended to virtual G-representations. On the other hand, any G/H-invariant H-representation V_{H} defines a “compatible family” of representations of the isotropy subgroups of E(G/H). It is not hard to find examples of G and H where Im[R(G) → R(H)] ⫋ R(H)^{G/H}, and hence of a compatible family which cannot be the fibers of a G-vector bundle over the finite G-CW-complex E(G/H)^{(n)}.

What we really would like to find is an example of an infinite discrete group G, such that E_{FIN} (G) has the homotopy type of a finite G-CW-complex, and for which not every compatible family {V_{H}} of representations of the finite subgroups can be realized as a Gvector bundle over E_{FIN} (G) (not even stably). Presumably such examples exist, but we have so far been unable to find any.

3. Equivariant K-theory for finite proper G-CW-complexes

The main result in this section is that when G is discrete, G-vector bundles define a ℤ/2-graded multiplicative cohomology theory K_{G}^{∗}(−) on the category of finite proper G-CW-complexes. This is summarized in Theorem 3.2 below.

The assumption here that G is discrete is essential, even in the case of finite proper CWcomplexes. So this will be assumed throughout most of the section. The problems arising in the case of positive dimensional Lie groups will be discussed in Section 5 below.

The usual way to define K_{G}(X, A), when G is finite, is as the reduced K-theory of the mapping cone of the inclusion of A in X (cf. [3] or [18]). That approach is not possible here, since the mapping cone of a map of proper G-CW-complexes has a G-fixed point, and hence is not proper if G is not compact. For the same reason, we are unable to use suspensions in this situation to define the groups K_{G}^{−n}(X, A). Instead, we make the following definitions:

Definition 3.1 For any Lie group G and any proper G-CW-complex X, let 𝕂_{G}(X) = 𝕂_{G}^{0}(X) be the Grothendieck group of the monoid of isomorphism classes of G-vector bundles over X. Define 𝕂_{G}^{−n}(X), for all n > 0, by setting

𝕂_{G}^{−n}(X) = Ker[𝕂_{G}(X × S^{n}) ⟶^{incl^{∗}} 𝕂_{G}(X)].

For any proper G-CW-pair (X, A), set

𝕂_{G}^{−n}(X, A) = Ker[𝕂_{G}^{−n}(X ∪_{A} X) ⟶^{i_{2}^{∗}} 𝕂_{G}^{−n}(X)].

When G is discrete and (X, A) is a finite proper G-CW-pair, write

K_{G}(X, A) = 𝕂_{G}(X, A)  and  K_{G}^{−n}(X, A) = 𝕂_{G}^{−n}(X, A).

The pullback construction makes 𝕂_{G}^{−n}(−) and K_{G}^{−n}(−) into contravariant functors on the categories of proper, or finite proper, G-CW-pairs.

Note that we get a natural isomorphism

pr_{X}^{∗} ⊕ i : K_{G}^{0}(X) ⊕ K_{G}^{−n}(X) ⟶^{≅} K_{G}^{0}(X × S^{n}),

where i is the inclusion and pr_{X} the projection.

We can now state the main theorem in this section.

Theorem 3.2 For any discrete group G, the groups K_{G}^{−n}(X, A) extend to a ℤ/2-graded multiplicative equivariant cohomology theory on the category of finite proper G-CW-pairs. In particular, K_{G}^{∗}(−) is a homotopy invariant contravariant functor, satisfies excision, and there is an exact sequence

K_{G}^{0}(X, A) ⟶ K_{G}^{0}(X) ⟶ K_{G}^{0}(A)
δ^{1}↑  _{δ^{0}}↓
K_{G}^{1}(A) ⟵ K_{G}^{1}(X) ⟵ K_{G}^{1}(X, A)

of K_{G}^{∗}(X)-modules for any finite proper G-CW-pair (X, A). For any pushout X = X_{1} ∪_{A} X_{2} where (X_{1}, A) is a finite proper G-CW-pair, all maps in the induced Mayer-Vietoris sequence are K_{G}^{∗}(X)-linear. For finite subgroups H ⊆ G, there are natural isomorphisms K_{G}^{0}(G/H) ≅ R(H), and K_{G}^{1}(G/H) = 0. If G is finite and X is compact, this construction agrees with the classicial definition.

The proof of Theorem 3.2 will occupy most of the rest of the section. We first show some of the properties of 𝕂_{G}^{∗}(−) which hold for any Lie group G and any proper G-CW-complex X, beginning with homotopy invariance.

Lemma 3.3 (Homotopy invariance) Let G be a Lie group. If f_{0}, f_{1} : (X, A) ⟶ (Y, B) are G-homotopic G-maps between proper G-CW-pairs, then

f_{0}^{∗} = f_{1}^{∗} : 𝕂_{G}^{−n}(Y, B) ⟶ 𝕂_{G}^{−n}(X, A)

for all n ≥ 0.

\underline{\underline{Proof}} : When n = 0 and A = B = ∅, this follows immediately from Theorem 1.2. The general case then follows from the definition of 𝕂_{G}^{−n}(X, A).

We next note the following relation between equivariant K-theory for different groups.

Lemma 3.4 (Induction) Let H ⊆ G be an inclusion of Lie groups and let (X, A) be a proper H-CW-pair. Then G ×_{H} (X, A) is a proper G-CW-pair, and there are isomorphisms

i_{H}^{G} : 𝕂_{H}^{−n}(X, A) ⟶^{≅} 𝕂_{G}^{−n} (G ×_{H} (X, A))

(for all n ≥ 0) defined by sending [E] to [G ×_{H} E].

\underline{\underline{Proof}} : This is clear when A = ∅. When A ≠ ∅, it follows since

G×_{H}(X ∪_{A} X) ≅ (G×_{H}X) ∪_{G×_{H}A} (G×_{H}X).

The next two lemmas are also very elementary.

Lemma 3.5 (Free quotients) Let G be a Lie group. Let (X, A) be a proper G-CW-pair for which the normal subgroup H ◁ G acts freely on X. Then the projection pr : X → X/H induces an isomorphism

pr^{∗} : 𝕂_{G/H}^{−n}(X/H, A/H) ⟶^{≅} 𝕂_{G}^{−n}(X, A).

\underline{\underline{Proof}} : This is quickly reduced to the case n = 0 and A = ∅, for which the inverse of the above map is defined by sending [E] ∈ 𝕂_{G}(X) to [E/H] ∈ 𝕂_{G/H}(X/H).

Lemma 3.6 Let G be a Lie group, and let (X, A) be a proper G-CW-pair. Suppose that X = ∐_{i∈I} X_{i}, the disjoint union of open G-invariant subspaces X_{i} (for any index set I), and set A_{i} = A ∩ X_{i}. Then there is a natural isomorphism

𝕂_{G}^{−n}(X, A) ⟶^{≅} ∏_{i∈I} 𝕂_{G}^{−n}(X_{i}, A_{i})

induced by the inclusions of the components.

We now assume, throughout (most of) the rest of the section, that G is a discrete group. When proving excision and constructing the exact sequences for equivariant K-theory, we need to know when a G-vector bundle E_{0} over a G-subspace A ⊆ X can be embedded as a summand of some bundle E over X. Suppose for simplicity that X/G is compact. If G is a finite group (or a compact Lie group), then it is easy to find E, since any G-vector bundle over a compact G-CW-complex is a summand of some product bundle. This is no longer the case when G is not compact, and instead of product bundles we will construct E using the bundles constructed in Corollary 2.8.

Note that the following lemma does not hold when G is a noncompact Lie group of positive dimension, even in the special case where X is finite. In fact, Phillips [15, §9] has shown that in this situation, equivariant K-theory defined via (finite-dimensional) G-vector bundles need not be an equivariant cohomology theory. We will discuss this in more detail in Section 5.

Lemma 3.7 Assume G is discrete, let φ : X → Y be an equivariant map between finite proper G-CW-complexes, and let E′ → X be a G-vector bundle. Then there is a G-vector bundle E → Y such that E′ is a summand of φ^{∗}E.

\underline{\underline{Proof}} : Let m be the maximum dimension of any fiber of E′. By Corollary 2.8, there is a G-vector bundle F → Y such that each fiber F |_{y} is a multiple of the regular G_{y}- representation. After possibly replacing F by some iterated direct sum with itself, we can assume that for each x ∈ X, (φ^{∗}F)|_{x} ≅ F |_{φ(x)} contains at least m copies of the regular representation of G_{x}; and hence that there is a G_{x}-linear injection of E′|_{x} into (φ^{∗}F)|_{x}. This extends to a monomorphism of G-vector bundles from E′|_{Gx} into (φ^{∗}F)|_{Gx}, which by Lemma 1.3 extends to a bundle map f_{x} : E′ → φ^{∗}F covering the identity on X. In particular, f_{x} is a monomorphism over some open G-invariant neighborhood U_{x} of Gx in X.

Since X/G is compact, we can choose x_{1}, …, x_{n} ∈ X such that X is covered by the sets U_{x_{1}}, …, U_{x_{n}}. The sum of the f_{x_{i}} is then a monomorphism f : E′ → φ^{∗}(F^{n}) of bundles covering the identity on X. The image of f is a G-invariant subbundle of φ^{∗}(F^{n}) (cf. [3, Lemma 1.3.1]). And via a Hermitian metric on F, it is seen to be a G-vector bundle summand.

The Mayer-Vietoris exact sequence follows as a consequence of Lemma 3.7.

Lemma 3.8 (Mayer-Vietoris sequence) Assume G is discrete. Let

\begin{matrix}A & ⟶^{i_{1}} & X_{1} \\ ^{i_{2}}↓ & & ^{j_{1}}↓ \\ X_{2} & ⟶^{j_{2}} & X \end{matrix}

be a pushout square of finite proper G-CW-complexes, where i_{1} and j_{2} are inclusions of subcomplexes. Then there is a natural exact sequence, infinite to the left

\begin{matrix}… ⟶^{d^{−n−1}} & K_{G}^{−n}(X) ⟶^{j_{1}^{∗}⊕j_{2}^{∗}} K_{G}^{−n}(X_{1}) ⊕ K_{G}^{−n}(X_{2}) ⟶^{i_{1}^{∗}−i_{2}^{∗}} K_{G}^{−n}(A) ⟶^{d^{−n}} \\ … & ⟶ K_{G}^{−1}(A) ⟶^{d^{−1}} K_{G}^{0}(X) ⟶^{j_{1}^{∗}⊕j_{2}^{∗}} K_{G}^{0}(X_{1}) ⊕ K_{G}^{0}(X_{2}) ⟶^{i_{1}^{∗}−i_{2}^{∗}} K_{G}^{0}(A). \end{matrix} (1)

\underline{\underline{Proof}} : We first show that the sequence

K_{G}(X) ⟶^{j_{1}^{∗}⊕j_{2}^{∗}} K_{G}(X_{1}) ⊕ K_{G}(X_{2}) ⟶^{i_{1}^{∗}−i_{2}^{∗}} K_{G}(A)  (2)

is exact, and hence that sequence (1) is exact at K_{G}^{−n}(X_{1}) ⊕ K_{G}^{−n}(X_{2}) for all n. Clearly the composite in (2) is zero. So fix an element (α_{1}, α_{2}) ∈ Ker(i_{1}^{∗} − i_{2}^{∗}). By Lemma 3.7, we can add an element of the form ([j_{1}^{∗}E′], [j_{2}^{∗}E′]) for some G-vector bundle E′ → X, and arrange that α_{1} = [E_{1}] and α_{2} = [E_{2}] for some pair of G-vector bundles E_{k} → X_{k}. Then i_{1}^{∗}E_{1} and i_{2}^{∗}E_{2} are stably isomorphic, and after adding the restrictions of another bundle over X (Lemma 3.7 again), we can arrange that i_{1}^{∗}E_{1} ≅ i_{2}^{∗}E_{2}. Lemma 1.5 now applies to show that there is a G-vector bundle E over X such that j_{k}^{∗}E ≅ E_{k} (k = 1, 2), and hence that ([E_{1}], [E_{2}]) ∈ Im(j_{1}^{∗} ⊕ j_{2}^{∗}).

Assume now that A is a retract of X_{1}. We claim that in this case,

Ker[K_{G}(X) ⟶^{j_{2}^{∗}} K_{G}(X_{2})] ⟶_{≅}^{j_{1}^{∗}} Ker[K_{G}(X_{1}) ⟶^{i_{1}^{∗}} K_{G}(A)]  (3)

is an isomorphism. It is surjective by the exactness of (2). So fix an element [E] − [E′] ∈ Ker(j_{1}^{∗} ⊕ j_{2}^{∗}). To simplify the notation, we write E|_{X_{1}} = j_{1}^{∗}E, E|_{A} = i_{2}^{∗}j_{2}^{∗}E, etc. (But we are not assuming that j_{1} and i_{2} are injective.) Let p_{1} : X_{1} → A be a retraction, and let p : X → X_{2} be its extension to X. Using Lemma 3.7, we can arrange that E|_{X_{k}} ≅ E′|_{X_{k}} for k = 1, 2. Upon applying Lemma 3.7 to the retraction p : X → X_{2}, we obtain a Gvector bundle F′ → X_{2} such that E′ is a summand of p^{∗}F′. Upon stabilizing again, we can assume that E′ ≅ p^{∗}F′, and hence that F′ ≅ E′|_{X_{2}} and E′|_{X_{1}} ≅ p_{1}^{∗}(F′|_{A}) ≅ p_{1}^{∗}(E′|_{A}). Fix isomorphisms ψ_{k} : E|_{X_{k}} → E′|_{X_{k}} covering Id_{X_{k}}. The automorphism (ψ_{2}|_{A})∘(ψ_{1}|_{A})^{−1} of E′|_{A} pulls back, under p_{1}, to an automorphism φ of E′|_{X_{1}}; and by replacing ψ_{1} by φ∘ψ_{1} we can arrange that ψ_{1}|_{A} = ψ_{2}|_{A}. Then ψ_{1} ∪ ψ_{2} is an isomorphism from E to E′, and this proves (3).

We now return to the general case. For each n ≥ 1,

K_{G}^{−n}(A) = Ker[K_{G}(A × S^{n}) → K_{G}(A)]
≅ Ker[K_{G}(X ∪_{A×•} (A × S^{n})) ⟶^{incl^{∗}} K_{G}(X)]  (by (3))
≅ Ker[K_{G}((X_{1} × D^{n}) ∪_{A×S^{n−1}} (X_{2} × D^{n})) ⟶^{(−,•)^{∗}} K_{G}(X)],  (hty. invar.)

the last step since ((X_{1}×•) ∪ (A×D^{n})) is a strong deformation retract of X_{1} × D^{n}. Define d^{−n} : K_{G}^{−n}(A) → K_{G}^{−n+1}(X) to be the homomorphism which makes the following diagram commute:

0 ⟶ K_{G}^{−n}(A) ⟶ K_{G}((X_{1}×D^{n}) ∪_{A×S^{n−1}} (X_{2}×D^{n})) ⟶^{(−,•)^{∗}} K_{G}(X) ⟶ 0
d^{−n}↓  ^{incl}^{∗}↓  ^{Id}↓
0 ⟶ K_{G}^{−n+1}(X) ⟶  K_{G}(X × S^{n−1})  ⟶^{(−,•)^{∗}} K_{G}(X) ⟶ 0

We have already shown that sequence (1) is exact at K_{G}^{−n}(X_{1}) ⊕ K_{G}^{−n}(X_{2}) for all n. To see its exactness at K_{G}^{−n+1}(X) and K_{G}^{−n}(A) (for any n ≥ 1), apply the exactness of (2) to the following split inclusion of pushout squares:

\begin{matrix}X_{1} ⨿ X_{2} & ⟶ & X_{1} ⨿ X_{2} \\ ↓ & & ↓ \\ X & ⟶ & X \\ ↓ & & ↓ \\ X & ⟶ & X \end{matrix} ⟶^{incl} \begin{matrix}(X_{1} ⨿ X_{2}) × S^{n−1} & ⟶ & (X_{1}×D^{n}) ⨿ (X_{2}×D^{n}) \\ ↓ & & ↓ \\ X × S^{n−1} & ⟶ & (X_{1}×D^{n}) ∪_{A×S^{n−1}} (X_{2}×D^{n}) \\ ↓ & & ↓ \\ X & ⟶ & (X_{1}×S^{n}) ∪_{A×•} (X_{2}×S^{n}). \end{matrix}

The upper pair of squares induces a split surjection of exact sequences whose kernel yields the exactness of (1) at K_{G}^{−n+1}(X). And since

Ker[K_{G}((X_{1}×S^{n}) ∪_{A×•} (X_{2}×S^{n})) ⟶ K_{G}(X)] ≅ Ker[K_{G}((X_{1}×S^{n}) ⨿ (X_{2}×S^{n})) ⟶ K_{G}(X_{1} ⨿ X_{2})] ≅ K_{G}^{−n}(X_{1}) ⊕ K_{G}^{−n}(X_{2})

by (3), the lower pair of squares induces a split surjection of exact sequences whose kernel yields the exactness of (1) at K_{G}^{−n}(A).

Excision, and the long exact sequence for a pair, follow as immediate consequences of the Mayer-Vietoris sequence.

Lemma 3.9 (Excision) Assume G is discrete. Let

φ : (X, A) ⟶ (Y, B)

be a map of finite proper G-CW-pairs, such that Y ≅ B∪_{φ|A}X. Then

φ^{∗} : K_{G}^{−n}(Y, B) ⟶ K_{G}^{−n}(X, A)

is an isomorphism for all n ≥ 0.

\underline{\underline{Proof}} : For each n, the square

X ⟶ Y
↓  ↓
X ∪_{A} X ⟶ Y ∪_{B} Y

is a pushout, and X is a retract of X ∪_{A} X. So its Mayer-Vietoris sequence splits into short exact sequences

0 ⟶ K_{G}^{−n}(Y ∪_{B} Y) ⟶ K_{G}^{−n}(X ∪_{A} X) ⊕ K_{G}^{−n}(Y) ⟶ K_{G}^{−n}(X) ⟶ 0.

And hence K_{G}^{−n}(Y, B) ≅ K_{G}^{−n}(X, A).

Lemma 3.10 (Exactness) Assume G is discrete, and let (X, A) be a finite proper G-CW-pair. Then the following sequence, extending infinitely far to the left, is natural and exact:

… ⟶^{δ^{−n−1}} K_{G}^{−n}(X, A) ⟶^{i^{∗}} K_{G}^{−n}(X) ⟶^{j^{∗}} K_{G}^{−n}(A) ⟶^{δ^{−n}} K_{G}^{−n+1}(X, A) ⟶^{i^{∗}}

… ⟶^{δ^{−1}} K_{G}^{0}(X, A) ⟶^{i^{∗}} K_{G}^{0}(X) ⟶^{j^{∗}} K_{G}^{0}(A).

\underline{\underline{Proof}} : This follows immediately from the Mayer-Vietoris sequence for the square

A ⟶ X
↓  ^{i_{2}}↓
X ⟶^{i_{1}} X ∪_{A} X.

In the nonequivariant case, 𝕂(X) ≅ K(X) for any finite dimensional CW-complex X: since any map X → BU factors through some BU(n). The following example shows that this is no longer true in the equivariant case, even for actions of finite groups: the Mayer-Vietoris sequence need not be exact in this situation.

Example 3.11 Fix any finite group G ≠ 1. Define X = (G × ℝ)/ ∼, where (g, n) ∼ (1, n) for any g ∈ G and any n ∈ ℤ. For each n ∈ ℤ, set A_{n} = (G × [n − \frac{1}{2}, n + \frac{1}{2}])/(G × \{n\}). Set X_{1} = ∐_{n∈ℤ} A_{2n}, X_{2} = ∐_{n∈ℤ} A_{2n+1}, and X_{0} = X_{1} ∩ X_{2}. Let i_{k} : X_{k} → X and j_{k} : X_{0} → X_{k} (k = 1, 2) denote the inclusions. Then the sequence

𝕂_{G}(X) ⟶^{(i_{1}^{∗},i_{2}^{∗})} 𝕂_{G}(X_{1}) ⊕ 𝕂_{G}(X_{2}) ⟶^{j_{1}^{∗}−j_{2}^{∗}} 𝕂_{G}(X_{0})

is not exact.

\underline{\underline{Proof}} : For each n, K_{G}(A_{n}) ≅ R(G) (each A_{n} is equivariantly contractible); and the kernel of the restriction map K_{G}(A_{n}) → K_{G}(A_{n}∩X_{0}) is (under this identification) the augmentation ideal IR(G). Choose representations V_{n}, W_{n} (all n ∈ ℤ) such that dim(V_{n}) = dim(W_{n}), Hom(V_{n}, W_{n}) = 0, and {dim(V_{n})} is unbounded. Then the element

({[V_{2n}×A_{2n}] − [W_{2n}×A_{2n}]}_{n∈ℤ}, {[V_{2n+1}×A_{2n+1}] − [W_{2n+1}×A_{2n+1}]}_{n∈ℤ})

lies in Ker(j_{1}^{∗} − j_{2}^{∗}), but not in Im(i_{1}^{∗}, i_{2}^{∗}).

We now consider products on K_{G}^{∗}(X) and on K_{G}^{∗}(X, A). For any proper G-CW-complex X, tensor product of G-vector bundles makes 𝕂_{G}(X) into a commutative ring, and all induced maps f^{∗} : 𝕂_{G}(Y) → 𝕂_{G}(X) are ring homomorphisms. For each n, m ≥ 0,

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

where the first isomorphism follows from the usual Mayer-Vietoris sequences. Hence

K_{G}(X × S^{n}) ⊗ K_{G}(X × S^{m}) ⟶^{mult ∘ (p_{1}^{∗}⊗p_{2}^{∗})} K_{G}(X × S^{n} × S^{m})

restricts to a homomorphism

K_{G}^{−n}(X) ⊗ K_{G}^{−m}(X) ⟶ K_{G}^{−m−n}(X).

By applying the above definition with n = 0 or m = 0, the multiplicative identity for K_{G}(X) is seen to be an identity for K_{G}^{∗}(X). Associativity of the graded product is clear, and graded commutativity follows upon showing (using a Mayer-Vietoris sequence) that composition with a degree −1 map S^{n} → S^{n} induces multiplication by −1 on K^{−n}(X). This product thus makes K_{G}^{∗}(X) into a ring. Clearly, f^{∗} : K_{G}^{∗}(Y) → K_{G}^{∗}(X) is a ring homomorphism for any G-map f : X → Y.

For a finite proper G-CW-pair (X, A), K_{G}^{∗}(X ∪_{A} X) → K_{G}^{∗}(X) is a split surjection and ring homomorphism (and split by a ring homomorphism), and so its kernel is a K_{G}^{∗}(X)- module. For any X = X_{1} ∪_{A} X_{2}, where (X_{1}, A) is a finite proper G-CW-pair, the boundary map in the corresponding Mayer-Vietoris sequence is K_{G}^{∗}(X)-linear, since it is defined via a certain map between spaces which commutes with their (split) projections onto X. And hence the boundary maps in the long exact sequence for a pair (X, A) are K_{G}^{∗}(X)-linear, since they are defined to be the boundary maps of a certain Mayer-Vietoris sequence all of whose spaces map to X.

It remains to prove Bott periodicity in this situation. Recall that K̃(S^{2}) =^{def} Ker[K(S^{2}) → K(pt)] ≅ ℤ, and is generated by the Bott element B ∈ K̃(S^{2}): the element [S^{2} × ℂ] − [H] ∈ K̃(S^{2}), where H is the canonical complex line bundle over S^{2} = ℂℙ^{1}. For any finite proper G-CW-complex X, there is an obvious pairing

K_{G}^{−n}(X) ⊗ K̃(S^{2}) ⟶^{⊗} Ker[K_{G}^{−n}(X × S^{2}) → K_{G}^{−n}(X × pt)] ≅ K_{G}^{−n−2}(X),

induced by (external) tensor product of bundles. Evaluation at the Bott element now defines a homomorphism

b = b(X) : K_{G}^{−n}(X) ⟶ K_{G}^{−n−2}(X),

which by construction is natural in X. And this is then extends to a homomorphism

b = b^{∗}(X, A) : K_{G}^{−n}(X, A) ⟶ K_{G}^{−n−2}(X, A)

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

Theorem 3.12 (Equivariant Bott periodicity) Assume G is discrete. Then the Bott homomorphism

b = b(X, A) : K_{G}^{−n}(X, A) ⟶ K^{−n−2}(X, A)

is an isomorphism for any discrete group G and any finite proper G-CW-pair (X, A) (and all n ≥ 0).

\underline{\underline{Proof}} : Assume first that X = Y ∪_{φ} (G/H × D^{m}), where H ⊆ G is finite and φ : G/H × S^{m−1} → Y is a G-map; and assume inductively that b(Y) is an isomorphism. Since

K_{G}^{−n}(G/H × S^{m−1}) ≅ K_{H}^{−n}(S^{m−1}) and K_{G}^{−n}(G/H × D^{m}) ≅ K_{H}^{−n}(D^{m}),

the Bott homomorphisms b(G/H ×S^{m−1}) and b(G/H ×D^{m}) are isomorphisms by the equivariant Bott periodicity theorem for actions of finite groups [4, Theorem 4.3]. The Bott map is natural, and compatible with the various boundary operators in the Mayer-Vietoris sequence (in non-positive degrees) for Y, X, G/H × S^{m−1}, and G/H × D^{m}; and so b(X) is an isomorphism by the 5-lemma. The proof that b(X, A) is an isomorphism for an arbitrary proper finite G-CW-pair (X, A) now follows immediately from the definitions of the relative groups.

We are now ready to prove the main theorem. Define, for all n ∈ ℤ,

K_{G}^{n}(X, A) = \begin{cases}K_{G}^{0}(X, A) & if n is even \\ K_{G}^{−1}(X, A) & if n is odd. \end{cases}

For any finite proper G-CW-pair (X, A), define the boundary operator δ^{n} : K_{G}^{n}(A) → K_{G}^{n+1}(X, A) to be δ : K_{G}^{−1}(A) → K_{G}^{0}(X, A) if n is odd, and to be the composite

K_{G}^{0}(A) ⟶_{≅}^{b} K_{G}^{−2}(A) ⟶^{δ^{−2}} K_{G}^{−1}(X, A)

if n is even.

\underline{\underline{Proof of Theorem 3.2:}} We have already proven excision (Lemma 3.9) and homotopy invariance (Lemma 3.3). The long exact sequence of a pair follows from that in negative degrees (Lemma 3.10), and the fact that the Bott map is natural and commutes with the boundary operators δ^{−n}. The same holds for the product structure which comes from that on K_{G}^{−n}(X, A). For any X = X_{1} ∪_{A} X_{2}, the boundary map in the corresponding Mayer-Vietoris sequence is K_{G}^{∗}(X)-linear, since it is defined via a certain map between spaces which commutes with their (split) projections onto X. And hence the boundary maps in the long exact sequence for a pair (X, A) are K_{G}^{∗}(X)-linear, since they are defined to be the boundary maps of a certain Mayer-Vietoris sequence all of whose spaces map to X. The other claims are immediate.

We next consider the Thom isomorphism theorem for proper actions of infinite discrete groups. This first requires a slight detour. The Thom class of a G-vector bundle E is an element in K_{G}(D(E), S(E)), where S(E) ⊆ D(E) denote the unit sphere and disk bundles in E (with respect to some G-invariant metric). This is most easily defined in terms of a chain complex of vector bundles over D(E), and we must first explain how such a chain complex determines an element in K-theory.

A G-vector bundle chain complex over a proper G-CW-pair (X, A) is a finite-dimensional chain complex (C_{∗}, c_{∗}) of G-vector bundles over X whose restriction to A is acyclic. In other words, for some N > 0,

0 ⟶ C_{N} ⟶^{c_{N}} C_{N−1} ⟶^{c_{N−1}} ⋯ ⟶^{c_{3}} C_{2} ⟶^{c_{2}} C_{1} ⟶^{c_{1}} C_{0} ⟶ 0

is a sequence of G-vector bundles and bundle maps, such that c_{n−1}∘c_{n} = 0 for all n, and such that restriction to the fibers over any x ∈ A is exact. When G is compact, the monoid of Gvector bundle chain complexes over (X, A), modulo an appropriate submonoid, is isomorphic to K_{G}(X, A) by a theorem of Segal [18, Proposition 3.1]. In a later paper, we will prove this in our present setting, for proper actions of infinite discrete groups. But for now, all we need to know is that any such complex defines an element of K_{G}(X, A) in a natural (functorial) way.

Fix a G-vector bundle chain complex (C_{∗}, c_{∗}) over (X, A). For each n, set C′_{n} = Im(c_{n+1}|_{A}) = Ker(c_{n}|_{A}). Each C′_{n} ⊆ C_{n} is a G-invariant subbundle: this follows by induction on n, since the kernel of the surjection C_{n}|_{A} −↠^{c_{n}} C′_{n−1} is a subbundle (cf. [12, Theorems 5.13 & 6.3]). Let C′′_{n} ⊆ C_{n}|_{A} be any G-invariant complementary bundle to C′_{n}; defined, for example, using a G-invariant Hermitian metric on C_{n}. Thus, for each n, c_{n} sends C′′_{n} isomorphically to C′_{n−1}. Set C_{odd} = ⊕_{n∈ℤ}C_{2n+1} and C_{ev} = ⊕_{n∈ℤ}C_{2n}, let f_{C} : C_{odd}|_{A} → C_{ev}|_{A} be the sum of the isomorphisms

C′_{2n+1} ⟶_{≅}^{(c_{2n+2})^{−1}} C′′_{2n+2} and C′′_{2n+1} ⟶_{≅}^{c_{2n+1}} C′_{2n}.

Finally, define

[C_{∗}, c_{∗}] = [C_{odd} ∪_{f_{C}} C_{ev}] − [C_{ev} ∪_{Id} C_{ev}] ∈ Ker[K_{G}(X ∪_{A} X) ⟶^{i_{2}^{∗}} K_{G}(X)] = K_{G}(X, A).

This is independent of the choice of C′′_{n}, since there is an affine structure on the space of all complementary bundles (and hence a homotopy between any two of them).

Now let p : E → X be an n-dimensional G-vector bundle over a proper G-CW-complex X, and set p_{D} = p|D(E). Consider the cochain complex of G-vector bundles (Λ^{k}p_{D}^{∗}E, δ) over (D(E), S(E)), which over any v ∈ D(E) takes the form

0 ⟶ Λ^{0}E_{p(v)} ⟶^{∧v} Λ^{1}E_{p(v)} ⟶^{∧v} Λ^{2}E_{p(v)} ⟶^{∧v} … ⟶^{∧v} Λ^{n}E_{p(v)} ⟶ 0.

Here, ∧v denotes the exterior product with the element v ∈ E_{p(v)}. One easily checks that this sequence is exact for all v not in the zero section of E.

There is a technical problem here: D(E) and S(E) do not have natural structures as G-CW-complexes, and so K_{G}^{∗}(D(E), S(E)) is not defined in Definition 3.1. It is not difficult, however, to modify the definitions (and the proof of Theorem 3.2) to include this case: either by showing that (D(E), S(E)) has the G-homotopy type of a finite proper G-CW-pair, or via a more general definition of equivariant cellular complexes, or by constructing K_{G}^{∗}(−) as an equivariant cohomology theory for all proper G-spaces with compact quotient. This last approach will be taken by the authors in a later, more technical, paper. For now, we just assume that equivariant K-theory has been defined, in some way or other, for disk and sphere bundles of G-vector bundles over finite proper G-CW-complexes.

Definition 3.13 For any G-vector bundle E over X, the Thom class of E is the element

λ_{E} ∈ K_{G}^{0}(D(E), S(E)),

defined to be the class of the cochain complex (Λ^{∗}(p_{D}^{∗}E), δ) over (D(E), S(E)) as defined above. The Thom homomorphism is the composite

T_{E} : K_{G}^{∗}(X) ⟶_{≅}^{p_{D}^{∗}} K_{G}^{∗}(D(E)) ⟶^{·λ_{E}} K_{G}^{∗}(D(E), S(E)),

where the second map is multiplication with the Thom class.

Theorem 3.14 (Thom isomorphism theorem) Assume G is discrete. Then for any Gvector bundle p : E → X over a finite proper G-CW-complex X, the Thom homomorphism

T_{E} : K_{G}^{∗}(X) ⟶^{≅} K_{G}^{∗}(D(E), S(E))

is an isomorphism.

\underline{\underline{Proof}} : Assume first that X = G/H × Y, where Y = S^{n−1} or D^{n}, and where E|_{Y} ≅ V × Y for some H-representation V. Then

K_{G}^{n}(X) ≅ K_{G}^{n}(G/H × Y) ≅ K_{H}^{n}(Y);

and

K_{G}^{n}(D(E), S(E)) ≅ K_{G}^{n}(G×_{H}(D(V) × Y), G×_{H}(S(V) × Y))
≅ K_{H}^{n}(D(V) × Y, S(V) × Y )

(the last step by Lemma 3.4). So in this case, T_{E} is an isomorphism by the Thom isomorphism theorem for actions of finite groups [4, Theorem 4.3].

Now assume that X = Y ∪_{φ} (G/H × D^{n}), where H is finite, φ : G/H × S^{n−1} → Y is a G-map, and T_{E|_{Y}} is an isomorphism. There is a relative Mayer-Vietoris sequence involving the groups K_{G}^{∗}(D(E|_{A}), S(E|_{A})) for A = X, Y, G/H×D^{n}, and G/H×S^{n−1}: this follows immediately from the usual Mayer-Vietoris sequence and our definition of the relative groups. Since all maps in both Mayer-Vietoris sequences — for X and for (D(E), S(E)) — are K_{G}^{∗}(X)-linear (Lemma 3.8), they commute with the Thom homomorphisms. So T_{E} is an isomorphism by the 5-lemma.

So far, we have worked entirely with complex K-theory. To finish the section, we note that the results of this section all hold in the real case as well. Define KO_{G}(X), for any discrete G and any finite proper G-CW-complex X, to be the Grothendieck group of real G-vector bundles over X, and extend this to a functor KO_{G}^{−n}(X, A) as in Definition 3.1. The key to proving the exactness properties of K_{G} was Lemma 3.7 (given φ : X → Y, any bundle over X is contained in the pullback of a bundle over Y); and this automatically holds in the real case using the forgetful and induction functors between complex and real G-vector bundles. Bott periodicity still holds, but with period eight: just as in the complex case, this reduces to the Bott periodicity theorem for KO_{G}(−) when G is a finite group, which was shown in [4, Theorem 6.1]. We thus get:

Theorem 3.15 For any discrete group G, the groups KO_{G}^{−n}(X, A) extend to a multiplicative equivariant cohomology theory on the category of finite proper G-CW-pairs. In particular, KO_{G}^{∗}(−) is a homotopy invariant contravariant functor, satisfies excision, and there are exact sequences

⋯ ⟶ KO_{G}^{n}(X, A) ⟶ KO_{G}^{n}(X) ⟶ KO_{G}^{n}(A) ⟶ KO_{G}^{n+1}(X, A) ⟶ · · · .

There are natural Bott periodicity isomorphisms KO_{G}^{n}(X) ⟶_{≅}^{b(X)} KO_{G}^{n−8}(X). For any

finite subgroup H ⊆ G, there are natural isomorphisms KO_{G}^{∗}(G/H) ≅ KO_{H}^{∗}(pt), and in particular KO_{G}^{0}(G/H) ≅ RO(H).

4. The completion theorem

Given a discrete group G, we prove here a completion theorem, analogous to that of Atiyah and Segal [6] for actions of compact Lie groups. We show that for any finite proper G-CW-complex X, K^{∗}(EG×_{G}X) is the completion of K_{G}^{∗}(X) with respect to a certain ideal. When the universal space E_{FIN} (G) (see Definition 2.1) for the family FIN of finite subgroups of G has the G-homotopy type of a finite-dimensional G-CW-complex, and there is an upper bound on the order of finite subgroups of G, then the ideal in question is that generated by the augmentation ideal of K_{G}(E_{FIN} (G)).

In fact, as in the theorem of Atiyah and Segal, we prove an isomorphism not just of inverse limits, but also of inverse systems. This has the advantage that it gives a stronger result (Theorem 4.3), and it easily implies that {(EG×_{G}X)^{(n)}} satisfies the Mittag-Leffler condition and has vanishing \underleftarrow{lim}^{1}. It is also needed in the proof of Theorem 4.3, which is carried out by induction over the number of cells using a version of the five-lemma.

We first fix our notation for handling pro-groups (by which we always mean pro-abelian groups). For the definitions in full generality, see [6, §2]. For simplicity, all pro-groups dealt with here will be indexed by the nonnegative (or positive) integers. We write (G, α) for the inverse system

⟶^{α_{3}} G_{3} ⟶^{α_{2}} G_{2} ⟶^{α_{1}} G_{1} ⟶^{α_{0}} G_{0},

and also write α_{i}^{j} = α_{i}∘ ⋯ ∘α_{j−1} : G_{j} → G_{i} for j ≥ i (α_{i}^{i} = Id_{G_{i}}). For the purposes here, it will suffice (and greatly simplify the notation) to work with “strict” pro-homomorphisms: homomorphisms f : (G, α) → (H, β) such that f_{i} : G_{i} → H_{i}, and β_{i−1}∘f_{i} = f_{i−1}∘α_{i−1}, for all i. Kernels and cokernels of strict homomorphisms are defined in the obvious way.

A pro-group will be called pro-trivial if for each i ≥ 0, there is some j ≥ i such that α_{i}^{j} = 0. A strict homomorphism f : (G, α) → (H, β) is a isomorphism of pro-groups if and only if Ker(f) and Coker(f) are both pro-trivial, or, equivalently, for each i ≥ 0 there is some j ≥ i such that Im(β_{i}^{j}) ⊆ Im(f_{i}) and Ker(f_{j}) ⊆ Ker(α_{i}^{j}). A sequence of strict homomorphisms

(G, α) ⟶^{f} (G′, α′) ⟶^{f′} (G′′, α′′)

will be called exact if f′_{i}∘f_{i} = 0 for each i, and if the pro-group {Ker(f′_{i})/ Im(f_{i})}_{i≥0} is pro-trivial. The following result will be needed.

Lemma 4.1 Fix any commutative noetherian ring A, and any ideal I ⊆ A. Then for any exact sequence M′ → M → M′′ of finitely generated A-modules, the sequence

{M′/I^{n}M′} ⟶ {M/I^{n}M} ⟶ {M′′/I^{n}M′′}

of pro-groups (pro-A-modules) is exact.

\underline{\underline{Proof}} : It suffices to prove this when the sequence is short exact. Regard M′ as a submodule of M, and consider the exact sequence

0 ⟶ \{\frac{(I^{n}M)∩M′}{I^{n}M′} \} ⟶ \{M′/I^{n}M′\} ⟶ \{M/I^{n}M\} ⟶ \{M′′/I^{n}M′′\} ⟶ 0.

By [5, Theorem 10.11 on page 107], the filtrations {(I^{n}M)∩M′} and {I^{n}M′} of M′ have “bounded difference”: i.e., there exists k > 0 such that (I^{n+k}M)∩M′ ⊆ I^{n}M′ for all n. The first term in the above exact sequence is thus pro-trivial, and so the remaining terms define a short exact sequence of pro-groups.

To avoid ambiguity, for any proper G-CW-complex X, the augmentation ideal I𝕂_{G}(X) ⊆ 𝕂_{G}(X) is defined to be the set of elements represented by virtual G-vector bundles of dimension zero on all connected components. In other words,

I𝕂_{G}(X) = Ker[𝕂_{G}(X) ⟶^{dim} ∏_{π_{0}(X)/G} ℤ],

where the ring homomorphism dim sends [E] to the map from π_{0}(X)/G ⟶ ℤ which assigns to the G-orbit through a path component C ⊂ X the dimension of the fiber E_{x} for any point x ∈ C. Given a G-map f : X ⟶ Y, the ring homomorphism f^{∗} : 𝕂_{G}(Y) ⟶ 𝕂_{G}(X) induces a map f^{∗} : I𝕂_{G}(Y) ⟶ I𝕂_{G}(X).

Lemma 4.2 Let X be a CW-complex of dimension n−1. Then any n-fold product of elements in IK^{∗}(X) is zero.

\underline{\underline{Proof}} : Write X = Y ∪ A, where Y and A are closed subsets, Y contains X^{(n−2)} as a deformation retract, and A is a disjoint union of (n−1)-disks. Fix elements v_{1}, v_{2}, …, v_{n} ∈ IK^{∗}(X). We can assume by induction that v_{1} ⋯ v_{n−1} vanishes after restricting to Y, and hence that it is the image of an element u ∈ K^{∗}(X, Y). Also, v_{n} clearly vanishes after restricting to A, and hence is the image of an element v ∈ K^{∗}(X, A). Their product is thus the image in K^{∗}(X) of the element uv ∈ K^{∗}(X, Y ∪ A) = 0 (cf. [12, II.5.8]), and so v_{1} ⋯ v_{n} = 0.

Fix any finite proper G-CW-complex X, and any map f : X → L to a finite dimensional proper G-CW-complex L whose isotropy subgroups have bounded order. Regard K_{G}^{∗}(X) as a module over the ring K_{G}(L). Set I = IK_{G}(L). For any n ≥ 0, the composite

I^{n}·K_{G}^{∗}(X) ⊆ K_{G}^{∗}(X) ⟶^{proj^{∗}} 𝕂_{G}^{∗}(EG×X) ⟶ 𝕂^{∗}(EG×_{G}X)

⟶^{res} 𝕂^{∗}((EG×_{G}X)^{(n−1)}) ⟶^{≅} K^{∗}((EG×_{G}X)^{(n−1)})

is zero, since the image is contained in IK^{∗}((EG×_{G}X)^{(n−1)})^{n} = 0 which vanishes by Lemma 4.2. (Recall that 𝕂^{∗}(Y) ≅ K^{∗}(Y) for finite dimensional Y, since any map Y → BU factors through some BU(n).) This thus defines a homomorphism of pro-groups

λ^{X,f} : {K_{G}^{∗}(X)/I^{n}·K_{G}^{∗}(X)}_{n≥1} ⟶ {K^{∗}((EG×_{G}X)^{(n−1)})}_{n≥1}.

As usual, (−)_{I} denotes completion with respect to an ideal I.

Theorem 4.3 (Completion theorem) Let G be a discrete group. Fix a finite proper G-CW-complex X, a finite dimensional proper G-CW-complex L whose isotropy subgroups have bounded order, and a G-map f : X → L. Regard K_{G}^{∗}(X) as a module over 𝕂_{G}(L), and let I = I𝕂_{G}(L) be the augmentation ideal. Then

λ^{X,f} : {K_{G}^{∗}(X)/I^{n}·K_{G}^{∗}(X)}_{n≥1} ⟶ {K^{∗}((EG×_{G}X)^{(n−1)})}_{n≥1}.

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

\underleftarrow{lim}^{1}K^{∗}((EG×_{G}X)^{(n)}) = 0,

and λ^{X,f} induces an isomorphism

K_{G}^{∗}(X)_{I} ⟶^{≅} K^{∗}(EG×_{G}X) ≅ \underleftarrow{lim}K^{∗}((EG×_{G}X)^{(n)}).

\underline{\underline{Proof}} : Assume that λ^{X,f} is an isomorphism. Then the system \{K^{∗}((EG×_{G}X)^{(n)})\}_{n≥1} satisfies the Mittag-Leffler condition because {K_{G}^{∗}(X)/I^{n}} does. In particular, \underleftarrow{lim}^{1}K^{∗}((EG×_{G}X)^{(n)}) = 0, and so K^{∗}(EG×_{G}X) ≅ \underleftarrow{lim}K^{∗}((EG×_{G}X)^{(n)}) (cf. [6, Proposition 4.1]).

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

Step 1 Assume first that X = G/H, for some finite subgroup H ⊆ G. Then the following diagram commutes

𝕂_{G}(L) ⟶^{f^{∗}} K_{G}^{∗}(G/H) ⟶^{pr_{1}} K^{∗}(EG×_{G}G/H)
ev_{f(eH)}↓  ^{ev}^{eH} ↓≅  ^{q}↓≅
R(H) ⟶_{≅} K_{H}^{∗}(∗) ⟶^{pr_{2}}  K^{∗}(BH),

where ev_{f(eH)} sends the class of a G-vector bundle E → L to the class of the fiber E|_{f(eH)} considered as an H-representation and the other maps are the obvious ones. Also, pr_{2} induces an isomorphism of pro-groups

{K_{H}^{∗}(∗)/IR(H)^{n}·K_{H}^{∗}(∗)}_{n≥1} ⟶ {K^{∗}((BH)^{(n−1)})}_{n≥1}

by the theorem of Atiyah and Segal [6] (where IR(H) denotes the augmentation ideal of R(H)). We want to show that pr_{1} induces an isomorphism of pro-groups

{K_{G}^{∗}(G/H)/I^{n}·K_{G}^{∗}(G/H)}_{n≥1} ⟶ {K^{∗}((EG ×_{G} G/H)^{(n−1)})}_{n≥1}.

So we must show that for some k, IR(H)^{k} ⊆ I′ =^{def} ev_{f(eH)}(I).

This means showing that the ideal IR(H)/I′ is nilpotent; or equivalently (since R(H) is noetherian) that it is contained in all prime ideals of R(H)/I′ (cf. [5, Proposition 1.8]). In other words, we must show that every prime ideal of R(H) which contains I′ also contains IR(H). Fix any prime ideal 𝔓 ⊆ R(H) which does not contain IR(H). Set ζ = exp(2πi/|H|), and A = ℤ[ζ]. By a result of Atiyah [2, 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}.

Also, s ≠ 1 since 𝔓 ⊉ IR(H). Set p = char(A/𝔭) (possibly p = 0). By [2, Lemma 6.3], we can assume that s has order prime to p.

By Corollary 2.9 (or Corollary 2.8 if p = 0), there is a G-vector bundle E → L such that p∤ dim(E), and such that (E|_{x})|_{⟨s⟩} is a multiple of the regular representation of ⟨s⟩. In particular, χ_{E|x}(s) = 0. Set k = dim(E), and v = [ℂ^{k}] − [E|_{x}] ∈ R(H). Then ([ℂ^{k}×L] − [E]) ∈ I, so v ∈ I′. Also, χ_{v}(s) = k ∉ 𝔭, so v ∉ 𝔓, and thus 𝔓 ⊉ I′.

Step 2 We now prove the theorem by induction over the dimension of X and the number of cells in a given dimension. It holds when dim(X) = 0 by Step 1. So assume that dim(X) = m > 0. Write X = Y ∪_{φ} (G/H×D^{m}), for some attaching map φ : G/H×S^{m−1} → Y, where λ^{Y,L} is an isomorphism. Consider the Mayer-Vietoris sequence of Lemma 3.8:

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

for X as a pushout of Y and G/H×D^{m} ≃ G/H over G/H×S^{m−1}. All terms in this sequence are K_{G}(X)-modules and all homomorphisms K_{G}(X)-linear, and the 𝕂_{G}(L)-module structure on each term is induced from the K_{G}(X)-module structure. So if we let I′ ⊆ K_{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, K_{G}(X) is noetherian (in fact, a finitely generated abelian group), and so the Mayer-Vietoris sequence above induces an exact sequence of pro-groups

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

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

by Lemma 4.1. (We write here M/I^{n} for M/I^{n}M for short.) Since the obvious strict map of progroups

{K^{∗}((EG×_{G}X)^{(n−1)})}_{n≥1} ⟶ {K^{∗}(EG^{(n−1)}×_{G}X)}_{n≥1}

is an isomorphism of progroups, the various long exact Mayer-Vietoris sequences of the pushouts EG^{(n−1)} ×_{G} X = EG^{(n−1)} ×_{G} Y ∪_{Id ×_{G}φ} (EG^{(n−1)} ×_{G} (G/H × D^{m})) yield a long exact sequence of progroups

⟶ {K^{∗}((EG ×_{G} X)^{(n−1)})}_{n≥1} ⟶

{K^{∗}((EG ×_{G} Y)^{(n−1)}) ⊕ K^{∗}((EG ×_{G} (G/H×D^{m}))^{(n−1)})}_{n≥1} ⟶

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

The 5-lemma for pro-groups (whose proof is essentially the same as that of the usual 5-lemma), together with the induction hypothesis applied to Y and G/H × S^{m−1}, and Step 1 applied to G/H × D^{m} ≃_{G} G/H, now proves that λ^{X,f} is an isomorphism of pro-groups.

As one immediate consequence of Theorem 4.3, we get:

Theorem 4.4 Let E_{FIN} (G) be the universal space for the family FIN of finite subgroups of G (introduced in Definition 2.1). Set I = IK_{G}(E_{FIN} (G)).

(a) If E_{FIN} (G) has the G-homotopy type of a finite dimensional G-CW-complex and there is an upper bound on the orders of the finite order subgroups of G, then for any finite proper G-CW-complex X,

K^{∗}(EG×_{G}X) ≅ K_{G}^{∗}(X)_{I}.

(b) If E_{FIN} (G) has the G-homotopy type of a finite G-CW-complex, then

K^{∗}(BG) ≅ K_{G}^{∗}(E_{FIN} (G))_{I}.

Note that when G is finite, E_{FIN} (G) ≃ ∗ and K_{G}(E_{FIN} (G)) ≅ R(G). So in this case, Theorem 4.4 is exactly the theorem of Atiyah and Segal. If G is torsion free, then any proper G-action is free, and so Theorems 4.3 and 4.4 follow immediately from Lemma 4.2.

Notice that the formulation in Theorem 4.4 doesn’t apply to all discrete groups G, but only to those with bounded torsion and for which E_{FIN} (G) is finite dimensional. One of the interesting features of Theorem 4.3 is that one doesn’t complete with respect to one single canonical ideal of K_{G}(X), but rather an ideal which depends on the choice of another space L . Thus, different choices of ideals yield the same result. In an attempt to give an intrinsic choice of ideal, we give the following third formulation of the completion theorem.

Theorem 4.5 Let X be any finite proper G-complex. Define

S = {x ∈ IK_{G}(X) | res_{X^{H}}(x) ∈ Im[R(H) → K_{H}(X^{H})], all finite H ⊆ G}.

Here, R(H) → K_{H}(X^{H}) sends a representation to the product bundle. Let I be the ideal generated by S. Then

K^{∗}(EG×_{G}X) ≅ K_{G}^{∗}(X)_{I}.

\underline{\underline{Proof}} : Let F be the family of subgroups of isotropy subgroups of X, and let L be the (dim(X) + 1)-skeleton of E_{F}(G). Choose any G-map f : X → L (unique up to homotopy by Lemma 2.2(b)). Let J ⊆ K_{G}(X) be the ideal generated by f^{∗}(I𝕂_{G}(L)). Then J ⊆ I ⊆ IK_{G}(X). So

{K^{∗}((EG×_{G}X)^{(n)})}_{n≥1} ≅ {K_{G}^{∗}(X)/J^{n}·K_{G}^{∗}(X)}_{n≥1} ≅ {K_{G}^{∗}(X)/IK_{G}(X)^{n}·K_{G}^{∗}(X)}_{n≥1},

as inverse systems by Theorem 4.3, and so they are all isomorphic to the inverse system

{K_{G}^{∗}(X)/I^{n}·K_{G}^{∗}(X)}_{n≥1}.

The result now follows upon taking inverse limits.

The completion theorem, in the above forms, also holds for KO_{G}(X), as described in Theorem 3.15. This can be proven in the same way as Theorem 4.3 above, but the classifying spaces for real G-vector bundles do not have simply connected fixed point sets, and hence a more complicated form of the obstruction theory used in Section 2 is needed. Instead of including those details here, we will prove this result in a different way in a later paper, using an equivariant version of the Chern character.

5. Proper actions of Lie groups

Recall (Definition 3.1) that we write 𝕂_{G}^{−n}(−) to denote the graded functor defined via G-vector bundles, also when G is a positive dimensional Lie group. Certain properties of 𝕂_{G}^{−n}(−), such as homotopy invariance (Lemma 3.3), were shown to hold in this generality. It was our proof of excision, and of the long exact sequence for a proper G-CW-pair, which required the assumption that G is discrete. In order to help explain exactly what goes wrong for Lie groups, we now exhibit an explicit example of a group G for which 𝕂_{G}^{−n} is not a cohomology theory, and for which the completion theorem fails: even after replacing 𝕂_{G}^{−n} by the “correct” equivariant K-theory.

Phillips [15] has constructed an equivariant cohomology theory K_{G}^{∗}(−), for any second countable locally compact group G, on the category of proper locally compact G-spaces. This is done using infinite dimensional G-vector bundles with Hilbert space fibers. When G is a Lie group (or in any situation where G-vector bundles are defined), there is an obvious natural transformation

φ_{G}(X) : 𝕂_{G}^{∗}(X) ⟶ K_{G}^{∗}(X)

for proper G-CW-complexes. Phillips [15, Example 9.11] also constructs G and X for which not all elements of K_{G}^{0}(X) are represented by (finite-dimensional) G-vector bundles over X; i.e., for which φ_{G}(X) is not surjective. In these terms, what we have shown in Section 3 is that φ_{G}(X) is an isomorphism whenever G is discrete and X is a finite G-CW-complex. (Since it is an isomorphism for orbits: 𝕂_{G}(G/H) ≅ R(H) ≅ K_{G}(G/H) for any finite H ⊆ G.) In a later, more technical paper, we will extend this result (still for discrete G) to arbitrary proper G-spaces with compact orbit space.

Phillips has also shown [16, Theorems 3.3 & 5.3] that φ_{G}(−) is an isomorphism whenever the space of connected components of G is compact, and that the completion theorem holds for K_{G}(−) whenever G is a Lie group with finitely many connected components. The key to showing this is a theorem of Abels [1], which says that any such group G contains a maximal compact subgroup K, such that any proper G-space X with paracompact orbit space maps to the quotient G/K. This means that X ≅ G×_{K}Y for some K-space Y, and hence that 𝕂_{G}(X) ≅ 𝕂_{K}(Y) ≅ K_{K}(Y) ≅ K_{G}(X) if X/G is compact.

In order to explain from our point of view what goes wrong for nondiscrete groups, we now construct a Lie group G, which is somewhat simpler than that used by Phillips, together with examples to show that 𝕂_{G}^{∗} satisfies neither excision nor exactness. In particular, 𝕂_{G}^{∗} is not a cohomology theory, and so φ_{G}(−) is not in general an isomorphism. We also show that the completion theorem fails for K_{G}(−).

Set T = S^{1} ×S^{1}, the 2-torus, and let α ∈ Aut(T) be the automorphism α(x, y) = (x, xy). Write G = T ⋊^{α} ℤ: the semidirect product determined by α. We also need to consider the subgroup

K = 1 × S^{1} = {g^{−1}·α(g) | g ∈ G}

(note that G/K ≅ (T/K) × ℤ).

Lemma 5.1 Let X be any proper connected (G/T)-CW-complex. Then for any G-vector bundle E → X, K acts via the identity on E. In particular, for any closed G-invariant subcomplex A ⊆ X, 𝕂_{G}^{∗}(X, A) ≅ 𝕂_{G/K}^{∗}(X, A).

\underline{\underline{Proof}} : For each x ∈ X, the fiber E|_{x} is a T -representation. Since X is connected, these representations are all isomorphic to some given T -representation V. Also, V and α^{∗}V are T -isomorphic, because multiplication with the generator z ∈ ℤ defines an α-equivariant linear isomorphism from E|_{x} to E|_{zx}. For any irreducible T -representation W, with character χ_{W} ∈ Hom(T, S^{1}), either the χ_{W} ∘α^{n} are all distinct, or χ_{W} = χ_{W} ∘α and K = {g^{−1}α(g) | g ∈ G} ⊆ Ker(χ_{W}). Since dim(V) < ∞, this shows that V must be a sum of irreducible (T/K)- representations, and thus that K acts on each fiber of E via the identity.

The last statement now follows immediately from the definitions.

It is now easy to see that Lemma 3.7 fails for G. For example, set (X, A) = (ℝ, ℤ) with the G-action induced by the translation action of ℤ = G/T, and let V be a T -representation upon which K does not act trivially. Then G×_{T} V is a G-vector bundle over A = ℤ ≅ G/T, and it cannot be embedded into any G-vector bundle over X = ℝ since K acts nontrivially.

More generally, let G be any Lie group and let (X, A) be any finite G-CW-pair. If E_{0} → A is any G-vector bundle, then an obvious necessary condition for being able to embed it in a G-vector bundle over X is to be able to choose the fibers: to find for each x ∈ X a G_{x}- representation V_{x}, such that E_{0}|_{x} embeds into V_{x} for each x ∈ A, and such that the V_{x} in any connected component of X^{H} (for any H ⊆ G) are all isomorphic as H-representations. For discrete G, we can always choose the V_{x} to be appropriate multiples of the regular representation of G_{x}, and this was the first (and easiest) step towards proving Lemma 3.7. In contrast, for the group G = T ⋊^{α} ℤ, the above example shows that this first step of the proof fails: we cannot even choose the representations V_{x}.

Given this example, it is not at all surprising that excision and exactness fail for 𝕂_{G}^{∗}(−). By homotopy invariance,

𝕂_{G}^{∗}(ℝ, ℤ) ≅ 𝕂_{G}^{∗}(ℝ, [0, \frac{1}{2}]+ℤ) and 𝕂_{G}^{∗}(ℤ × I, ℤ × ∂I) ≅ 𝕂_{G}^{∗}([\frac{1}{2}, 1]+ℤ, \{0, \frac{1}{2}\}+ℤ),

and hence these should be isomorphic if excision holds for 𝕂_{G}^{∗}(−). However:

Example 5.2 For G as above,

𝕂_{G}^{−n}(ℝ, ℤ) ≅ \begin{cases}R(T/K) & if n is odd \\ 0 & if n is even \end{cases} and 𝕂_{G}^{−n}(ℤ × I, ℤ × ∂I) ≅ \begin{cases}R(T) & if n is odd \\ 0 & if n is even. \end{cases}

Thus, excision does not hold for 𝕂_{G}^{∗}(−). Furthermore,

𝕂_{G}^{−n}(ℝ) ≅ R(T/K) (for all n)  and  𝕂_{G}^{−n}(ℤ) ≅ \begin{cases}R(T) & if n is even \\ 0 & if n is odd, \end{cases}

and so there is no long exact sequence in 𝕂_{G}^{∗} for the pair (ℝ, ℤ).

\underline{\underline{Proof}} : By Lemma 3.5 and Lemma 5.1,

𝕂_{G}^{−n}(ℝ, ℤ) ≅ 𝕂_{G/K}^{−n}(ℝ, ℤ) ≅ 𝕂_{(T/K)×ℤ}^{−n}(ℝ, ℤ)

≅ 𝕂_{T/K}^{−n}(ℝ/ℤ, pt) ≅ K_{T/K}^{−n−1}(pt) ≅ \begin{cases}R(T/K) & if n is odd \\ 0 & if n is even; \end{cases}

and

𝕂_{G}^{−n}(ℝ) ≅ 𝕂_{G/K}^{−n}(ℝ) ≅ 𝕂_{(T/K)×ℤ}^{−n}(ℝ) ≅ 𝕂_{T/K}^{−n}(ℝ/ℤ) ≅ K_{T/K}^{−n}(pt) ⊕ K_{T/K}^{−n−1}(pt) ≅ R(T/K)

(for all n). Also, by Lemma 3.4,

𝕂_{G}^{−n}(ℤ × I, ℤ × ∂I) ≅ 𝕂_{T}^{−n}(I, ∂I) ≅ K_{T}^{−n−1}(pt) ≅ \begin{cases}R(T) & if n is odd \\ 0 & if n is even; \end{cases}

and similarly

𝕂_{G}^{−n}(ℤ) ≅ 𝕂_{T}^{−n}(pt) ≅ \begin{cases}R(T) & if n is even \\ 0 & if n is odd. \end{cases}

All of these groups are R(T/K)-modules (via the isomorphism G/(K×ℤ) ≅ T/K), and all natural maps between them are R(T/K)-linear. But there is no R(T/K)-linear exact sequence 𝕂_{G}^{0}(ℝ) → 𝕂_{G}^{0}(ℤ) → 𝕂_{G}^{−1}(ℝ, ℤ), since the middle term is infinitely generated and the others finitely generated.

In fact, φ_{G}(ℝ) is an isomorphism in this case; i.e., K_{G}^{0}(ℝ) ≅ 𝕂_{G}^{0}(ℝ) ≅ R(T/K). To see this, consider the composite

𝕂_{G}^{0}(ℝ) ⟶^{φ_{G}(ℝ)} K_{G}^{0}(ℝ) ⟶^{i^{∗}} K_{G}^{0}(ℤ),

where i : ℤ → ℝ denotes the inclusion. Under the identifications 𝕂_{G}^{0}(ℝ) ≅ R(T/K) (see Example 5.2) and K_{G}^{0}(ℤ) = K_{G}^{0}(G/T) ≅ R(T), i^{∗}∘φ_{G}(ℝ) corresponds to the inclusion R(T/K) ↪ R(T) defined by regarding T/K-representations as T -representations. We have seen, in the proof of Lemma 5.1, that R(T/K) = R(T)^{α}: the subgroup of elements fixed by composition with α ∈ Aut(T). The map i^{∗} is injective since K_{G}^{0}(ℝ, ℤ) ≅ K_{G}^{0}(ℤ×I, ℤ×∂I) = 0. Hence it remains to show that the image of i^{∗} is contained in R(T)^{α}. This follows since the action of α on K_{G}^{0}(ℤ) = K_{G}^{0}(G/T) ≅ R(T) corresponds to the map t^{∗} : K_{G}^{0}(ℤ) ⟶ K_{G}^{0}(ℤ) induced by the G-map t : n ↦ n+1, and since the analogous G-map t : ℝ → ℝ is G-homotopic to the identity.

This can now be used to show that the completion theorem (stated in terms of K_{G}^{∗} or 𝕂_{G}^{∗}) fails for this group G. Consider the space X = ℤ. Since ℝ^{H} = ℝ is contractible for any compact subgroup H ⊆ G, ℝ ≃ E_{CPCT} (G) is a universal proper G-CW-complex. Then

Im[K_{G}(ℝ) ⟶^{i^{∗}} K_{G}(ℤ) ≅ R(T)] = R(T/K),

while

K^{0}(EG×_{G}ℤ) ≅ K^{0}(BT) ≅ R(T)_{IR(T)} ≠ R(T)_{IR(T/K)} ≅ K_{G}^{0}(ℤ)_{IR(T/K)}.

Here, the completions of R(T) with respect to IR(T) and IR(T/K) are distinct since

R(T)_{IR(T)} ≅ ℤ[[x, y]] ⫌ ℤ[[x]][t, t^{−1}] ≅ R(T)_{IR(T/K)} (where y = t − 1).

In the example constructed above, the reason for the failure of representing K_{G}(−) by G-vector bundles and of the completion theorem comes down to the fact that the isotropy subgroup in question is positive dimensional, and hence has infinitely many irreducible representations. For example, one of the key lemmas which makes possible our results for actions of a discrete group G is that any finite proper G-CW-complex has a G-vector bundle over it whose fibers are free as representations of the isotropy subgroups — and this makes sense only if the isotropy subgroups are finite. It is thus natural to ask whether these results hold for any Lie group G and any finite proper G-complex X all of whose isotropy subgroups are finite. This does, in fact, turn out to be the case — 𝕂_{G}(X) ≅ K_{G}(X) and K(EG×_{G}X) is a completion of K_{G}(X) — but proving it requires working out the details of obstruction theory for Lie group actions, which is more complicated than that used in Section 2.

References

[1] H. Abels, Parallizability of proper actions, global K-slices and maximal compact subgroups,, Math. Ann. 212,(1974), 1 - 19

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

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[16] N. C. Phillips, Equivariant K-theory for proper actions II: Some cases in which finite dimensional bundles suffice, Index theory of elliptic operators, foliations, and operator algebras, Contemp. Math. \underline{70} (1988), 205–227

[17] N. C. Phillips, Equivariant K-theory for proper actions III: Discrete groups and uniform bundles, preprint

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[19] C. Weibel, An introduction to homological algebra, Cambridge Univ. Press (1994)

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Addresses:

Wolfgang Lück Bob Oliver Institut für Mathematik und Informatik LAGA — UMR 7539 du CNRS 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
wwwmath.uni-muenster.de/math/u/lueck          zeus.math.univ-paris13.fr/~bob