EQUIVARIANT VECTOR BUNDLES OVER CLASSIFYING SPACES FOR PROPER ACTIONS

DIETER DEGRIJSE AND IAN J. LEARY

Abstract. Let G be an infinite discrete group and let \underline{E}G be a classifying space for proper actions of G. Every G-equivariant vector bundle over \underline{E}G gives rise to a compatible collection of representations of the finite subgroups of G. We give the first examples of groups G with a cocompact classifying space for proper actions \underline{E}G admitting a compatible collection of representations of the finite subgroups of G that does not come from a G-equivariant (virtual) vector bundle over \underline{E}G. This implies that the Atiyah-Hirzeburch spectral sequence computing the G-equivariant topological K-theory of \underline{E}G has non-zero differentials. On the other hand, we show that for right angled Coxeter groups this spectral sequence always collapes at the second page and compute the K-theory of the classifying space of a right angled Coxeter group.

1. Introduction

Let G be an infinite discrete group and F be the family of finite subgroups of G. Recall that a classifying space for proper actions of G, denoted by \underline{E}G, is a proper G-CW-complex such that the fixed point set \underline{E}G^{H} is contractible for every H ∈ F. The space \underline{E}G is said to be cocompact if the orbit space G \ \underline{E}G = \underline{B}G is compact. Many interesting classes of groups G have cocompact models for \underline{E}G, for example cocompact lattices in Lie groups, mapping class groups of surfaces, Out(F_{n}), CAT(0)-groups and word-hyperbolic groups. We refer the reader to [7] for more examples and details.

Now assume G is an infinite discrete group admitting a cocompact classifying space for proper actions \underline{E}G. If

ξ : E → \underline{E}G

is a G-equivariant complex vector bundle over \underline{E}G (see Definition 2.2) and x is a point of \underline{E}G, then the fiber ξ^{−1}(x) is a complex representation of the finite isotropy group G_{x}. The connectivity of the fixed point sets of \underline{E}G ensures that these representations are compatible (see Defintition 2.1) with one another as x and hence G_{x} varies. Therefore, every G-equivariant complex vector bundle over \underline{E}G gives rise to a compatible collection of complex represesentations of the finite subgroups of G, and hence to an element of

lim_{G/H∈O_{F} G} R(H).

Here, lim_{G/H∈O_{F} G} R(H) is the limit over the orbit category O_{F}G of the represenation ring functor

R(−) : O_{F}G → Ab : G/H ↦ R(H).

Date: April 28, 2015.

The first author was supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92) .

Denoting the Grothendieck completion of the monoid of isomorphism classes of complex Gvector bundles over \underline{E}G by K_{G}^{0}(\underline{E}G), one obtains a map

ε_{G} : K_{G}^{0}(\underline{E}G) → lim_{G/H∈O_{F} G} R(H)

that maps a formal difference of (isomorphism classes) vector bundles (i.e. a virtual vector bundle) to a formal difference of (isomorphism classes) of compatible collections of representations of the finite subgroups of G. We say a compatible collection of represesentations of the finite subgroups of G can be realized as a (virtual) G-equivariant vector bundle over \underline{E}G if there exists a (virtual) G-equivariant vector bundle over \underline{E}G that maps to this collection under ε_{G}. One can also look at the corresponding situation for real (orthogonal) vector bundles and real (orthogonal) represenations and obtain the map

ε_{G} : KO_{G}^{0}(\underline{E}G) → lim_{G/H∈O_{F} G} RO(H).

The maps ε_{G} are equal to the edge homomorphisms of certain Atiyah-Hirzebruch spectral sequences converging to K_{G}^{∗}(\underline{E}G) and KO_{G}^{∗}(\underline{E}G) (see (1) and (2)). Lück and Oliver proved that (see Proposition 2.4) the map ε_{G} (real or complex) is rationally surjective, meaning that a high enough multiple of every element in the target of ε_{G} is contained in the image of ε_{G}. In the last paragraph of [10, p. 596] Lück and Oliver ask for an example of a group G admitting a cocompact classifying space for proper actions \underline{E}G such that ε_{G} is not surjective. In Section 3 of this paper we give the first example of such a group in the complex case. In Section 4 we give the first example of such a group in the real case. In both cases we also explain how to construct an example of a group G admitting a cocompact classifying space for proper actions that admits a compatible collection of representations for its finite subgroups that cannot be realized as a G-vector-bundle over \underline{E}G. However, we cannot exclude the possibility that for this example there exists a virtual vector bundle that maps to this collection of representations under ε_{G}. On the other hand, these examples are more explicit and lower dimensional.

In the last section we show that for a right angled Coxeter group W, every compatible collection of represenations of the finite subgroups of W can be realized as a W -equivariant vector bundle over \underline{E}W, so that the map

ε_{W} : K_{W}^{0} (\underline{E}W) → lim_{W/H∈O_{F} W} R(H).

is always surjective. Morever, we show that this map is actually an isomorphism and that the higher equivariant K-theory (see Theorem 2.3) vanishes, i.e.

K_{W}^{1} (\underline{E}W) = 0.

Using a version of the Atiyah-Segal completion theorem for infinite discrete groups proven by Lück and Oliver, we use these results to compute the complex K-theory of BW, the classifying space of W (see Corollary 5.5).

2. G-vector bundles and isotropy representations

Throughout, let G be a discrete group and let X be a cocompact proper G-CW-complex, i.e. X has finite isotropy and the orbit space G \ X has a finite number of cells. The family of finite subgroups of G will be denoted by F. Although it is not necessary for most of the statements in this section, we will also assume that for every H ∈ F, the fixed point set X^{H} is non-empty and connected. For H ∈ F, we also fix a zero cell e_{H} ∈ X^{H}. Recall that the orbit category O_{F}G is the category whose objects are the cosets G/H, for all H ∈ F, and whose morphism are all G-equivariant maps between the objects.

Definition 2.1. Let Γ be either the unitary group U(n) or the real orthogonal group O(n, ℝ). For H ∈ F, let

Rep_{Γ}(H) = Hom(H, Γ)/Inn(Γ).

Note that if Γ = U(n) (Γ = O(n)), then Rep_{Γ}(H) is the set of isomorphism classes of ndimensional complex (real) representations of H. One can consider Rep_{Γ}(−) as a functor from O_{F}G to Sets. An element of the limit

α = ([α_{H}])_{H∈F} ∈ lim_{G/H∈O_{F} G} Rep_{Γ}(G/H)

is a called a compatible collection of complex or real n-dimensional representations of the finite subgroups of G. For H ∈ F, let R(H) (RO(H)) be the complex (real) representation ring of H, i.e. the Grothendieck completion of the abelian cancellative monoid of isomorphism classes of finite dimensional complex (real) representations of H. Note that Rep_{U(n)}(H) is naturally a subset of R(H) and Rep_{O(n)}(H) is naturally a subset of RO(H). One can consider R(−) as a functor from O_{F}G to Ab. An element of the inverse limit

α = ([α_{H}])_{H∈F} ∈ lim_{G/H∈O_{F} G} R(H)

is a called a compatible collection of complex virtual representations of the finite subgroups of G. One has

lim_{G/H∈O_{F} G} Rep_{U(n)}(G/H) ⊂ lim_{G/H∈O_{F} G} R(H)

and every element of lim_{G/H∈O_{F} G} R(H) can be written as the difference of elements in lim_{G/H∈O_{F} G} Rep_{U(n)}(G/H) and lim_{G/H∈O_{F} G} Rep_{U(m)}(G/H), for n and m large enough. The analogous statements for O(n, ℝ) and RO also hold.

Definition 2.2. A complex (real) G-vector bundle over X is a complex (real) vector bundle π : E → X such that π is G-equivariant and each g ∈ G acts on E and X via a bundle isomorphism. An isomorphism of G-vector bundles over X is just an isomorphism of vector bundle that is G-equivariant. The set of isomorphims classes of complex (real) G-vector bundles over X will be denoted by Bdl_{G}(X) (OBdl_{G}(X)). For every x ∈ X, the fiber π^{−1}(x) is denoted by E_{x}. We refer the reader to [10, Section 1] and [14, Section I.9] for elementary properties of G-vector bundles over proper (cocompact) G-CW complexes.

Theorem 2.3. [10, Th. 3.2 and 3.15] There exists a 2-periodic (8-periodic) equivariant cohomology theory K_{G}^{∗}(X, A) (KO_{G}^{∗}(X, A)) on the category of proper G-CW-pairs such that K_{G}^{0}(X) (KO_{G}^{0}(X)) is the Grothendieck completion of the monoid of isomorphism classes of complex (real) G-vector bundles over X. In particular, for every H ∈ F, K_{G}^{0}(G/H) (KO_{G}^{0}(G/H) ) equals R(H) (RO(H)).

As usual (see [11, Section 6] and [4, Th. 4.7]), the skeletal filtration of X induces Atiyah-Hirzebruch spectral sequences

(1)  E_{2}^{p,q} = H_{G}^{p}(X, K_{G}^{q}(G/−)) ⟹ K_{G}^{p+q}(X).

and

(2)  E_{2}^{p,q} = H_{G}^{p}(X, KO_{G}^{q}(G/−)) ⟹ KO_{G}^{p+q}(X)

where H_{G}^{p}(X, −) denotes Bredon cohomology of X (see [1]).

Proposition 2.4. [11, Prop 5.8] The spectral sequences (1) and (2) above rationally collapse, meaning that the images of all differentials in these spectral sequences consist of torsion elements.

By our assumptions on X, the degree zero Bredon cohomology group H_{G}^{0}(X, R(−)) (resp. H_{G}^{0}(X, RO(H))), equals the limit of the functor R(−) (resp. RO(−)), over the orbit category O_{F}G. Consider the edge homomorphisms

ε_{G} : K_{G}^{0}(X) → H_{F}^{0}(X, R(−))

and

ε_{G} : KO_{G}^{0}(X) → H_{F}^{0}(X, RO(−))

of the spectral sequences (1) and (2). If [π] is the isomorphism class of an n-dimensional complex G-vector bundle π : E → X, then ε_{G}([π]) equals

([E_{e_{H}}])_{H∈F} ∈ lim_{G/H∈O_{F} G} Rep_{U(n)}(H) ⊂ H_{G}^{0}(X, R(−))

where [E_{e_{H}}] denotes the isomorphism class in R(H) of the H-representation E_{e_{H}}. The corresponding statement for real G-vector bundles also holds. Note that it follows from Proposition 2.4 that a suitable multiple of every compatible collection of (virtual) real or complex representations of the finite subgroups of G is contained in the image of the edge homomorphism ε_{G}.

Recall that the classifying space for proper actions \underline{E}G is a terminal object in the homotopy category of proper G-CW complexes (e.g. [7, Th. 1.9]). Hence, if X is any proper cocompact G-CW complex such that X^{H} is non-empty and connected for each H ∈ F, then there exists a G-map X → \underline{E}G that is unique up to G-homotopy and induces commutative diagrams

\xymatrix{K_{G}^{0}(X) \ar[r] & lim_{G/H∈O_{F} G} R(H) & and & KO_{G}^{0}(X) \ar[r] & lim_{G/H∈O_{F} G} RO(H) \\ K_{G}^{0}(\underline{E}G) \ar[u] \ar@{-->}[ur] &  &  & KO_{G}^{0}(\underline{E}G). \ar[u] \ar@{-->}[ur]}

Hence, if a compatible collection α of representations can be realized as a G-vector bundle over \underline{E}G, it can also be realized as a G-vector bundle over X.

3. Complex vector bundles

The purpose of this section is to construct a group G with a cocompact classifying space for proper actions \underline{E}G admitting a compatible collection of complex representations of the finite subgroups of G that cannot be realized as G-equivariant virtual complex vector bundle over \underline{E}G, i.e. so that the edge homomorphism

ε_{G} : K_{G}^{0}(\underline{E}G) → lim_{G/h∈O_{F} G} R(H).

is not surjective.

Let F = C_{4} ⋊ C_{2} be the dihedral group of order 8 where σ is generator for C_{4} and ε is a generator of C_{2}. Let H = ⟨σ^{2}⟩ be the center of F, which has order two and denote the n-skeleton of the universal F/H-space E(F/H) by X^{n}. We let F act on X and X^{n} via the projection onto F/H. Consider the complex 1-dimensional representation

λ : H = ⟨σ^{2}⟩ → U(1) = S^{1} : σ^{2} ↦ −1.

Lemma 3.1. The isomorphism class [λ] is contained in R(H)^{F/H}. For k ∈ ℤ, the multiple [λ^{k}] is contained in the image of the restriction map res : R(F) → R(H) if and only if k is even.

Proof. Since H is the center of F it follows that the conjugation action of F/H on R(H) is trivial, hence [λ] ∈ R(H)^{F/H} = R(H). One easily verifies that the representation

μ : F → U(2)

defined by

μ(σ) = ( \begin{matrix}0 & i \\ i & 0 \end{matrix} ) and μ(ε) = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} )

satisifies res([μ]) = [λ^{2}]. Hence, [λ^{k}] is contained in the image of res for every even k ∈ ℤ. Note that, as a free abelian group, R(H) is generated by [λ] and the isomorphism class of the 1-dimensional complex trivial representation [tr]. Now suppose k is odd and there exists an element [μ] − [ρ] ∈ R(F) such that res([μ] − [ρ]) = [λ^{k}]. There are integers l, m, n and such that res([μ]) = [tr^{l}] + [λ^{m}], res([ρ]) = [tr^{l}] + [λ^{n}] and m − n = k. By changing the representative of [μ], we may also assume that

μ : F → U(l + m)

where μ(σ) is a diagonal matrix. Since μ(σ^{2}) has an m-dimensional eigenspace with eigenvalues −1 and an l-dimensional eigenspace with eigenvalue 1, it follows that μ(σ) has an s-dimensional eigenspace with eigenvalue i and a t-dimensional eigenspace with eigenvalue −i such that s + t = m. Morever, μ(σ^{3}) has an s-dimensional eigenspace with eigenvalue −i and a t-dimensional eigenspace with eigenvalue i. Since σ and σ^{3} are conjugate in F, it follows that s = t proving that m is even. A similiar argument shows that n is also even. But this contradicts the fact that k = m − n is odd. Hence, there does not exist an element [μ] − [ρ] ∈ R(F) such that res([μ] − [ρ]) = [λ^{k}], if k is odd.

□

The following lemma uses the notation introduced above and will be cited in the next section.

Lemma 3.2. Every complex one-dimensional F -bundle over X^{3} is isomorphic to the pullback of a complex one-dimensional F -bundle over E(F/H) along the inclusion i : X^{3} → E(F/H).

Proof. Let S be the family of subgroups of F containing only H and the trivial subgroup. Note that isomorphism classes of complex one-dimensional F -vector bundles are the same as isomorphism classes of F -equivariant principal S^{1}-bundles, or (F, S^{1} = U(1))-bundles, in the sense of [10, Section 2] and [12, Section 2,3]. Let π : E → X^{3} be a complex onedimensional F -bundle and let [α_{H} : H → U(1) = S^{1}] be the isomorphism class in Rep_{S^{1}}(H) of the H-represenation induced on the fibers of π. If we set α_{{e}} : {e} → S^{1}, then A = ([α_{K}])_{K∈S} ∈ lim_{S∈S} Rep_{S^{1}}(K). It follows from [10, Lemma 2.4(b)] for Γ = S^{1}, that in order to show that p is the pullback of a complex one-dimensional F -bundle over E(F/H) along the inclusion i : X^{3} → E(F/H), if suffices to show that every F -map from X^{3} to B_{S}(F, A) can be extended to an F -map from E(F/H) to B_{S}(F, A). Here B_{S}(F, A) is a certain F - CW-complex such that B_{S}(F, A)^{S} is homotopy equivalent to BS^{1} = ℂP^{∞} for all S ∈ S, by [10, Lemma 2.4(a)] for Γ = S^{1}. It follows from Bredon’s equivariant obstruction theory (see [1]) that the potential obstruction for doing this lies in the relative Bredon cohomology groups H_{F}^{n+1}(E(F/H), X^{3}; π_{n}(B_{S}(F, A)^{−})) for n ≥ 3. Since π_{n}(ℂP^{∞}) is zero unless n = 2, the lemma is proven. □

The idea for the following lemma is contained in [10, p 596].

Lemma 3.3. There exists an n ≥ 1 such that [λ] is not contained in the image of the edge homomorphism

K_{F}^{0} (X^{n}) → R(H)^{F/H}.

Proof. By [5, Theorem 5.1] for X = {∗}, F = {e, H} and EF = E(F/H), there are maps

α_{n} : {R(F)/I^{n}}_{n≥0} ⟶^{≅} {K_{F}^{0} (X^{n})}_{n≥0}

that induce an isomorphism of pro-rings. Here I is the kernel of the restriction map R(F) → R(H). This implies that for sufficiently large n ≥ 1 there exists a map β_{1} : K_{F}^{0} (X^{n}) → R(F)/I making the following diagram commute

\xymatrix{R(F)/I^{n} \ar[r]^{α_{n}} \ar[dd] & K_{F}^{0} (X^{n}) \ar[ddl]^{β_{1}} \ar[dd] \ar[dr]^{ε_{F}} \\  &  & R(H)^{F/H} \\ R(F)/I \ar[r]^{α_{1}} & K_{F}^{0} (X^{1}). \ar@{-->}[ur]^{ε_{F}}}

This shows that the image of the restriction map

R(F) → R(H)^{F/H}

coincides with the image of the edge homomorphism

K_{F}^{0} (X^{n}) → R(H)^{F/H}.

Since [λ] does not lie in the image of R(F) → R(H)^{F/H} by Lemma 3.1, the lemma follows. □

Let n ≥ 3. By [6, Th. A & Th. 8.3] there exists a compact n-dimensional locally CAT(0)- cubical complex T_{X^{n}} equipped with a free cellular F/H-action and an F/H-equivariant map t_{X^{n}} : T_{X^{n}} → X^{n} that induces an isomorphism

(3)  H_{F}^{∗} (X^{n}) ⟶^{≅} H_{F}^{∗} (T_{X^{n}})

for any equivariant cohomology theory H_{?}^{∗}(·) (e.g. see [9, section 1]). (We remark that [6, Th. 8.3] is stated for equivariant homology theories, but the analogous statement holds for equivariant cohomology theories by essentially the same proof.) The action of F on T_{X^{n}} in the above is via the projection F → F/H. Now let Y^{n} be the univeral cover of T_{X^{n}} and let Γ_{n} be the group of self-homeomorphisms of Y^{n} that lift the action of F/H on T_{X^{n}}. Since F/H acts freely on T_{X^{n}}, Γ_{n} acts freely on Y^{n}. In particular, Γ_{n} is torsion-free and Y^{n} is an n-dimensional CAT(0)-cubical complex on which Γ_{n} acts freely, compactly and cellulary. By construction there is a surjection Γ_{n} → F/H whose kernel N_{n} is the torsion-free group of deck transformation of the covering Y^{n} → T_{X^{n}}. Now define the group G_{n} to be the pullback of π_{n} : Γ_{n} → F/H along F → F/H. Then G_{n} acts on Y^{n} via the quotient map G_{n} → G_{n}/H = Γ_{n} and fits into the short exact sequence

1 → N_{n} → G_{n} ⟶^{p_{n}} F → 1.

Note that the only non-trivial finite subgroup of G_{n} is H ≅ C_{2} and that since N_{n} acts freely on Y^{n}, the F -equivariant quotient map Y^{n} → N_{n} \ Y^{n} = T_{X^{n}} induces an isomorphism ([10, Lemma 3.5])

(4)  K_{F}^{∗} (T_{X^{n}}) ⟶^{≅} K_{G}^{∗}(Y^{n}).

Applying (3) and (4) to the compostion Y^{n} → T_{X^{n}} → X^{n} and the equivariant cohomology theories K_{?}^{∗}(·) and H_{?}^{∗}(·, R(−)) with ∗ = 0, we obtain a commutative diagram

\xymatrix{K_{F}^{0} (X^{n}) \ar[r]^{≅} \ar[d]^{ε_{F}} & K_{G_{n}}^{0}(Y^{n}) \ar[d]^{ε_{Gn}} \\ R(H)^{F/H} \ar[r]^{≅} & lim_{G_{n}/S∈O_{F} G_{n}} R(S).}

Since we proved in Lemma 3.3 that, for n large enough, the isomorphism class of λ does not lie in the image of the edge homomorphism

K_{F}^{0} (X^{n}) → R(H)^{F/H}

it follows from the commutative diagram above that the compatible system of representations

(λ ∘ p_{n|S})_{S∈F} ∈ lim_{G_{n}/S∈O_{F} G_{n}} R(S) = H_{F}^{0}(G_{n}, R(−)).

does not lie in the image of the edge homomorphism

ε_{G_{n}} : K_{G_{n}}^{0}(Y^{n}) → lim_{G_{n}/S∈O_{F} G_{n}} R(S).

Recall from [2] that non-empty CAT(0)-cube complexes are contractible and that the fixed point set for a finite group action on a CAT(0)-cube complex is contractible. Since G_{n} acts cellularly properly and cocompactly on the CAT(0)-cube complex Y_{n}, we deduce that Y_{n} is a cocompact model for \underline{E}G_{n}. To summarize, we have constructed a group G = G_{n} with a cocompact classifying space for proper actions \underline{E}G admitting a compatible collection of complex representations of the finite subgroups of G that cannot be realized as G-equivariant virtual complex vector bundle over \underline{E}G.

4. Real vector bundles

One could apply the same technique of the previous section in the real setting to obtain a group G with cocompact classifying space for proper actions \underline{E}G so that the edge homomorphism

ε_{G} : KO_{G}^{0}(\underline{E}G) → lim_{G/H∈O_{F} G} RO(H)

is not surjective. Instead we give an explicit description of a group G that admits ℝ^{2} as a cocompact model for \underline{E}G and admits a compatible collection of real representations of its finite subgroups that cannot be realized as a real G-vector bundle over ℝ^{2}.

We start by describing a related group Γ that is a 2-dimensional crystallographic group, or wallpaper group; this group is known as p2gg, but we will describe it explicitly. Endow ℝ^{2}

Figure 1. A wallpaper pattern for Γ = p2gg

with the CW-structure coming from the standard tesselation by unit squares with vertices at ℤ^{2}, and let Γ be the group of automorphisms of this CW-structure that preserves the pattern shown in Figure 1. The stabilizer of a 2-cell is clearly trivial, and so the 2-cells form a single free Γ-orbit. There are two orbits of 1-cells, the vertical and horizontal edges, and again each orbit is free. There are two orbits of 0-cells, and the stabilizer of a 0-cell is cyclic of order two, generated by the rotation of order two fixing the point. Since the stabilizer of each cell acts trivially on that cell, the given CW-structure makes ℝ^{2} into a Γ-CW-complex.

The translation subgroup T of Γ has index four, and consists of the elements (x, y) ↦ (x + 2m, y + 2n). The orientation-preserving subgroup N of Γ has index two, and consists of T together with the rotations through π about some point of ℤ^{2}, which are of the form (x, y) ↦ (2m − x, 2n − y). Finally the elements of Γ − N are the glide reflections whose axes bisect the sides of the 2-cells: (x, y) ↦ (2m + 1 − x, 2n + 1 + y) and (x, y) ↦ (2m + 1 + x, 2n + 1 − y). The quotients T \ℝ^{2}, N\ℝ^{2} and Γ\ℝ^{2} are respectively a torus consisting of four squares, an S^{2} obtained by identifying the boundaries of two squares, and a copy of ℝP^{2} obtained by identifying the edges of a square in pairs. The fact that Γ − N contains no torsion elements is reflected in the fact that Γ/N acts freely on the sphere N\ℝ^{2}.

Now let F be a copy of C_{4} and let H ≅ C_{2} be the index two subgroup of F. The group G is defined as the pullback of the two maps Γ → Γ/N ≅ C_{2} and F → F/H ≅ C_{2}. By construction the group G admits ℝ^{2} as a cocompact model for \underline{E}G, and fits into a short exact sequence

1 → N → G ⟶^{p} F → 1

such that p(F) = H.

Now let

λ : H → O(1, ℝ) = C_{2}

be the 1-dimensional real sign represenation of H, i.e. λ is the identity map. The isomorphism class [λ] is clearly contained in RO(H)^{F/H}, since F is abelian.

Lemma 4.1. The isomorphism class [λ^{k}] is contained in the image of the restriction map

RO(F) → RO(H)^{F/H}.

if and only if k is even.

Proof. Recall that the irreducible real representations of C_{4} are up to isomorphism the onedimensional trivial representation represenation, the one dimensional sign representation of F/H = C_{2} and one 2-dimensional faithful representation in which the elements of order four act as rotations by ±\frac{π}{2} . The restriction of the first two of the represenations to H gives the trivial one dimensional representation of H, while the restriction of H of the third is λ ⊕ λ. We therefore conclude that the image of RO(F) → RO(H)^{F/H} consists of element of the form 2n[λ] + m[tr], where tr is the trivial one dimensional representation of H and n, m ∈ ℤ This shows that [λ^{k}] is contained in the image of the restriction map RO(F) → RO(H)^{F/H} if and only if k is even. □

Lemma 4.2. Let F act on the infinite dimensional sphere S^{∞} by first projecting onto F/H = C_{2} and then acting via the antipodal map. View S^{2} as the 2-skeleton of S^{∞}. Every F - equivariant orthogonal real bundle over S^{2} is isomorphic to the pullback of a F -equivariant orthogonal real bundle over S^{∞} along the inclusion S^{2} → S^{∞}.

Proof. Let S be the family of subgroups of F containing H and the trivial subgroup. Note that F -equivariant orthogonal real bundles are the same as F -equivariant O(1, ℝ) = C_{2}- bundles, or (F, C_{2})-bundles, in the sence of [10, section 2]. Now let ξ be an (F, C_{2})-bundle over S^{3} with fibers A = (ξ_{S}) ∈ lim_{S∈S} Rep_{C_{2}}(S). Note that B_{S}(F, A)^{S} ≅ BC_{2} = ℝP^{∞} for all S ∈ S (see [10, Def. 2.3 and Lemma 2.4]). By [10, Lemma 2.4] it suffices to show that every F -map f : S^{2} → B_{S}(F, A) can be extended to an F -map f̃: S^{∞} → B_{S}(F, A). It follows from Bredon’s equivariant obstruction theory that the obstruction for doing this lies in the relative Bredon cohomology groups H_{F}^{n+1}(S^{∞}, S^{2}; π_{n}(B_{S}(F, A)^{−})) for n ≥ 2. Since π_{n}(ℝP^{∞}) is zero unless n = 1, the lemma is proven. □

Lemma 4.3. Let F act on S^{2} by first projecting onto F/H = C_{2} and then acting via the antipodal map. There does not exist a real F -vector bundle ξ : E → S^{2} such that the represensation of H on the fibers of ξ is isomorphic to λ.

Proof. Consider the infinite dimensional sphere S^{∞} as a the universal C_{2}-space EC_{2}, where C_{2} acts via the antipodal map and let F act on S^{∞} via first projection onto F/H = C_{2} and then acting via C_{2}. Now assume that there exists a real F -vector bundle ξ : E → S^{2} such that the represensation of H on the fibers of ξ is isomorphic to λ. By Lemma 4.2 there also exists a real F -vector bundle ξ : E → S^{∞} such that the represensation of H on the fibers of ξ is isomorphic to λ. By [5, Corollary 5.2] (and the comments below) for X = {∗} and EF = S^{∞}, there is an isomorphism

p : RO(F) ⟶^{≅} KO_{F}^{0} (S^{∞}),

where the completion is the I-adic completion with respect to the kernel of the restriction map RO(F) → RO(H). Moreover, as explained in [5, Example 5.5], the edge homomorphism of the equivariant Atiyah-Hirzebruch spectral sequence provides a map

KO_{F}^{0} (S^{∞}) → RO(H)^{C_{2}}

such that composition with p is the restriction map RO(F) → RO(H)^{F/H}. This implies that the fibers of any F -vector bundle over S^{∞}, considered as H-representations, always lie in the image of the restriction map RO(F) → RO(H). Since λ does not lie in the image of RO(F) → RO(H) by Lemma 4.1 we arrive at a contradiction and conclude that there does not exist a real F -vector bundle ξ : E → S^{2} such that the represensation of H on the fibers of ξ is isomorphic to λ. □

Now consider the compatible system of real orthogonal representions

(λ ∘ p_{|S})_{S∈F} ∈ lim_{G/S∈O_{F} G} RO(S) = H_{F}^{0}(G, RO(−))

and assume that there exists a real G-vector bundle ξ : E → ℝ^{2} that realizes it. Since the kernel of p is N it follows from the lemma below and our observations above that N \ ξ : N \ E → N \ X is an F -vector bundle over S^{2}, where F acts on S^{2} via projection onto F/H = C_{2}, followed by the antipodal map. Morever, the represenation of H on the fibers of N \ ξ is by construction exactly λ. This however contradicts Lemma 4.3, so we conclude that there does not exist a real G-vector bundle ξ : E → ℝ^{2} that realizes the compatible system of real orthogonal represenations (λ ∘ p_{|S})_{S∈F}.

Lemma 4.4. If ξ : E → X is a G-vector bundle over X and N is a normal subgroup of G such that N ∩ G_{x} acts trivially on ξ^{−1}(x) for every x ∈ X, then

N \ ξ : N \ E → N \ X

is a G/N-vector bundle over N \ X.

Proof. Denote the projection G → G/N = Q by π. Let us first consider the case where ξ is trivial (trivial in the sense of [8, Section 6.1.]), i.e. assume ξ is a pullback

\xymatrix{G ×_{H} V \ar[r] & G/H \\ E \ar[u]^{r} \ar[r]^{ξ} & X \ar[u]^{p}}

of the G-vector bundle G ×_{H} V → G/H along the G-map p : X → G/H where H is some point stablizer of X. Note that H ∩ N acts trivially on V and consider the pullback diagram

\xymatrix{Q ×_{π(H)} V \ar[r] & Q/π(H) \\ P \ar[u]^{w} \ar[r]^{q} & N \ X \ar[u]^{N\p}}

of the Q-vector bundle Q ×_{π(H)} V → Q/π(H) along the Q-map N \ p : N \ X → Q/π(H). We define the map

ψ : N \ E → P : \overline{(g, v, x)} ↦ (π(g), v, \overline{x}).

It is easy to check that ψ is a well-defined Q-equivariant open map. We claim that ψ is a bijection. To prove surjectivity, let (π(g), v, \overline{x}) ∈ Q ×_{π(H)} V × (N \ X) be an element of P. This means that there exists an n ∈ N such that ngH = p(x). Hence, \overline{(ng, v, x)} ∈ N \ E and ψ(\overline{(ng, v, x)}) = (π(g), v, \overline{x}). To prove injectivity, consider \overline{(g, v, x)} and \overline{(g′, v′, x)} in N \ E and assume that ψ(\overline{(g, v, x)}) = ψ(\overline{(g′, v′, x′)}), i.e. (π(g), v, \overline{x}) = (π(g′), v′, \overline{x′}). This implies that there exists an h ∈ H such that π(h)v = v′ and π(g) = π(g′h). Also, there exists an n ∈ N such that nx = x′. Since \overline{(g, v, x)} and \overline{(g′, v′, x)} in N \ E we have gH = p(x) and g′H = p(x′), which implies that g′h′ = ng for some h′ ∈ H. Now

n(g, v, x) = (ng, v, x′)
= (g′h′, v, x′)
= (g′, h′v, x′).

We also compute,

ng = g′h′
= g′hh^{−1}h′
= n′gh^{−1}h′

for some n′ ∈ N. Multiplying this last equation by g^{−1} on the left and using normality of N, we conclude that h^{−1}h′ ∈ H ∩ N and hence h^{−1}h′v = v. Since hv = v′, this implies that n(g, v, x) = (g′, v′, x′). Hence \overline{(g, v, x)} = \overline{(g′, v′, x)}, proving injectivity and the claim. Since q ∘ ψ = N \ ξ and w ∘ ψ = N \ r, we conclude that the lemma holds in case ξ is trivial.

Now consider the general case. Let \overline{x} ∈ N \ X. Since ξ : E → X is locally trivial, x ∈ X has an open G-neighourhood U such that there is a G-map p : U → G/H where H = G_{x} and ξ_{|U} is (homeomorphic to) the pullback

\xymatrix{G ×_{H} V \ar[r] & G/H \\ ξ_{|U} \ar[u] \ar[r]^{ξ} & U \ar[u]^{p}}

of the G-vector bundle G ×_{H} V → G/H along the G-map p : X → G/H, where V = ξ^{−1}(x). By the above, the quotient diagram

\xymatrix{Q ×_{π(H)} V \ar[r] & Q/π(H) \\ N \ ξ_{|U} \ar[u] \ar[r]^{N\ξ} & N \ U \ar[u]^{N\p}}

is a pullback diagram. Since N \ U is an open Q-neighbourhood of \overline{x} and Q\overline{x} = π(H), it follows that N \ ξ : N \ E → N \ X is a Q-vector bundle. □

We finish this section by noting that a similar approach to the above together with Lemma 3.2 (which is the complex version of Lemma 4.2) and an application of [6, Th. A] can be used to produce a group G admitting a three dimensional cocompact model for \underline{E}G that has a compatible system of one-dimensional complex represenations that cannot be realized as a complex G-vector bundle over \underline{E}G.

5. Right angled Coxeter groups

Let Γ be a finite graph. We denote the vertex set of Γ by S = V (Γ) and the set edges of Γ by E(Γ) ⊆ V (Γ) × V (Γ). The right angled Coxeter group determined by Γ is the Coxeter group W with presentation

W = ⟨S | s^{2} for all s ∈ V (Γ) and (st)^{2} if (s, t) ∈ E(Γ)⟩.

Note that W fits into the short exact sequence

1 → N → W ⟶^{p} F = ⨁_{s∈S} C_{2} → 1

where p takes s ∈ S to the generator of the C_{2}-factor corresponding to s. A subset J ⊆ S is called spherical if the subgroup W_{J} = ⟨J⟩ is finite (and hence isomorphic to ⨁_{s∈J} C_{2}). The empty subset of J is by definition spherical. We denote the poset of spherical subsets of S orderded by inclusion by S. If J ∈ S, then W_{J} is called a spherical subgroup of W, while a coset wW_{J} is called spherical coset. We denote the poset of spherical cosets, ordered by inclusion, by W S. Note that W acts on W S by left multiplication, preserving the ordering. The Davis complex Σ of W is the geometric realization of W S. One easily sees that Σ is a proper cocompact W -CW-complex. Since Σ admits a complete CAT(0)-metric such that W acts by isometries, it follows that Σ is a cocompact model for \underline{E}W (see [3, Th. 12.1.1 & Th. 12.3.4]). A consequence of this fact is that every finite subgroup of W is subconjugate to some spherical subgroup of W. This implies that the group N defined above is torsion-free. Since the quotient space W \ Σ is homeomorphic to the geometric realization of the poset S, which is contractible since it has a minimal element, another consequence is that the quotient \underline{B}W = W \ \underline{E}W is contractible. We refer the reader to [3] for more details and information about these groups and the spaces on which they act.

Let F be the family of finite subgroups of W and let Γ be either the orthogonal group O(n, ℝ) or the unitary group U(n). Given an abelian group A, we denote by

\underline{A} : O_{F}W → Ab

the trivial functor that takes all objects to A and all morphism to the identity map. One can verify that

(5)  H_{G}^{∗}(\underline{E}G, \underline{A}) ≅ H^{∗}(\underline{B}G, A).

Lemma 5.1. Every element of

lim_{W/H∈O_{F} W} Rep_{Γ}(H)

is of the form ([λ ∘ p_{|H}])_{H∈F} for some group homomorphism λ : F → Γ.

Proof. Every finite subgroup H of W is isomorphic to a finite direct sum of C′_{2} s. Since every element of order 2 in Γ is conjugate in Γ to a diagonal matrix with ±1 on the diagonal and commuting matrices can be simultanuoulsy diagonalized, it follows that the image of every homomorphism H → Γ is conjugate to a finite subgroup of Γ consisting of diagonal matrices with diagonal entries equal to ±1. It follows that every element of lim_{W/H∈O_{F} W} Rep_{Γ}(H) is of the form ([α_{H}])_{H∈F} where α_{H} : H → Γ is a homomophism whose image lands in the finite abelian subgroup of Γ consisting of diagonal matrices with diagonal entries equal to ±1. Since every finite subgroup of W is subconjugate to a special subgroup W_{J}, the compatibility of the represenations tells us that ([α_{H}])_{H∈F} is completely determined by the homomorphisms α_{⟨s⟩} : ⟨s⟩ → Γ, for s ∈ S. Now define the homomorphism

λ : F = ⨁_{s∈S} C_{2} → Γ : (σ_{s})_{s∈S} ↦ ∑_{s∈S} α_{⟨s⟩}(σ_{s}).

Then the compatibility of the represenations implies that

([λ ∘ p_{|H}])_{H∈F} = ([α_{H}])_{H∈F},

proving the lemma. □

The following theorem applies to both complex and real represenations and vector bundles.

Theorem 5.2. Let W be a right angled Coxeter group. Every compatible collection of represenations of the finite subgroups of W can be realized as a W -equivariant vector bundle over the Davis complex Σ = \underline{E}W.

Proof. Let A = ([α_{H}])_{H∈F} ∈ lim_{W/H∈O_{F} W} Rep_{F} (H). It follows from [10, Lemma 2.4(b)] that the existence of a (W, A)-bundle over Σ (see [10, p. 591]) follows from the existence a W -map Σ → B_{F}(G, A). Since B_{F}(G, A)^{H} is homotopic to BC_{F} (α_{H}) = BF for all H ∈ F, it follows that the functor

π_{k}(B_{F}(W, A)^{−}) : O_{F}(W) → Ab : W/H ↦ π_{k}(B_{F}(G, A)^{H})

equals the trivial functor \underline{π_{k}(BF)} for all k ≥ 0. From (5) and the contractibility of \underline{B}W, it follows that the Bredon cohomology groups

H_{W}^{k+1}(Σ, π_{k}(B_{F}(W, A)^{−}))

are zero for all k ≥ 0. Since there certainly exists a W -map from the 0-skeleton of Σ to B_{F}(G, A), it follows from Bredon’s equivariant obstruction theory that there exists a W -map Σ → B_{F}(G, A). In particulcar, we conclude that there exists a (W, A)-bundle over Σ, where A = ([α_{H}])_{H∈F} ∈ lim_{W/H∈O_{F} W} Rep_{F} (H).

Now consider a compatible collection of represenations of the finite subgroups of W. By Lemma 5.1(i) above, this collection is of the form

([λ ∘ p_{|H}])_{H∈F} ∈ lim_{W/H∈O_{F} W} Rep_{Γ}(H)

for some group homomorphism λ : F → Γ. Since

A = ([p_{|H}])_{H∈F} ∈ lim_{W/H∈O_{F} W} Rep_{F} (H)

it follows from the above that there exists a (W, A)-bundle ξ : E → Σ. If Γ = O(n, ℝ) then

ξ : E ×_{F} ℝ^{n} → Σ

is a real W -vector bundle over Σ that realizes ([λ ∘ p_{|H}])_{H∈F}, and if Γ = U(n) then

ξ : E ×_{F} ℂ^{n} → Σ

is a complex W -vector bundle over Σ that realizes ([λ ∘ p_{|H}])_{H∈F}. Here F acts on ℝ^{n} or ℂ^{n} via the map λ. □

Lemma 5.3. If W is a right angled Coxeter group, then H_{W}^{n} (Σ, R(−)) = 0 for all n > 0, and H_{W}^{0} (Σ, R(−)) is free abelian of rank equal to the number of spherical subgroups of W.

Proof. This is proven in much the same way as the corresponding result for homology in [13]. In more detail, one uses the cubical structure on Σ, in which there is one orbit of n-cubes with stabilizer isomorphic to (C_{2})^{n} for each n-tuple of commuting elements of S. Since the stabilizer of a cube of strictly positive dimension acts non-trivially on the cube, this is not a W -CW-structure on Σ. However, its barycentric subdivision is isomorphic to the realization of the poset W S as described in the introduction to this section.

Let Σ^{n} denote the n-skeleton of Σ with this cubical structure. Firstly, Σ^{0} consists of a single free W -orbit of vertices, so H_{W}^{∗} (Σ^{0}; R(−)) is isomorphic to the ordinary cohomology of a point; since W acts freely the calculation reduces to an equivariant cohomology calculation for the trivial group action.

Let I = [−1, 1] be an interval, with C_{2} acting by x ↦ −x (i.e., swapping the ends of the interval). Note that I is isomorphic to the Davis complex for the Coxeter group C_{2}. Let ∂I denote the two end points {−1, 1}. A direct computation shows that H_{C_{2}}^{m}(I, ∂I; R(−)) is isomorphic to ℤ for m = 0 and is zero for m > 0.

Next consider I^{n} with C_{2}^{n} acting as the direct product of n copies of the above action of C_{2} on I. This is the Davis complex for the Coxeter group C_{2}^{n}. Since the representation ring of a direct product of finite groups is naturally identified with the tensor product of the representation rings, the C_{2}^{n}-Bredon cochain complex for the pair (I^{n}, ∂I^{n}) with coefficients in R(−) is naturally isomorphic to the tensor product of n copies of the C_{2}-Bredon cochain complex for (I, ∂I) with coefficients in R(−). Since H_{C_{2}}^{∗}(I, ∂I; R(−)) is free abelian one obtains a Künneth formula

H_{C_{2}^{n}}^{∗}(I^{n}, ∂I^{n}, R(−)) ≅ ⨂_{i=1}^{n} H_{C_{2}}^{∗}(I, ∂I; R(−)).

It follows that for each n, H_{C_{2}^{n}}^{m}(I^{n}, ∂I^{n}; R(−)) is isomorphic to ℤ for m = 0 and is zero for m > 0.

From these computations, it follows easily that H_{W}^{m}(Σ^{n}, Σ^{n−1}; R(−)) is zero for m > 0 and is isomorphic to a direct sum of copies of ℤ indexed by the W -orbits of n-cubes in Σ. By induction on n one sees that H_{W}^{m}(Σ^{n}; R(−)) is zero for m > 0 and isomorphic to a direct sum of copies of ℤ indexed by the W -orbits of cubes of dimension at most n for m = 0. The claimed result follows, since the W -orbits of cubes in Σ are in bijective correspondence with the spherical subgroups of W. □

Theorem 5.4. Let W be the right angled Coxeter group determined by a finite graph Γ. Then K_{W}^{1} (\underline{E}W) = 0 and there is a ring isomorphism

K_{W}^{0} (\underline{E}W) ≅ ℤ[V (Γ)]/(s^{2} − 1, st − s − t + 1 | s ∈ V (Γ),(s, t) ∉ E(Γ)).

Here, ℤ[V (Γ)] is the polynomial ring with variables in V (Γ) and integer coefficients. In particular,

K_{W}^{0} (\underline{E}W) ≅ ℤ^{d}

as an abelian group, where d is the number of spherical subgroups of W.

Proof. Consider the Atiyah-Hirzebruch spectral sequence

E_{2}^{p,q} = H_{W}^{p} (\underline{E}W, K_{W}^{q} (W/−)) ⟹ K_{W}^{p+q}(\underline{E}W)

where K_{W}^{q} (W/−) = R(−) if q is even and K_{W}^{q} (W/−) = 0 q is odd. In the lemma above, we proved that H_{W}^{k} (Σ, R(−)) = 0 for k > 0. It therefore follows that

K_{W}^{n} (\underline{E}W) = \{ \begin{matrix}H_{W}^{0} (\underline{E}W, R(−)) = lim_{W/H∈O_{F} W} R(H) & if n = 0 \\ 0 & if n = 1. \end{matrix}

Let I be the ideal

(s^{2} − 1, st − s − t + 1 | s ∈ V (Γ), (s, t) ∉ E(Γ))

in the polynomial ring ℤ[V (Γ)]. Note that as an abelian group ℤ[V (Γ)]/I is free, with basis elements the commuting products s_{1} … s_{k}, for all J = {s_{1} …, s_{k}} ∈ S (J = ∅ corresponds to the unit of ℤ[V (Γ)]/I ). This shows that

ℤ[V (Γ)]/I ≅ ℤ^{d}

as an abelian group, where d is the number of spherical subgroups of W.

We claim there is an isomorphism of rings

lim_{W/H∈O_{F} W} R(H) ≅ ℤ[V (Γ)]/I.

Since lim_{W/H∈O_{F} W} R(H) = H_{W}^{0} (Σ, R(−)) = H_{W}^{0} (Σ^{1}, R(−)), one can use the explicit description of Σ to see that

lim_{W/H∈O_{F} W} R(H) ≅ lim_{J∈S} R(W_{J}).

as rings. Since W_{J} = ⨁_{s∈J} C_{2} and R(C_{2}) = ℤ[X]/(X^{2} − 1), we have that

R(W_{J}) = ⨂_{s∈J} ℤ[s]/(s^{2} − 1).

as rings. For each J ⊆ S, we define the unital ring homomorphism

ρ_{J} : ℤ[V (Γ)]/I → R(W_{J})

by setting

ρ_{J}(s) = \{ \begin{matrix}1 ⊗ … ⊗ 1 ⊗ s ⊗ 1 ⊗ … ⊗ 1 ∈ R(W_{J}) & if s ∈ J \\ 1 ⊗ … ⊗ 1 ∈ R(W_{J}) & if s ∉ J. \end{matrix}

One easily verifies that the maps ρ_{−} are compatible with the restriction maps of R(−), i.e. for J ⊂ T ∈ S, we have a commutative diagram

\xymatrix{ℤ[V (Γ)]/I \ar[r]^{ρ_{T}} \ar[dr]^{ρ_{J}} & R(W_{T}) \ar[d]^{R_{J}^{T}} \\  & R(W_{J}).}

This implies that there is a ring homomorphism

ρ : ℤ[V (Γ)]/I → lim_{J∈S} R(W_{J}) ⊆ ⨁_{J∈S} R(W_{J}) : f ↦ (ρ_{J}(f))_{J∈S}.

Using the explicit basis for ℤ[V (Γ)]/I given above, one easily verfies that ρ is an isomorphism. □

Corollary 5.5. Let W be a right angled Coxeter group determined by a finite graph Γ. Then there is an isomorphism of rings

K^{n}(BW) ≅ \{ \begin{matrix}ℤ[[V (Γ)]]/(s(s + 2), st | s ∈ V (Γ), (s, t) ∉ E(Γ)) & if n = 0 \\ 0 & if n = 1. \end{matrix}

Here, ℤ[[V (Γ)]] is the formal power series ring with variables in V (Γ) and integer coefficients.

Proof. The version of the Atiyah-Segal completion theorem that is proven for infinite discrete groups admitting a cocompact model for the classifying space for proper actions in [10, Theorem 4.4.(b)] implies that

K^{n}(BW) = K_{W}^{n} (\underline{E}W)Ĵ,

where the ideal J is the kernel of the augmentation map K_{W}^{n} (\underline{E}W) → ℤ that maps vector bundles to their dimension. It therefore follows from the theorem above that

K^{n}(BW) = \{ \begin{matrix}(ℤ[V (Γ)]/I)Ĵ & if n = 0 \\ 0 & if n = 1. \end{matrix}

where the ideal J is the kernel of the ring homomorphism

ℤ[V (Γ)]/I → ℤ

that takes s ∈ V (Γ) to 1 ∈ ℤ and

I = (s^{2} − 1, st − s − t + 1 | s ∈ V (Γ), (s, t) ∉ E(Γ)).

Using the map s ↦ s + 1, we obtain an isomorphism

(ℤ[V (Γ)]/I)Ĵ ≅ (ℤ[V (Γ)]/(s(s + 2), st | s ∈ V (Γ), (s, t) ∉ E(Γ))){ \atop (s,s∈V (Γ))}.

Since there is an inclusion of ideals

(s(s + 2), st | s ∈ V (Γ), (s, t) ∉ E(Γ)) ⊆ { \atop (s, s ∈ V (Γ))},

and it is well-known that ℤ[V (Γ)]{ \atop (s,s∈V (Γ))} = ℤ[[V (Γ)]], we obtain the isomorphism

(ℤ[V (Γ)]/I)Ĵ ≅ ℤ[[V (Γ)]]/(s(s + 2), st | s ∈ V (Γ), (s, t) ∉ E(Γ)).
□

References

[1] Bredon, G.E., Equivariant cohomology theories , Lecture Notes in Mathematics 34, Springer (1967)

[2] Bridson, M.R. and Haefliger, A., Metric spaces of non-positive curvature Springer Verlag Vol. 319 (1999)

[3] Davis, M., The geometry and topology of Coxeter groups, Princeton University Press, vol. 32 (2008)

[4] Davis, J. F. and Lück, W., Spaces over a category and assembly maps in isomorphism conjectures in Kand L- theory, K-theory 15 (1998), 201–252.

[5] Jackowski, S., Families of subgroups and completion, Journal of Pure and Applied Algebra 37 (1985), 167–179

[6] Leary, I.J., A metric Kan-Thurston theorem, Journal of Topology 6(1) (2013), 251–284

[7] Lück, W., Survey on classifying spaces for families of subgroups, Infinite Groups: Geometric, Combinatorial and Dynamical Aspects, Springer (2005), 269–322.

[8] Lück, W, The Burnside ring and equivariant stable cohomotopy for infinite groups, Journal of Pure and Applied Mathematics Quaterly 1(3) (2005), 479–541

[9] Lück, W, Equivariant cohomological Chern characters, International Journal of Algebra and Computation 15 (2005), 1025–1052

[10] Lück, W, and Oliver, B., The completion theorem in K-theory for proper actions of a discrete group, Topology 40 (2001), 585–616

[11] Lück, W, and Oliver, B., Chern characters for the equivariant K-theory of proper G-CW-complexes, Cohomological methods in homotopy theory (2001), 217–24

[12] Lück, W, and Uribe, B., Equivariant principal bundles and their classifying spaces, Algebraic and Geometric Topology 14 (2014), 1925-1995

[13] Sánchez-García, R., Equivariant K-homology for some Coxeter groups, Journal of the London Mathematical Society, 75(3), (2007), 773–790

[14] t. Dieck, T., Transformation groups, Studies in Math. 8, de Gruyter (1987)