EQUIVARIANT VECTOR BUNDLES OVER CLASSIFYING SPACES FOR
PROPER ACTIONS

arXiv:1504.07358v3 [math.AT] 19 May 2016

DIETER DEGRIJSE AND IAN J. LEARY
Abstract. Let G be an infinite discrete group and let EG be a classifying space for proper
actions of G. Every G-equivariant vector bundle over EG 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 EG admitting a compatible collection of
representations of the finite subgroups of G that does not come from a G-equivariant (virtual)
vector bundle over EG. This implies that the Atiyah-Hirzeburch spectral sequence computing
the G-equivariant topological K-theory of EG has non-zero differentials. On the other hand,
we show that for right angled Coxeter groups this spectral sequence always collapses 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 the orbit category OF G is a category whose objects are the transitive G-sets G/H,
for all H ∈ F, and whose morphism are all G-equivariant maps between the objects. A
classifying space for proper actions of G, denoted by EG, is a proper G-CW-complex such
that the fixed point set EGH is contractible for every H ∈ F. The space EG is said to be
cocompact if the orbit space G \ EG = BG is compact. Many interesting classes of groups
G have cocompact models for EG, for example cocompact lattices in Lie groups, mapping
class groups of surfaces, Out(Fn ), CAT(0)-groups and word-hyperbolic groups. We refer the
reader to [9] for more examples and details.
Now assume G is an infinite discrete group admitting a cocompact classifying space for
proper actions EG. If
ξ : E → EG
is a G-equivariant complex vector bundle over EG (see Definition 2.3) and x is a point of
EG, then the fiber ξ −1 (x) is a complex representation of the finite isotropy group Gx . The
connectivity of the fixed point sets of EG ensures that these representations are compatible
(see Definition 2.1) with one another as x and hence Gx varies. Therefore, every G-equivariant
complex vector bundle over EG gives rise to a compatible collection of complex representations
of the finite subgroups of G, and hence to an element of
lim

G/H∈OF G

R(H).

Date: May 20, 2016.
The first author was supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).
1

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DIETER DEGRIJSE AND IAN J. LEARY

Here, limG/H∈OF G R(H) is the limit over the orbit category OF G of the contravariant representation ring functor
R(−) : OF G → Ab
G/H 7→ R(H).
Denoting the Grothendieck group of the abelian monoid of isomorphism classes of complex
G-vector bundles over EG by K0G (EG), one obtains a map
εG : K0G (EG) →

lim

G/H∈OF 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 EG if
there exists a (virtual) G-equivariant vector bundle over EG that maps to this collection under εG . One can also look at the corresponding situation for real (orthogonal) vector bundles
and real (orthogonal) representations and obtain the map
εG : KO0G (EG) →

lim

G/H∈OF G

RO(H).

The maps εG are equal to the edge homomorphisms of certain Atiyah-Hirzebruch spectral
sequences converging to K∗G (EG) and KO∗G (EG) (see (1) and (2)). Lück and Oliver proved
that (see Proposition 2.5) the map εG (real or complex) is rationally surjective, meaning that
a high enough multiple of every element in the target of εG is contained in the image of εG . In
the last paragraph of [12, p. 596] Lück and Oliver ask for an example of a group G admitting
a cocompact classifying space for proper actions EG such that εG 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. We also construct examples of groups
G admitting a cocompact EG with the following weaker property: G admits a compatible
collection of representations for its finite subgroups that cannot be realized as a G-vector
bundle over EG. However, for these examples we cannot exclude the possibility that there
exists a virtual vector bundle that maps to this collection of representations under εG . On
the other hand, these examples are more explicit and lower dimensional.
In the final section we show that for a right angled Coxeter group W , every compatible
collection of representations of the finite subgroups of W can be realized as a W -equivariant
vector bundle over EW , so that the map
εW : K0W (EW ) →

lim

W/H∈OF W

R(H).

is always surjective. Morever, we show that this map is actually an isomorphism and that
(see Theorem 2.4)
K1W (EW ) = 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 5.6).
Acknowledgement
We thank the referee, whose comments on an earlier version of this article were extremely
helpful.

EQUIVARIANT VECTOR BUNDLES OVER CLASSIFYING SPACES FOR PROPER ACTIONS

3

2. G-vector bundles and isotropy representations
Let G be a discrete group and let Γ be a Lie group. Let S be a family of finite subgroups
of G, i.e. any collection of finite subgroups of G that is closed under conjugation and passing
to subgroups. The orbit category OS G is a category whose objects are the transitive G-sets
G/H, for all H ∈ S, and whose morphism are all G-equivariant maps between the objects.
Definition 2.1. [12, p. 590] Let X be a G-CW-complex. A (G, Γ)-bundle over X is a Γprincipal bundle p : E → X, where E is a left G-space such that p is G-equivariant and
such that the left G-action and the right Γ-action on E commute. We denote the set of
isomorphism classes of (G, Γ)-bundles over X by Bdl(G,Γ) (X). For H ∈ F, let
RepΓ (H) = Hom(H, Γ)/Inn(Γ).
One can consider RepΓ (−) as a contravariant functor from OS G to Sets. An element of the
limit
A = ([αH ])H∈S ∈
lim RepΓ (H)
G/H∈OS G

is a called an S-compatible collection of Γ-representations. Given such an element A, let SA
be the family of subgroups of G × Γ consisting of conjugates of the subgroups of the form
{(h, αH (h)) |h ∈ H}
for all H ∈ S and let ES (G, A) be the universal G × Γ-CW-complex for the family SA .
Lemma 2.2. [12, Lemma 2.4] For every S-compatible collection of Γ-representations A =
([αH ])H∈S there exists a G-CW-complex BS (G, A) with isotropy in S satisfying the following
properties.
- The quotient map
π : ES (G, A) → Γ \ ES (G, A) = BS (G, A)
is a (G, Γ)-bundle over the G-CW-complex BS (G, A).
- The (G, Γ)-bundle π : ES (G, A) → BS (G, A) is universal in the sense that for every
G-CW-complex X with isotropy in S there is an isomorphism
∼
=

[X, BS (G, A)]G −
→ Bld(G,Γ) (X)
given by pulling back the univeral bundle π along a G-map X → BS (G, A).
- For every S ∈ S, the fixed point set BF (G, A)H is homotopy equivalent to BCΓ (αH ),
the classifying space of the centralizer of the image of αH in Γ.
If Γ = U(n) (Γ = O(n)) and S = F, the family of all finite subgroups of G, then RepΓ (H)
is the set of isomorphism classes of n-dimensional complex (real) representations of H. In
this case, an element of the limit
A = ([αH ])H∈F ∈

lim

G/H∈OF G

RepΓ (H)

is a called is called a compatible collection of complex (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 group of the abelian cancellative monoid of isomorphism
classes of finite dimensional complex (real) representations of H. Note that RepU(n) (H) is

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DIETER DEGRIJSE AND IAN J. LEARY

naturally a subset of R(H) and RepO(n) (H) is naturally a subset of RO(H). One can consider
R(−) as a functor from OF G to Ab. An element of the inverse limit
α = ([αH ])H∈F ∈

lim

G/H∈OF G

R(H)

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

G/H∈OF G

RepU(n) (H) ⊂

lim

G/H∈OF G

R(H).

The analogous statements for O(n, R) and RO also hold.
Now let X be a proper cocompact G-CW-complex, i.e. X has finite isotropy and the orbit
space G \ X has a finite number of cells, such that for every H ∈ F, the fixed point set X H
is non-empty and connected.
Definition 2.3 ([18]). 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
bundles that is G-equivariant. The set of isomorphims classes of complex (real) G-vector
bundles over X will be denoted by BdlG (X) (OBdlG (X)). For every x ∈ X, the fiber π −1 (x)
is denoted by Ex . We refer the reader to [12, Section 1] and [20, Section I.9] for elementary
properties of G-vector bundles over proper (cocompact) G-CW complexes.
Theorem 2.4. [12, 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
when X is cocompact, K0G (X) (KO0G (X)) is the Grothendieck group of the abelian monoid
of isomorphism classes of complex (real) G-vector bundles over X. In particular, for every
H ∈ F, K0G (G/H) (KO0G (G/H) ) is canonically isomorphic to R(H) (RO(H)).
As usual (see [13, Section 6] and [5, Th. 4.7]), the skeletal filtration of X induces AtiyahHirzebruch spectral sequences
(1)

E2p,q = HpG (X, KqG (G/−)) =⇒ Kp+q
G (X).

and
(2)

E2p,q = HpG (X, KOqG (G/−)) =⇒ KOp+q
G (X)

where HpG (X, −) denotes Bredon cohomology of X (see [2]).
Proposition 2.5. [13, Prop 5.8] If X is a cocompact G-CW complex then 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 zeroth Bredon cohomology group H0G (X, R(−)) (resp.
R(−) (resp. RO(−)), over the orbit cat-

H0G (X, RO(H))), equals the limit of the functor
egory OF G. Consider the edge homomorphisms

εG : K0G (X) → H0G (X, R(−))
and
εG : KO0G (X) → H0G (X, RO(−))

EQUIVARIANT VECTOR BUNDLES OVER CLASSIFYING SPACES FOR PROPER ACTIONS

5

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 εG ([π]) equals
([EeH ])H∈F ∈

lim

G/H∈OF G

RepU (n) (H) ⊂ H0G (X, R(−))

where [EeH ] denotes the isomorphism class in R(H) of the H-representation EeH . The corresponding statement for real G-vector bundles also holds. Note that it follows from Proposition
2.5 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 εG .
Recall that the classifying space for proper actions EG is a terminal object in the homotopy
category of proper G-CW complexes (e.g. [9, 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 → EG that is unique up to G-homotopy and induces commutative diagrams
K0G (X)
O

K0G (EG)

/ limG/H∈O G R(H)
F
♥♥6
♥
♥
♥♥
♥♥♥
♥♥♥

and

KO0G (X)
O

/ limG/H∈O G RO(H)
F
❧❧6
❧
❧
❧❧❧
❧❧❧
❧❧❧

KO0G (EG).

Hence, if a compatible collection α of virtual representations can be realized as a virtual
G-vector bundle over EG, it can also be realized as a virtual 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 EG 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 EG, i.e. so that the edge homomorphism
εG : K0G (EG) →

lim

G/H∈OF G

R(H).

is not surjective.
Let F = C4 ⋊ C2 be the dihedral group of order 8 where σ is generator for C4 and ε is
a generator of C2 . Let H = hσ 2 i be the center of F , which has order two and denote the
n-skeleton of the universal F/H-space X = 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 = hσ 2 i → U(1) = S 1 : σ 2 7→ −1.
Lemma 3.1. The isomorphism class [λ] is contained in R(H)F/H . For k ∈ Z, the multiple
k[λ] 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
τ : F → U (2)

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DIETER DEGRIJSE AND IAN J. LEARY

defined by





0 i
−1 0
τ (σ) =
and τ (ε) =
i 0
0 1
satisifies res([τ ]) = 2[λ]. Hence, k[λ] 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] (e.g. see [19]). Now suppose k is odd and
there exists an element [µ] − [ρ] ∈ R(F ) such that res([µ] − [ρ]) = k[λ]. There are integers
l, m, n and such that res([µ]) = l[tr] + m[λ], res([ρ]) = l[tr] + n[λ] 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]) = k[λ], if k is odd.

The following lemma uses the notation introduced above and will be cited in the next
section.
Lemma 3.2. Every F -equivariant complex line bundle over X 3 is isomorphic to the pullback
of an F -equivariant complex line 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 F -equivariant complex line bundles correspond to isomorphism classes of (F, S 1 = U(1))-bundles. Let π : E → X 3 be an F -equivariant complex line
bundle over and let [αH : H → U(1) = S 1 ] be the isomorphism class in RepS 1 (H) of the Hrepresentation induced on the fibers of π. If we set α{e} : {e} → S 1 , then A = ([αK ])K∈S ∈
limK∈S RepS 1 (K). It follows from Lemma 2.2 for Γ = S 1 , that in order to show that π
is the pullback of an F -equivariant complex line bundle over E(F/H) along the inclusion
i : X 3 → E(F/H), it suffices to show that every F -map from X 3 to BS (F, A) can be extended to an F -map from E(F/H) to BS (F, A). Here BS (F, A) is homotopy equivalent to
BS 1 = CP ∞ for all K ∈ S, again by Lemma 2.2. It follows from Bredon’s equivariant obstruction theory (see [2, Section II.1],[15, Th. I.5.1]) that the potential obstructions for extending
3
−
such a map lie in the relative Bredon cohomology groups Hn+1
F (E(F/H), X ; πn (BS (F, A) ))
for n ≥ 3. Since πn (CP ∞ ) is zero unless n = 2, the lemma is proven.

The idea for the following lemma is contained in [12, p 596].
Lemma 3.3. There exists an n ≥ 1 such that [λ] is not contained in the image of the edge
homomorphism
K0F (X n ) → R(H)F/H .
Proof. By [7, Theorem 5.1] for X = {∗}, F = {e, H} and EF = E(F/H), there are maps
αn : R(F )/I n → K0F (X n )

EQUIVARIANT VECTOR BUNDLES OVER CLASSIFYING SPACES FOR PROPER ACTIONS

7

that induce a map of inverse systems from {R(F )/I n }n≥0 to {K0F (X n )}n≥0 that induces 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 β1 : K0F (X n ) → R(F )/I making
the following diagram commute
αn

/ K0 (X n )
F
▲▲▲
▲▲▲εF
☎☎
☎
▲▲▲
☎
☎
▲&
☎
☎
☎
☎
R(H)F/H
☎☎ β1
r8
☎
r
☎
εF rr
☎
r
rr
☎☎
rrr

 ☎☎
α1
/ K0 (X 1 ).
R(F )/I

R(F )/I n

F

This shows that the image of the restriction map
R(F ) → R(H)F/H
coincides with the image of the edge homomorphism
K0F (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 [8, Th. A & Th. 8.3] there exists a compact n-dimensional locally CAT(0)cubical complex TX n equipped with a free cellular F/H-action and an F/H-equivariant map
tX n : TX n → X n that induces an isomorphism
(3)

∼
=

→ HF∗ (TX n )
HF∗ (X n ) −

for any equivariant cohomology theory H?∗ (·) (e.g. see [11, section 1]). (We remark that [8,
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 TX n in the
above is via the projection F → F/H. Now let Y n be the universal cover of TX n and let Γn
be the group of self-homeomorphisms of Y n that lift the action of F/H on TX n . Since F/H
acts freely on TX n , Γn acts freely on Y n . We conclude that Y n is an n-dimensional CAT(0)cubical complex on which Γn acts freely, cocompactly and cellularly. Since Yn is contractible,
this implies that Γn is torsion-free. By construction there is a surjection Γn → F/H whose
kernel Nn is the torsion-free group of deck transformation of the covering Y n → TX n . Now
define the group Gn to be the pullback of πn : Γn → F/H along F → F/H. Then Gn acts
on Y n via the quotient map Gn → Gn /H = Γn and fits into the short exact sequence
pn

1 → Nn → Gn −→ F → 1.
Note that the only non-trivial finite subgroup of Gn is H ∼
= C2 and that since Nn acts freely
on Y n , the Gn -equivariant quotient map Y n → Nn \ Y n = TX n induces an isomorphism ([12,
Lemma 3.5])
(4)

∼
=

→ K∗Gn (Y n ).
K∗F (TX n ) −

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DIETER DEGRIJSE AND IAN J. LEARY

Applying (3) and (4) to the compostion Y n → TX n → X n and the equivariant cohomology
theories K∗? (·) and H∗? (·, R(−)) with ∗ = 0, we obtain a commutative diagram
∼
=

K0F (X n )

/ K0 (Y n )
Gn
εGn

εF



∼
=

R(H)F/H



/ limG /S∈O G R(S).
n
F n

The fact that this diagram commutes can be seen as follows. Using equivariant cellular
approximation, we may assume that the map X n → Y n is cellular. By considering the
inclusion of zero-skeleta in n-skeleta, naturality yields a commutative diagram
K0F (X n )

∼
=

/ K0 (Y n )
Gn

/ K0 (Y 0 ).
Gn



K0F (X 0 )

The edge homomorphism εF : K0F (X n ) → R(H)F/H ⊆ K0F (X 0 ) coincides by construction
with K0F (X n ) → K0F (X 0 ) once we restrict the codomain, and similarly for εGn . Therefore,
commutativity follows.
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
K0F (X n ) → R(H)F/H
it follows from the commutative diagram above that the compatible system of representations
(λ ◦ pn|S )S∈F ∈

lim

Gn /S∈OF Gn

R(S) = H0F (Gn , R(−)).

does not lie in the image of the edge homomorphism
εGn : K0Gn (Y n ) →

lim

Gn /S∈OF Gn

R(S).

Recall from [3] 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 Gn acts
cellularly properly and cocompactly on the CAT(0)-cube complex Yn , we deduce that Yn is
a cocompact model for EGn . To summarize, we have constructed a group G = Gn with
a cocompact classifying space for proper actions EG 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 EG.
We remark that Wolfgang Lück has shown us another quite different way to find a finite
group F and an F -CW-complex X that satisfy Lemma 3.3; any such pair could be used to
construct a group with similar properties to the group G = Gn .
4. Real vector bundles
One could apply the techniques of the previous section in the real setting to obtain a group
G with cocompact classifying space for proper actions EG so that the edge homomorphism
εG : KO0G (EG) →

lim

G/H∈OF G

RO(H)

EQUIVARIANT VECTOR BUNDLES OVER CLASSIFYING SPACES FOR PROPER ACTIONS

9

is not surjective. Here one would need the real version of [7, Theorem 5.1], which also holds
as explained in the paragraph below [7, Theorem 5.1].
Instead we give an explicit description of a group G that admits R2 as a cocompact model
for EG and admits a compatible collection of real representations of its finite subgroups that
cannot be realized as a real G-vector bundle over R2 .
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 R2
with the CW-structure coming from the standard tesselation by unit squares with vertices at
Z2 , 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 R2 into a Γ-CW-complex.
The translation subgroup T of Γ has index four, and consists of the elements (x, y) 7→
(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 Z2 , which are of the form (x, y) 7→
(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) 7→ (2m + 1 − x, 2n + 1 + y) and (x, y) 7→ (2m + 1 + x, 2n + 1 − y).
The quotients T \R2 , N \R2 and Γ\R2 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 \R2 .
Now let F be a copy of C4 and let H ∼
= C2 be the index two subgroup of F . The group
G is defined as the pullback of the two maps Γ → Γ/N ∼
= C2 and F → F/H ∼
= C2 . By
2
construction the group G admits R as a cocompact model for EG, and fits into a short exact
sequence
p
1→N →G−
→F →1
such that every finite subgroup of G maps onto a subgroup of H under p.
Now let
λ : H → O(1, R) = C2
be the 1-dimensional real sign representation 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 k[λ] 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 C4 are up to isomorphism the onedimensional trivial representation representation, the one dimensional sign representation of
F/H = C2 and one 2-dimensional faithful representation in which the elements of order four
act as rotations by ± π2 . The restriction of the first two of the representations to H gives the
trivial one dimensional representation of H, while the restriction to 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.

10

DIETER DEGRIJSE AND IAN J. LEARY

Figure 1. A wallpaper pattern for Γ = p2gg
This shows that k[λ] 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 = C2 and then acting via the antipodal map. View S 2 as the 2-skeleton of S ∞ . Every F -equivariant orthogonal real line bundle over S 2 is isomorphic to the pullback of an
F -equivariant orthogonal real line 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
isomorphism classes of F -equivariant orthogonal real line bundles correspond to isomorphism
classes of (F, C2 )-bundles. Now let ξ be an (F, C2 )-bundle over S 2 with fibers A = (ξS ) ∈
limS∈S RepC2 (S). By Lemma 2.2, it suffices to show that every F -map f : S 2 → BS (F, A) can
be extended to an F -map f˜: S ∞ → BS (F, A). Again by Lemma 2.2, BS (F, A)S ∼
= BC2 =
RP ∞ for all S ∈ S. It follows from Bredon’s equivariant obstruction theory (see [2, Section
II.1],[15, Th. I.5.1]) that the potential obstructions for extending such a map lie in the relative
∞
2
−
∞
Bredon cohomology groups Hn+1
F (S , S ; πn (BS (F, A) )) for n ≥ 2. Since πn (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 = C2 and then acting via the
antipodal map. There does not exist a real F -vector bundle ξ : E → S 2 such that the representation of H on the fibers of ξ is isomorphic to λ.
Proof. Consider the infinite dimensional sphere S ∞ as a the universal C2 -space EC2 , where
C2 acts via the antipodal map and let F act on S ∞ via first projection onto F/H = C2 and
then acting via C2 . Now assume that there exists a real F -vector bundle ξ : E → S 2 such
that the representation of H on the fibers of ξ is isomorphic to λ. By Lemma 4.2 there exists
a real F -vector bundle ξ : E → S ∞ such that the representation of H on the fibers of ξ is
isomorphic to λ. By pulling back this bundle along the inclusion S n → S ∞ , there also exists
a real F -vector bundle ξ : E → S n such that the representation of H on the fibers of ξ is
isomorphic to λ, for every n ≥ 2.

EQUIVARIANT VECTOR BUNDLES OVER CLASSIFYING SPACES FOR PROPER ACTIONS

11

By the real version of [7, Theorem 5.1] (see comments below [7, Theorem 5.1] ), there are
maps
αn : RO(F )/I n → KO0F (S n )
that induce a map of inverse systems from {RO(F )/I n }n≥0 to {KO0F (S n )}n≥0 that in turn
induces an isomorphism of pro-rings. Here I is the kernel of the restriction map RO(F ) →
RO(H). This implies that for sufficiently large n ≥ 1 there exists a map β1 : KO0F (S n ) →
R(F )/I making the following diagram commute
αn

/ KO0 (S n )
F
◆◆◆
◆◆◆εF
✁✁
✁
◆◆◆
✁
✁
◆◆'
✁
✁
✁
✁✁
RO(H)F/H
✁✁ β1
♣7
✁
✁
εF ♣♣♣♣
✁
♣♣♣
✁✁

♣♣♣
 ✁✁
α1
/ KO0 (S 1 ).
RO(F )/I

RO(F )/I n

F

This shows that the image of the restriction map
RO(F ) → RO(H)F/H
coincides with the image of the edge homomorphism
KOF0 (S n ) → RO(H)F/H ,
implying that the H-representations coming from the fibers of any real F -vector bundle over
S n can be extended to virtual F -representations. However, 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 representation of H on the fibers
of ξ is isomorphic to λ.

Consider the projection p : G → F and the compatible system of real orthogonal representations
([λ ◦ p|S ])S∈F ∈ lim RO(S) = H0G (EG, RO(−)),
G/S∈OF G

and assume that there exists a real G-vector bundle ξ : E → R2 that realizes it. Since the
kernel of p : G → F 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 = C2 , followed by the antipodal map. Morever, the representation 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 → R2 that realizes the compatible system
of real orthogonal representations (λ ◦ p|S )S∈F .
Lemma 4.4. Let G be any discrete group with normal subgroup N and let X be a proper
G-CW-complex. If ξ : E → X is a G-vector bundle over X such that N ∩ Gx 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.

12

DIETER DEGRIJSE AND IAN J. LEARY

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 [10, Section 6.1.]), i.e. assume ξ is a pullback
G ×OH V

/ G/H
O
p

r

E

ξ

/X

of the G-vector bundle G ×H V → G/H along the G-map p : X → G/H where H is some
finite subgroup of G and V is a finite dimensional real H-representation such that H ∩ N acts
trivially on V . Consider the pullback diagram
Q ×π(H) V

/ Q/π(H)
O

O

w

N \p

P

q

/ N \X

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 : (g, v, x) 7→ (π(g), v, x).
It is easy to check that ψ yields a well-defined morphism of Q-equivariant bundles over N \ X.
Moreover, since ψ is a fiberwise linear map of Q-vector bundles that is a fiberwise isomorphism,
it follows that ψ is a homeomorphism.
Now consider the general case. Let 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 is finite
subgroup of G and ξ|U is (homeomorphic to) the pullback
G ×OH V

/ G/H
O
p

ξ|U

ξ

/U

of the G-vector bundle G ×H V → G/H along the G-map p : U → G/H. By the above, the
quotient diagram
Q ×π(H) V

/ Q/π(H)
O

O

N \p

N \ ξ|U

N \ξ

/ N \U

is a pullback diagram. Since N \ U is an open Q-neighbourhood of x, it follows that N \
ξ : N \ E → N \ X is a Q-vector bundle.

We finish this section by explaining how a similar approach to the one above can be used
to produce a group G admitting a three dimensional cocompact model for EG that has a
compatible system of one-dimensional complex representations that cannot be realized as a
complex G-vector bundle over EG. As in Section 3, let F = C4 ⋊ C2 be the dihedral group of
order 8 where σ is a generator for C4 . Let H = hσ 2 i be the center of F , which has order two

EQUIVARIANT VECTOR BUNDLES OVER CLASSIFYING SPACES FOR PROPER ACTIONS

13

and denote the 3-skeleton of the universal F/H-space X = E(F/H) by X 3 . We let F act on
X and X 3 via the projection onto F/H. Consider the complex 1-dimensional representation
λ : H = hσ 2 i → U(1) = S 1 : σ 2 7→ −1.
By [8, Th. A & Th. 8.3] there exists a compact 3-dimensional locally CAT(0)-cubical complex
TX 3 equipped with a free cellular F/H-action, an F/H-equivariant map tX 3 : TX 3 → X 3 and
an isometric cellular involution τ on TX 3 that commutes with the F/H-action on TX 3 and
the map tX 3 such the induced F/H-equivariant map
hτ i \ TX 3 → X 3
is a homotopy equivalence. Note that F/H acts freely on hτ i \ TX 3 since it acts freely on X 3 .
Hence TX 3 is also the 3-skeleton of a universal F/H-space Z. So we may continue assuming
that Z = X and hτ i \ TX 3 = X 3 .
Now let Y be the univeral cover of TX 3 and let Γ be the group of self-homeomorphism of Y
that lifts the action of F/H ⊕ hτ i on TX 3 . Then Y is a 3-dimensional CAT(0)-cubical complex
on which Γ acts properly, compactly and cellularly. By construction there is a surjection
α : Γ → F/H ⊕ hτ i whose kernel Ker(α) is the torsion-free group of deck transformations of
Y → TX 3 . Let π denote the composition of α with the projection of F/H ⊕ hτ i onto F/H.
Since F/H acts freely on TX 3 and every finite subgroup of Γ must fix a point of Y since Y is
CAT(0), it follows that every finite subgroup of Γ is contained in the kernel of π, which we
denote by N . Now define the group G to be the pullback of π : Γ → F/H along F → F/H.
Then G acts on Y via the quotient map G → G/H = Γ that fits into the short exact sequence
p

1→N →G−
→ F → 1.
such that p maps all the finite subgroup of G onto a finite subgroup of H and N \ Y = X 3 .
Let F be the family of finite subgroups of G, note that Y is a three dimensional cocompact
model for EG and suppose that there exists a G-vector ξ : E → Y whose fibers give rise to
the compatible system of representations
([λ ◦ p|S ])S∈F ∈

lim

G/S∈OF G

R(S).

Applying Lemma 4.4, we obtain an F -equivariant complex line bundle N \ ξ : N \ E → X
such that the representation of H on the fibers of N \ ξ is isomorphic to λ. By Lemma 3.2,
this bundle can be extended to an F -equivariant complex line bundle over X = E(F/H).
We now continue in a similar fashion as in the proof of Lemma 4.3 to conclude that [λ] is
contained in the image of the restriction map R(F ) → R(H)F/H , which contradicts Lemma
3.1. We conclude that the bundle ξ cannot exist.
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 = hS | s2 for all s ∈ V (Γ) and (st)2 if (s, t) ∈ E(Γ)i.
Note that W fits into the short exact sequence
p

1→N →W −
→F =

M
s∈S

C2 → 1

14

DIETER DEGRIJSE AND IAN J. LEARY

where p takes s ∈ S to the generator of the C2 -factor corresponding to s. A subset
J ⊆ S is
L
called spherical if the subgroup WJ = hJi is finite (and hence isomorphic to s∈J C2 ). The
empty subset of S is by definition spherical. We denote the poset of spherical subsets of S
orderded by inclusion by S. If J ∈ S, then WJ is called a spherical subgroup of W , while
a coset wWJ 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 EW (see [4, 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
BW = W \ EW is contractible. We refer the reader to [4] 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 . Given an abelian group A, we denote by
A : OF W → Ab
the trivial functor that takes all objects to A and all morphism to the identity map. One can
verify that
(5)
H∗ (EW, A) ∼
= H∗ (BW, A).
W

Lemma 5.1. Let A = ([p|H ])H∈F ∈ limW/H∈OF W RepF (H). For every k ≥ 0, the contravariant functor
OF W → Ab : W/H 7→ πk (BF (W, A)H )
equals the trivial functor πk (BF ).
Proof. Let EF be a contractible F -CW -complex with free F -action and consider the product
space EW × EF . This space becomes a (W × K) − CW -complex by letting (w, f ) ∈ W × F
act on (x, y) ∈ EW × EF as
(w, f ) · (x, y) = (w · x, p(w)f · y).
One checks that with this action EW × EF is a model for EF (W, A), i.e. (EW × EF )K is
contractible when K ∈ FA and empty otherwise. By definition, it follows that EW × BF
is a model BF (W, A), where W acts on trivially on the second coordinate. Since EW H is
contractible for every H ∈ F, the lemma follows easily.

Let Γ be either the orthogonal group O(n, R) or the unitary group U(n).
Lemma 5.2. Every element of
lim

W/H∈OF W

RepΓ (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 C2 ’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 simultaneously diagonalized (e.g. see [6, Th. 1.3.12]), it follows
that the image of every homomorphism H → Γ is conjugate to a finite subgroup of Γ consisting

EQUIVARIANT VECTOR BUNDLES OVER CLASSIFYING SPACES FOR PROPER ACTIONS

15

of diagonal matrices. Hence, every element of limW/H∈OF W RepΓ (H) is of the form ([αH ])H∈F
where αH : H → Γ is a homomorphism whose image lands in the finite abelian subgroup of
Γ consisting of diagonal matrices. Since every finite subgroup of W is subconjugate to a
spherical subgroup WJ , the compatibility of the representations tells us that ([αH ])H∈F is
completely determined by the homomorphisms αhsi : hsi → Γ, for s ∈ S. Since the images of
the αhsi are diagonal, they commute. Therefore, one can define the homomorphism
M
X
λ: F =
C2 → Γ : (σs )s∈S 7→
αhsi (σs ).
s∈S

s∈S

The compatibility of the representations implies that
([λ ◦ p|H ])H∈F = ([αH ])H∈F ,
proving the lemma.



The following theorem applies to both complex and real representations and vector bundles.
Theorem 5.3. Let W be a right angled Coxeter group. Every compatible collection of representations of the finite subgroups of W can be realized as a W -equivariant vector bundle over
the Davis complex Σ = EW .
Proof. Consider A = ([p|H ])H∈F ∈ limW/H∈OF W RepF (H). It follows from Lemma 2.2 that
the existence of a (W, A)-bundle over Σ follows from the existence a W -map Σ → BF (W, A).
Since by Lemma 5.1, the contravariant functor
πk (BF (W, A)− ) : OF (W ) → Ab : W/H 7→ πk (BF (W, A)H )
equals the trivial functor πk (BF ) for all k ≥ 0, it follows from (5) and the contractibility of
BW that the Bredon cohomology groups
−
Hk+1
W (Σ, πk (BF (W, A) ))

are zero for all k ≥ 0. Since there certainly exists a W -map from the 0-skeleton of Σ to
BF (W, A), it follows from Bredon’s equivariant obstruction theory that there exists a W -map
Σ → BF (W, A).
Now consider a compatible collection of representations of the finite subgroups of W . By
Lemma 5.2, this collection is of the form
([λ ◦ p|H ])H∈F ∈

lim

W/H∈OF W

RepΓ (H)

for some group homomorphism λ : F → Γ. Letting A = ([p|H ])H∈F , it follows from the above
that there exists a (W, A)-bundle ξ : E → Σ. If Γ = O(n, R) then
ξ : E ×F Rn → 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 Cn → Σ
is a complex W -vector bundle over Σ that realizes ([λ ◦ p|H ])H∈F . Here F acts on Rn or Cn
via the map λ.

Lemma 5.4. If W is a right angled Coxeter group, then HnW (Σ, R(−)) = 0 for all n > 0, and
H0W (Σ, R(−)) is free abelian of rank equal to the number of spherical subgroups of W .

16

DIETER DEGRIJSE AND IAN J. LEARY

Proof. This is proven in much the same way as the corresponding result for homology in [17].
In more detail, one uses the cubical structure on Σ, in which there is one orbit of n-cubes
with stabilizer isomorphic to (C2 )n for each n-tuple of commuting elements of S. (For each
n ≥ 0, for each spherical subgroup WJ ∼
= (C2 )n and for each w ∈ W , the subposet consisting
of all special cosets contained in wWJ is order isomorphic to the poset of faces of an n-cube.
Furthermore this isomorphism is equivariant for the stabilizer subgroup wWJ w−1 ∼
= (C2 )n ,
acting on the n-cube as the group generated by reflections in its coordinate planes. The
realizations of these subposets are the cubes that make up the cubical structure on Σ. For
more details concerning the cubical structure on Σ see [4, Ch. 1.1-1.2 or Ch. 7].) Since
the stabilizer of a cube of strictly positive dimension acts non-trivially on the cube, this
cubical structure is not a W -CW-structure on Σ. However, its barycentric subdivision is a
simplicial complex naturally isomorphic to the realization of the poset W S as described in
the introduction to this section.
Let Σn denote the n-skeleton of Σ with the cubical structure. Firstly, Σ0 consists of a
single free W -orbit of vertices, so H∗W (Σ0 ; 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 C2 acting by x 7→ −x (i.e., swapping the ends of the
interval). Note that I is equivariantly isomorphic to the Davis complex for the Coxeter group
C2 . Let ∂I denote the two end points {−1, 1}. Make I into a C2 -CW-complex, for example
by taking three 0-cells in two orbits at the points −1, 0 and 1, and one free orbit of 1-cells
consisting of the two intervals [−1, 0] and [0, 1]. The cellular C2 -Bredon cochain complex for
the pair (I, ∂I) with coefficients in R(−) is a cochain complex of free abelian groups in which
the degree zero term has rank two, the degree one term has rank one, and all other terms
are trivial. A direct computation with this cochain complex shows that HCm2 (I, ∂I; R(−)) is
isomorphic to Z for m = 0 and is zero for m > 0.
Next consider I n with C2n acting as the direct product of n copies of the above action
of C2 on I. This is the Davis complex for the Coxeter group C2n . Since the representation
ring of a direct product of finite groups is naturally identified with the tensor product of
the representation rings [19, Ch. 3.2], the C2n -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 C2 Bredon cochain complex for (I, ∂I) with coefficients in R(−). (If one wants to think about
this cochain complex geometrically, it arises from the (C2 )n -CW-structure on I n in which
the cells are the direct products of the cells arising in the C2 -CW-structure on I.) Since
these cochain complexes consist of finitely generated free abelian groups, there is a Künneth
formula as described in for example [16, Thrm 60.3]. Since HC∗ 2 (I, ∂I; R(−)) is free abelian
the Künneth formula implies that
HC∗ 2n (I n , ∂I n , R(−))

∼
=

n
O

HC∗ 2 (I, ∂I; R(−)).

i=1

It follows that for each n, HCmn (I n , ∂I n ; R(−)) is isomorphic to Z for m = 0 and is zero for
2
m > 0.
m (Σn , Σn−1 ; R(−)) is zero for m > 0 and
From these computations, it follows easily that HW
is isomorphic to a direct sum of copies of Z indexed by the W -orbits of n-cubes in Σ. By
m (Σn ; R(−)) is zero for m > 0 and isomorphic to a direct
induction on n one sees that HW
sum of copies of Z indexed by the W -orbits of cubes of dimension at most n for m = 0. The

EQUIVARIANT VECTOR BUNDLES OVER CLASSIFYING SPACES FOR PROPER ACTIONS

17

claimed result follows, since the W -orbits of cubes in Σ are in bijective correspondence with
the spherical subgroups of W .

Before stating our theorem concerning K∗W (EW ), we make some remarks concerning the
representation ring of a direct sum of copies of the cyclic group C2 , indexed by a (finite) set
S. For any finite group G, the collection of all isomorphism types of 1-dimensional complex
representations of G is an abelian group, with product given by taking the tensor product of
representations. Furthermore, this group is naturally isomorphic to the group Hom(G, U(1)).
In the case when G is abelian, every irreducible representation of G is 1-dimensional, and so
Hom(G, U(1)) forms a basis for the additive group of the representation ring. In this way the
representation ring R(G) is naturallyL
isomorphic to the integral group algebra of the group
Hom(G, U(1)). In the case when G = s∈S C2 is a direct sum of copies of C2 indexed by S, we
may view G as a vector space over the field of two elements, in which case Hom(G, U(1)) may
be identified with the dual space. For s ∈ S, let s∗ denote the 1-dimensional representation
of G with the properties that s∗ (s) = −1 and s∗ (t) = 1 for t ∈ S − {s}. let S ∗ denote the set
of these representations: S ∗ := {s∗ | s ∈ S}. In terms of vector spaces over the field of two
elements, S ∗ ⊆ Hom(G, U(1)) is the dual basis to the set S ⊆ G. The set S ∗ generates the
representation ring of G, giving rise to the following presentation:
R(G) = Z[S ∗ ]/(s∗2 − 1 | s ∈ S),
in which the monomials s∗1 s∗2 · · · s∗k for all subsets {s1 , . . . , sk } ⊆ S correspond to the irreducible representations.
L
Suppose now that L
J is a subset of S. The inclusion J ⊆ S identifies H = s∈J C2 with
a subgroup of G =
s∈J C2 . The induced map R(G) → R(H) of representation rings is
described easily in terms of the above ring presentation: for s ∈ J, s∗ ∈ R(G) restricts to
s∗ ∈ R(H), while for s ∈
/ J, s∗ ∈ R(G) restricts to 1 ∈ R(H).
Now suppose that Γ is a graph with vertex set V (Γ) = S, and let W be the right angled
Coxeter
group associated to Γ. The abelianization of W is naturally identified with G =
L
C
. There is a unique equivariant map α : EW → ∗, from the W -space EW toLa point
2
s∈S
∗, viewed as a G-space with trivial action. If J is a spherical subset
L of S then WJ = s∈J C2
maps isomorphicallyLto the corresponding subgroup of G =
s∈S C2 . If x ∈ EW is a 0cell fixed by WJ = s∈J C2 , then α(x) = ∗, and this map is WJ -equivariant. The induced
map α∗ : K∗G (∗) → K∗W (EW ), and the composite map K∗G (∗) → K∗WJ ({x}) will be used in
the statement and proof of our theorem. If we identify R(G) with K0G (∗) and R(WJ ) with
K0WJ ({x}), then the composite is identified with the restriction map.
Theorem 5.5. Let W beL
the right angled Coxeter group determined by a finite graph Γ, with
vertex set S, and let G = s∈S be the abelianization of W . The map α∗ : K∗G (∗) → K∗W (EW )
is surjective in each degree. In particular, K1W (EW ) = 0 and there is a ring isomorphism
2
K0W (EW ) ∼
/ E(Γ)).
= Z[S ∗ ]/(s∗ − 1, s∗ t∗ − s∗ − t∗ + 1 | s ∈ S = V (Γ),(s, t) ∈

It follows that K0W (EW ) ∼
= Zd as an abelian group, where d is the number of spherical subgroups of W .
Proof. Consider the Atiyah-Hirzebruch spectral sequence (1)
E2p,q = HpW (EW, KqW (W/−)) =⇒ Kp+q
W (EW )

18

DIETER DEGRIJSE AND IAN J. LEARY

where KqW (W/−) = R(−) if q is even and KqW (W/−) = 0 if q is odd (see [12, Th. 3.2]). In
the lemma above, we proved that HkW (Σ, R(−)) = 0 for k > 0. It therefore follows that
n H0 (EW, R(−)) = lim
W/H∈OF W R(H) if n = 0
W
KnW (EW ) =
0
if n = 1.
Let I be the ideal
(s∗ 2 − 1, s∗ t∗ − s∗ − t∗ + 1 | s ∈ S, (s, t) ∈
/ E(Γ))
in the polynomial ring Z[S ∗ ]. Note that as an abelian group Z[S ∗ ]/I is free, with basis
elements the commuting products s∗1 . . . s∗k , for all J = {s1 . . . , sk } ∈ S (The case J = ∅
corresponds to the unit of Z[V (Γ)]/I ). This shows that
Z[S ∗ ]/I ∼
= Zd
as an abelian group, where d is the number of spherical subgroups of W .
We claim there is an isomorphism of rings
lim
R(H) ∼
= Z[S ∗ ]/I.
W/H∈OF W

Since every finite subgroup of W is subconjugate to a spherical subgroup of W , it follows that
lim
R(H) ∼
= lim R(WJ )
J∈S

W/H∈OF W

as rings. By the remarks in the paragraph preceeding the statement of the theorem, there
are ring isomorphisms
R(WJ ) = Z[J ∗ ]/(s∗ 2 − 1 | s ∈ J),

R(G) = Z[S ∗ ]/(s∗ 2 − 1 | s ∈ S),

which are natural for inclusions J ⊆ J ′ ⊆ S. From this it follows that the natural ring
homomorphism
ρ : R(G) →
lim
R(H)
W/H∈OF W

is surjective, and that limW/H∈OF W R(H) is isomorphic to the ring described in the statement;
in particular its additive group is free abelian of the same rank as K0W (EW ). Since ρ factors
through K0W (EW ), the claimed isomorphism follows.

Before stating our corollary concerning
K∗ (BW ), we recall some facts from [1] concernL
ing K∗ (BG), where as above G =
s∈S C2 . For any finite group H, Atiyah showed that
Ki (BH) = 0 for i odd, and that K2i (BH) is naturally isomorphic to the completion of the
representation ring R(H) at its augmentation ideal. To discuss the case of G, it is convenient to take new generators for R(G); replace the irreducible representation s∗ by the degree
zero virtual representation s = s∗ − 1. With respect to these generators one obtains the
presentation
R(G) = Z[S]/(s(s + 2) | s ∈ S),
L
where S = {s | s ∈ S}. If H = s∈J C2 , then of course there is a similar description of R(H),
which is natural for the inclusion J ⊆ S. Note that if s ∈
/ J, then the image of s under the
restriction map R(G) → R(H) is zero.
Completing R(G), as described above, with respect to its augmentation ideal gives rise to
the following presentation for the ring K0 (BG):
K0 (BG) = Z[[S]]/(s(s + 2) | s ∈ S),

EQUIVARIANT VECTOR BUNDLES OVER CLASSIFYING SPACES FOR PROPER ACTIONS

19

which is natural for the inclusion J ⊆ S, and so also describes the induced map K0 (BG) →
K0 (BH). The additive group of this ring is the direct sum of one copy of Z, generated by 1,
and for each non-empty subset J Q
⊆ S, one copy of the 2-adic integers, Z2 , consisting of the
set of power series in the element s∈J s with zero constant term.

Corollary 5.6. Let W be the right angled
Coxeter group determined by a finite graph Γ with
L
vertex set S = V (Γ), and let G =
C
s∈S 2 be the abelianization of W . The induced map
∗
∗
K (BG) → K (BW ) is surjective in each degree. In particular K1 (BW ) = 0 and there is a
ring isomorphism
K0 (BW ) ∼
/ E(Γ)).
= Z[[S]]/(s(s + 2), st | s ∈ S, (s, t) ∈
Here, Z[[S]] is the formal power series ring with Z coefficients in the variables S = {s | s ∈ S}.

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 [12,
Theorem 4.4.(b)] implies that
Kn (BW ) = KnW (EW )Jˆ,
where the ideal J is the kernel of the augmentation map KnW (EW ) → Z that maps vector
bundles to their dimension. Changing variables in the above theorem to s = s∗ − 1, we see
that Ki (BW ) = 0 for i odd and that K0 (BW ) is the completion of the ring
/ E(Γ))
Z[S]/(s(s + 2), st | s ∈ S, (s, t) ∈
with respect to the ideal generated by the set S = {s | s ∈ S}. This completion is the ring
described in the statement.

There is an alternative proof of Corollary 5.6 that does not use Theorem 5.5 or results
from [12]. Instead one uses a description of W as a free product with amalgamation. If the
graph Γ is not a complete graph, then there is an expression Γ = Γ1 ∪ Γ2 , Γ3 = Γ1 ∩ Γ2 , in
which each Γi is a full subgraph of Γ and has fewer vertices than Γ. This gives an expression
for W as a free product with amalgamation W = W1 ∗W3 W2 . From this one obtains a
Mayer-Vietoris sequence that can be used to compute K∗ (BW ). To establish Corollary 5.6,
one shows by induction on |S| that K∗ (BW ) is as described and that for each J ⊆ S, the
map K∗ (BW ) → K∗ (BWJ ) is a split surjection.
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DIETER DEGRIJSE AND IAN J. LEARY

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