Equivariant K-homology for some Coxeter groups
Rub'en S'anchez-Garc'ia*
20th April 2006
Abstract
We obtain the equivariant K-homology of the classifying space E_W for W
a right-angled or, more generally, an even Coxeter group. The key result
is a formula for the relative Bredon homology of E_W in terms of Coxeter
cells. Our calculations amount to the K-theory of the reduced C*-algebra
of W , via the Baum-Connes assembly map.
1 Introduction
Consider a discrete group G. The Baum-Connes conjecture [1] identifies the K-
theory of the reduced C*-algebra of G, C*r(G), with the equivariant K-homology
of a certain classifying space associated to G. This space is called the classi*
*fying
space for proper actions, written E_G. The conjecture states that a particular
map between these two objects, called the assembly map,
~i: KGi(E_G) -! Ki(C*r(G)) i 0 ,
is an isomorphism. Here the left hand side is the equivariant K-homology of
E_G and the right hand side is the K-theory of C*r(G). The conjecture can be
stated more generally [1, Conjecture 3.15].
The equivariant K-homology and the assembly map are usually defined in
terms of Kasparov's KK-theory. For a discrete group G, however, there is
a more topological description due to Davis and L"uck [4], and Joachim [10]
in terms of spectra over the orbit category of G. We will keep in mind this
topological viewpoint (cf. Mislin's notes in [14]).
Part of the importance of this conjecture is due to the fact that it is rela*
*ted
to many other relevant conjectures [14, x7]. Nevertheless, the conjecture itself
allows the computation of the K-theory of C*r(G) from the KG -homology of E_G.
In turn, this K-homology can be achieved by means of the Bredon homology of
E_G, as we explain later.
In this article we focus our attention on (finitely generated) Coxeter group*
*s.
These groups are well-known in geometric group theory as groups generated
by reflections (elements of order 2) and only subject to relations in the form
(st)n = 1. Coxeter groups have the Haagerup property [2] and therefore satisfy
the Baum-Connes conjecture [8].
For W a Coxeter group, we consider a model of the classifying space E_W
called the Davis complex. We obtain a formula for the relative Bredon homology
____________________________*
Funded by the EPSRC and the School of Mathematics, University of Southampton
1
of this space in terms of Coxeter cells (Theorem 5.2). From this, we can de-
duce the Bredon homology of E_W in same cases; for instance, for right-angled
Coxeter groups and, more generally, even Coxeter groups (Theorems 7.1 and
8.1). The equivariant K-homology of E_W follows immediately. Since Coxeter
groups satisfy Baum-Connes, our results also amount to the K-theory of the
corresponding reduced C*-algebra.
These results appear in the author's PhD thesis [17, Chapter 5]. I would like
to thank my PhD supervisor Ian Leary for his guidance through this research
project, and particularly for suggesting the use of the Coxeter cell structure *
*and
relative Bredon homology.
2 Preliminaries
2.1 Classifying spaces
The classifying space E_G that appears in the Baum-Connes conjecture is a
particular case of a more general construction.
Let G be a discrete group. A G-CW-complex is a CW-complex with a G-
action permuting the cells and such that if a cell is sent to itself, it is don*
*e by
the identity map. Let F be a non-empty family of subgroups of G closed under
conjugation and passing to subgroups. A model for EFG is a G-CW-complex
X such that (1) all cell stabilizers are in F; (2) for any G-CW-complex Y with
all cell stabilizers in F, there is a G-map Y ! X, unique up to G-homotopy
equivalence. The last condition is equivalent to the statement that for each
H 2 F, the fixed point subcomplex XH is contractible.
For the family with just the trivial subgroup, we obtain EG, a contractible
free G-CW-complex whose quotient BG is the classifying space for principal
G-bundles; when F = Fin(G), the family of all finite subgroups of G, this is the
definition of E_G.
It can be shown that general classifying spaces EFG always exists. They are
clearly unique up to G-homotopy. See [1, x2] or [14] for more information and
examples of E_G.
2.2 Bredon (co)homology
Given a group G and a family F of subgroups, we will write OFG for the orbit
category. The objects are left cosets G=K, K 2 F, and morphisms the G-
maps OE : G=K ! G=L. Such a G-map is uniquely determined by its image
OE(K) = gL, and we have g-1Kg L. Conversely, such g 2 G defines a G-map,
which will be written Rg.
A covariant (resp. contravariant) functor M :OFG ! Ab is called a left
(resp. right) Bredon module. The category of left (resp. right) Bredon modules
and natural transformations is written G-Mod F (resp. Mod F-G). It is an abelian
category, and we can use homological algebra to define Bredon homology (see
[14, pp. 7-10]). Nevertheless, we give now a practical definition.
Consider a G-CW-complex X, and write Iso(X) for the family of isotropy
subgroups {stab(x), x 2 X}. Let F be a family of subgroups of G containing
Iso(X), and M a left Bredon module. The Bredon homology groups HFi(X; M)
can be obtained as the homology of the following chain complex (C*, @*). Let
2
{eff} be orbit representatives of the d-cells (d 0) and write Sfffor stab(eff*
*) 2 F.
Define M
Cd = M (G=Sff).
ff
If g . e0 is a typical (d - 1)-cell in the boundary of effthen g-1 . stab(eff) *
*. g
stab(e0), giving a G-map (write S0 for stab(e0))
Rg : G=Sff! G=S0,
which induces a homomorphism M(OE): M (G=Sff)! M (G=S0), usually written
(Rg)*. This yields a differential @d: Cd ! Cd-1, and the Bredon homology
groups HFi(X; M) correspond to the homology of (C*, @*). Observe that the
definition is independent of the family F as long as it contains Iso(X).
Bredon cohomology is defined analogously, for M a right Bredon module (a
contravariant M will reverse the arrows (Rg)* = M(OE): M(G=S0) ! M(G=Sff)
so that @ : Cd-1 ! Cd).
The Bredon homology of a group G with coefficients in M 2 G-Mod F can
be defined in terms of a Tor functor ([14, Def. 3.12]). If F is closed under
conjugations and taking subgroups then
HFi(G; M) ~=HFi(EFG; M) ,
which may as well be taken as a definition.
We are interested in the case X = E_G, F = Fin(G) and M = R the complex
representation ring, considered as a Bredon module as follows. On objects we
set
R(G=K) = RC(K), K 2 Fin(G)
the ring of complex representations of the finite group K (viewed just as an
abelian group), and for a G-map Rg : G=K ! G=L, we have g-1Kg L so
that (Rg)* : RC(K) ! RC(L) is given by induction from g-1Kg to L after
identifying RC(g-1Kg) ~=RC(K).
2.3 Equivariant K-homology
There is an equivariant version of K-homology, denoted KGi(-) and defined in
[4] (see also [10]) using spaces and spectra over the orbit category of G. It w*
*as
originally defined in [1] using Kasparov's KK-theory. We will only recall the
properties we need.
Equivariant K-theory satisfies Bott mod-2 periodicity, so we only consider
KG0and KG1. For any subgroup H of G, we have
KGi(G=H) = Ki(C*r(H)),
that is, its value at one-orbit spaces corresponds to the K-theory of the reduc*
*ed
C*-algebra of the typical stabilizer. If H is a finite subgroup then C*r(H) = CH
and ae
KGi(G=H) = Ki(CH) = RC(H)0 ii==01,.
This allows us to view KGi(-) as a Bredon module over OFinG.
3
We can use an equivariant Atiyah-Hirzebruch spectral sequence to compute
the KG -homology of a proper G-CW-complex X from its Bredon homology (see
[14, pp. 49-50] for details), as
G G
E2p,q= HFinpX; Kq (-) ) Kp+q (X) .
In the simple case when Bredon homology concentrates at low degree, it coin-
cides with the equivariant K-homology:
Proposition 2.1. Write Hi for HFini(X; R) and KGi for KGi(X). If Hi = 0
for i 2 then KG0= H0 and KG1= H1.
Proof.The Atiyah-Hirzebruch spectral sequence collapses at the 2-page. }
3 K"unneth formulas for Bredon homology
We will need K"unneth formulas for (relative) Bredon homology. We devote this
section to state such formulas, for the product of spaces and the direct product
of groups. Theorem 3.1 holds in more generality (see [7]) but we state the resu*
*lt
we need and sketch a direct proof; details can be found in [17, Chapter 3].
Results similar to those explained here are treated in [11].
K"unneth formula for X xY
Let X be a G-CW-complex and Y a H-CW-complex. Let F (resp. F0) be a
family of subgroups of G (resp. of H) containing Iso(X) (resp. Iso(Y )). Then
X xY is a (GxH)-CW-complex (with the compactly generated topology) and
Iso(X xY ) = Iso(X) x Iso(Y ) F x F0.
Moreover, the orbit category OFxF0(GxH) is isomorphic to OFG x OF0H.
Given M 2 G-Mod F, N 2 H-Mod F0 we define their tensor product over Z
as the composition of the two functors
M N : OFG x OF0H MxN_//_AbxAb___Z_//Ab
considered a Bredon module over OFxF0(GxH) ~=OFG x OF0H. We can easily
extend this tensor product to chain complexes of Bredon modules.
Theorem 3.1 (K"unneth Formula for Bredon Homology). Consider X, Y , F,
F0, M, N as above, with the property that M(G=K) and N(H=K0) are free for
all K 2 Iso(X), K0 2 Iso(Y ). Then for every n 0, there is a split exact
sequence
M i 0 j 0
0 ! HFi(X; M) Z HFj(Y ; N) ! HFxFn(X xY ; M N)
i+j=n
M i 0 j
! Tor HFi(X; M), HFj(Y ; N) ! 0
i+j=n-1
4
Proof.(sketch) Consider the chain complexes of abelian groups
D* = C*(X)_ F M and D0*= C*(Y_)_ F0N ,
where C*(-)_is the chain complex of Bredon modules defined in [14, p. 10-11],
and - F - is the tensor product defined in [14, p. 14]. Then, by Definition
3.13 in [14],
0
Hi(D*) = HFi(X; M) , Hj(D0*) = HFj(Y ; N) .
We now use the following lemmas, which follow from the definitions of C*(-)_
and - F -.
Lemma 3.2. Given C*_a chain complex in Mod F-G, C0*_a chain complex in
Mod F0-H and Bredon modules M 2 G-Mod F, N 2 H-Mod F0, the following chain
complexes of abelian groups are isomorphic
i j i j i j i j
C*_ F M C0*_ F0N ~= C*_ C0*_ FxF0 M N .
Lemma 3.3. The following chain complexes in Mod FxF0-(GxH) are isomorphic
C*(X_xY_)_~=C*(X)_ C*(Y_)_.
By Lemma 3.2,
i j
D* Z D0*~= C*(X)_ C*(Y_)_ FxF0 (M N)
and therefore, by Lemma 3.3,
0
Hn(D* Z D0*) = HFxFn(X xY ; M N) .
The result is now a consequence of the ordinary K"unneth formula of chain
complexes of abelian groups. Observe that the chain complexes D* and D0*are
free since M
Dn = Cn(X)_ F M ~= M(G=Sff)
ff
(see [14, p. 14-15]), and similarly for D0n. }
There is also a K"unneth formula for relative Bredon homology (see [14, p. 1*
*2]
for definitions).
Theorem 3.4 (K"unneth Formula for Relative Bredon Homology). Under the
same hypothesis of Theorem 3.1, plus A X a G-CW-subcomplex and B Y a
H-CW-subcomplex, we have that for every n 0 there is a split exact sequence
M i 0 j
0 -! HFi(X, A; M) Z HFj(Y, B; N)
i+j=n
0
-! HFxFn(X xY, AxY [ X xB; M N)
M i 0 j
-! Tor HFi(X, A; M), HFj(Y, B; N) -! 0 .
i+j=n-1
The proof is the same as the non-relative formula but using the chain complexes
D* = C*(X,_A)_ F M
D0*= C*(Y,_B)_ F0M
and a relative version of Lemma 3.3.
5
K"unneth formula for GxH
We will work in general with respect to a class of groups F (formally, F is a
collection of groups closed under isomorphism, or a collection of isomorphism
types of groups). Our main example will be Fin, the class of finite groups.
Given a specific group G, let us denote by F(G) the family of subgroups of
G which are in F, and by HF*(-) the Bredon homology with respect to F(G).
Suppose now that F is closed under taking subgroups (note that it is always
closed under conjugation). Then so is F(G) and we will write EFG for EF(G)G.
We want to know when EFG x EFH is a model for EF(G x H).
Proposition 3.5. Let F be a class of groups which is closed under subgroups,
finite direct products and homomorphic images. Then, for any groups G and
H, and X and Y models for EFG respectively EFH, the space X x Y is a model
for EF(GxH).
Proof.If (x, y) 2 X x Y then
stabGxH (x, y) = stabG(x) x stabH(y) 2 F(G) x F(H) F(G x H) .
Consider the projections ss1 : GxH ! G and ss2 : GxH ! H. If K 2 F(GxH),
(X x Y )K = Xss1(K)x Y ss2(K)
contractible since ss1(K) 2 F(G) and ss2(K) 2 F(H). }
Remark 1. The proposition is not true if we remove any of the two extra con-
ditions on F.
Remark 2. Not every family of subgroups of G can be written as F(G) (for
instance, F(G) is always closed under conjugation). To be more general, one
can prove that given two families F1 and F2 of subgroups of G respectively H,
EF1G x EF2H is a model for EF(G x H), where F is the smallest family closed
under subgroups and containing F1 x F2. Nevertheless, we are interested in
group theoretic properties (as `being finite'), so we think of F as the class of
groups with the required property.
Thus we can apply the K"unneth formula (Theorem 3.1) to EF(GxH). We
obtain Bredon homology groups with respect to the family F(G)xF(H) instead
of F(GxH), but both families contain the isotropy groups of EFG x EFH, so
the Bredon homology groups are the same (cf. Section 2.2).
Theorem 3.6 (K"unneth formula for GxH). Let F be a class of groups closed
under taking subgroups, direct product and homomorphic images. For every
n 0 there is a split exact sequence
M i j
0 ! HFi(G; M) Z HFj(H; N) ! HFn(GxH; M N) !
i+j=n
M i j
Tor HFi(G; M), HFj(H; N) ! 0 .
i+j=n-1
Remark 3. The analogous result for relative Bredon homology also holds, by
Theorem 3.4.
6
Application to proper actions and coefficients in the repre-
sentation ring
The original motivation for stating a K"unneth formula was the Bredon homology
of GxH for proper actions (i.e. F = Fin) and coefficients in the representation
ring. We only need a result on the coefficients.
For finite groups P and Q, the representation ring RC(P xQ) is isomorphic
to RC(P ) RC(Q). The same is true at the level of Bredon coefficient systems.
Given groups G and H, consider the left Bredon modules
RG 2 G-Mod Fin(G), RH 2 H-Mod Fin(H) and RGxH 2 (GxH)-Mod Fin(GxH).
Let us denote by eRGxH the restriction of RGxH to Fin(G)xFin(H).
Proposition 3.7. These two Bredon modules are naturally isomorphic
RG RH ~=ReGxH .
Proof.Consider the isomorphism P,Q : RC(P xQ) ~=RC(P ) RC(Q) and the
universal property of the tensor product to obtain a map ff as
RC(P )xRC(Q) _____//RC(P ) RC(Q)
RRR
RRRR ~=| P,Q
ffRRRRR((R fflffl||
RC(P xQ) .
That is, P,Q(ae o) = ff(ae, o), where the latter denotes the ordinary tensor
product of representations and ae o is their formal tensor product in RC(P )
RC(Q). Extend this map to a natural isomorphism : RG RH ! eRGxH. It
only remains to check naturality, that is, the commutativity of
P,Q
RC(P ) RC(Q) ______//_RC(P x Q)
(Rg)* (Rh)*|| |(R(g,h))*|
fflffl| P0,Q0 fflffl|
RC(P 0) RC(Q0)____//_RC(P 0x Q0) ,
for any (G x H)-map R(g,h)= Rg x Rh. To do this, one can compute both
sides of the square and show that the two representations have the same char-
acter, using the following two observations. Denote by O(-) the character of a
representation. Since ff(-, -) is the tensor product of representations we have
O ff(o1, o2)= O(o1) . O(o2). If Rg : G=K ! G=L is a G-map, ae 2 RC(K) and
t 2 G then X
O (Rg)*ae(t) = _1_|K| O(ae)(k) ,
*
where the sum * is taken over the s 2 G such that k = gsts-1g-1 2 K. We
leave the details to the reader. }
For an arbitrary group G, write HFini(G; R) for HFini(G; RG ).
7
Corollary 3.8. For every n 0 there is a split exact sequence
M i j
0 ! HFini(G; R) Z HFinj(H; R) ! HFinn(GxH; R) !
i+j=n
M i j
Tor HFini(G; R), HFinj(H; R) ! 0 .
i+j=n-1
Remark 4. Again, there is a similar statement for relative Bredon homology.
4 Coxeter groups and the Davis complex
We briefly recall the definitions and basic properties of Coxeter groups, and
describe a model of the classifying space for proper actions. Most of the mater*
*ial
is well-known and can be found in any book on the subject, as [9].
4.1 Coxeter groups
Suppose W is a group, S = {s1, . .,.sN } a finite subset of elements of order 2
which generate W . Let mijdenote the order of sisj. Then 2 mij= mji 1
if i 6= j and mii= 1. We call the pair (W, S) a Coxeter system and W a Coxeter
group if W admits a presentation
< s1, . .,.sN | (sisj)mij = 1 > .
We call N the rank of (W, S). The Coxeter diagram is the graph with vertex
set S and one edge joining each pair {si, sj}, i 6= j, with label mij, and the
conventions: if mij = 2 we omit the edge; if mij = 3 we omit the label. A
Coxeter system is irreducible if its Coxeter diagram is connected. Every Coxeter
group can be decomposed as a direct product of irreducible ones, corresponding
to the connected components of its Coxeter diagram. All finite irreducible
Coxeter systems have been classified; a list of their Coxeter diagrams can be
found, for instance, in [9, p. 32].
Every Coxeter group can be realized as a group generated by reflections in
RN . Namely, there is a faithful canonical representation W ! GLN (R) (see [9,
x5.3] for details). The generators in S correspond to reflections with respect *
*to
hyperplanes. These hyperplanes bound a chamber, which is a strict fundamental
domain (examples below).
Coxeter cells
Suppose now that W is finite. Take a point x in the interior of the chamber.
The orbit W x is a finite set of points in RN . Define the Coxeter cell CW as *
*the
convex hull of W x. It is a complex polytope by definition.
Examples:
(1) Rank 0: W = {1} the trivial group and CW is a point.
(2) Rank 1: W = C2 the cyclic group of order 2, RN = R, the W -action con-
sists on reflecting about the origin, and there are two chambers, (-1, 0]
and [0, +1). Take a point x in the interior of, say, [0, +1); then W x =
{-x, x} and CW = [-x, x], an interval.
8
(3) Rank 2: W = Dn dihedral group of order 2n, RN = R2 and the canonical
representation identifies W with the group generated by two reflections
respect to lines through the origin and mutual angle ss=n. A chamber is
the section between the two lines and the W -orbit W x is the vertex set of
a 2n-gon (see Figure 1). Therefore, CW is a 2n-gon.
Figure 1: Coxeter cell for a dihedral group.
(4) If W = W1 x W2 then CW = CW1 x CW2 .
The Davis complex
It is a standard model of E_W introduced by M. Davis in [5]. Given T S,
the group WT = generated by T is called a special subgroup. It can be
shown that (WT , T ) is a Coxeter system, and that WT \ WT0 = WT\T0. The
subset T is called spherical is WT is finite. We will write S for the set {T
S | T is spherical}.
A spherical coset is a coset of a finite special subgroup, that is, wWT for
some w 2 W , T 2 S. Note that wWT = w0WT0 if and only if T = T 0and
w-1w02 WT (use the so-called deletion condition [9, x5.8]). Denote by W S the
set of all spherical cosets in W , that is, the disjoint union
[
W S = W=WT .
T2S
This is a partially ordered set (poset) by inclusion, and there is a natural W -
action whose quotient space is S. The Davis complex is defined as the geo-
metric realization |W S| of the poset W S, that is, the simplicial complex with
one n-simplex for each chain of length n
w1WT1 . . .wnWTn , Ti2 S , (4.1)
and obvious identifications. The W -action on W S induces an action on for
which this is a proper space: the stabilizer of the simplex corresponding to the
chain (4.1) is w1WT1w-11, finite. Moreover, admits a CAT(0)-metric [16] and
therefore it is a model of E_W , by the following result.
Proposition 4.1. If a finite group H acts by isometries on a CAT(0)-space X,
then the fixed point subspace XH is contractible.
9
Proof.The idea is to find a fixed point for H and then use that if x, y 2 X are
fixed by H then so is the geodesic between x and y (from which the contractibil-
ity of XH follows). The existence of a fixed point for H is a consequence of t*
*he
Bruhat-Tits fixed point theorem [3, p. 157]. }
Examples:
(1) Finite_groups.-_Suppose that W is a finite Coxeter group and C is its
Coxeter cell, defined as the convex hull of certain orbit W x. Denote by
F(C) the poset of faces of the convex polytope C. The correspondence
w 2 W 7! wx 2 W x = vertex set of C
allows us to identify a subset of W with a subset of vertices of C. In fac*
*t,
a subset of W corresponds to the vertex set of a face of C if and only if
it is a coset of a special subgroup ([3, III]); see Figure 1 for an exampl*
*e.
Hence, we have the isomorphism of posets W S ~= F(C) and the Davis
complex = |W S| is the barycentric subdivision of the Coxeter cell.
(2) Triangle_groups.-_These are the Coxeter groups of rank 3,
< a, b, c | a2 = b2 = c2 = (ab)p = (bc)q = (ca)r = 1 > .
Suppose that p, q, r 6= 1 and 1_p+ 1_q+ 1_r 1. All subsets T ( S are
spherical, giving the poset (where arrows stand for inclusions) of Figure
2. It can be realized as the barycentric subdivision of an euclidean or
hyperbolic triangle with interior angles ss_p, ss_qand ss_r, and a, b and *
*c acting
as reflections through the corresponding sides. The whole model consists
on a tessellation of the euclidean or hyperbolic space by these triangles.
Figure 2: Davis complex (quotient) for a triangle group.
A second definition of the Davis complex
There is an alternative description of the Davis complex in terms of Coxeter
cells. Given a poset P and an element X 2 P, we denote by P X the subposet
of elements in P less or equal to X. Consider a spherical subset T S and an
element w 2 W .
10
Proposition 4.2. There is an isomorphism of posets (W S) wWT ~=WT S.
Proof.Firstly, observe that the poset (W S) wWT is equivalent to (W S) WT via
the isomorphism induced by multiplication by w-1. So it suffices to show that
W S WT = WT S. A standard element in the right hand side is a coset wWT0
with w 2 WT and T 0 T so we have WT0 WT and wWT0 wWT = WT . On
the other hand, if wWT0 WT then w = w . 1 2 WT . Finally, T 0 T : if s02 T 0
then ws02 WT so s02 WT and, using the deletion condition, s02 T . }
Now, since WT is a finite Coxeter group, we can identify the subcomplex
| (W S) wWT | = |WT S|
with the barycentric subdivision of the associated Coxeter cell CWT . From
this point of view, the poset W S is the union of subposets (W S) wWT with T
spherical and w 2 W=WT . That is, can be viewed as the union of Coxeter
cells [
CwWT
wW T2S
T2W=WT
where CwWT is the copy of CWT corresponding to the coset wWT . This union
is obviously not disjoint, and the inclusions and intersections among subposets
are precisely the following.
Lemma 4.3. (i) wWT w0WT0 if and only if T T 0and w-1w02 WT0.
(ii)wWT \ w0WT0 = w0WT\T0 if there is any w0WT 2 wWT \ w0WT0, empty
otherwise.
Proof.(i) One implication is obvious, the other uses the deletion condition in
the fashion of the proof of Proposition 4.2.
(ii) Straightforward, since WT \ WT0 = WT\T0 . }
Consequently, the intersection of two Coxeter cells is
ae 0
CwWT \ Cw0WT0 = Cw0WT\T0? ifwWTo\twhWT0e=rw0WT\T0wise. (4.2)
Denote by @CW the boundary of the Coxeter cell, that is, the topological
boundary in the ambient space RN . We have the following explicit description.
Proposition 4.4. [
@CW = wCWT
T_S
wWT2W=WT
Proof.CW is a convex polytope whose faces correspond to cosets of special
subgroups (Example on page 10). The barycentric subdivision sd(CW ) = |W S|
is indeed a cone over its center (the vertex corresponding to the coset WS =
W ). }
11
5 Relative Bredon homology of the Davis com-
plex
The aim of this section is to deduce a formula for the relative Bredon homology
of the n-skeleton of the Davis complex with respect to the (n - 1)-skeleton,
HFin*( n, n-1), in terms of the Bredon homology of the Coxeter cells.
We will denote by the Davis complex with the Coxeter cell decomposition
explained on page 10 and by 0the original definition as the nerve of W S. Note
that both are W -homeomorphic spaces and 0 = sd( ) (barycentric subdivi-
sion) but is not a W -CW-complex while 0is (recall that in a G-CW-complex
a cell is sent to itself only by the identity map).
As an example, see Figure 3: it is the tessellation of the euclidean plane
induced by the triangle group (2, 4, 4) together with the dual tessellation gi*
*ven
by squares and octagons. The latter is , the Davis complex given as union of
Coxeter cells. The skeleton filtration correspond to Coxeter cells of rank 0
(points), rank 1 (intervals) and rank 2 (2n-gons).
Figure 3: Tessellation of the euclidean plane given by the triangle group
(2, 4, 4), and its dual tessellation.
Definition 5.1. For n -1 define n as the union of Coxeter cells
corresponding to finite WT with rank(T ) n. That is,
[
n = CwWT .
T2S, |T| n
wWT2W=WT
Then -1 = ?, 0 is a free orbit of points and 1 is indeed the Cayley
graph of W with respect to S (see Figure 3 for an example or compare with
Proposition 5.4).
We have that n is a W -subspace (w0CwWT = Cw0wWT ) and, since the
dimension of CWT is rank(T ), n is indeed the n-skeleton of .
Our aim is to prove the following theorem.
12
Theorem 5.2. For any dimension n 0, any degree i and any Bredon coeffi-
cient system M, we have the isomorphism
M
HFini( n, n-1; M)~= HFini(CWT , @CWT ; M).
raT2Snk(T)=n
Comments on the statement of the theorem:
(i) We have defined Bredon homology on G-CW-complexes so we implicitly
assume that we take the barycentric subdivision on the spaces.
(ii) The Bredon homology on the left-hand side is with respect to the W -
action and the family Fin(W ). The Bredon homology on the right-hand side
is with respect to the WT -action and the family Fin(WT ) = All(WT ) (all sub-
groups), for each Coxeter cell.
Firstly, we state some properties of the Coxeter cells that we will use.
Lemma 5.3. (a) Suppose T 2 S with rank(T ) n. Then
ae
CWT \ n-1 = CWT@C ifrank(T ) < n ,
WT ifrank(T ) = n .
(b) The intersection of any two different Coxeter cells in n is included in
n-1.
(c) The Coxeter cell CWT is a WT -subspace of n and stabW(x) is contained
in WT for all x 2 CWT .
Proof.(a) If rank(T ) < n then CWT n-1. If rank(T ) = n then (Proposition
4.4) [
@CWT = wCWT0 n-1 .
T0_T
wWT02WT=WT0
On the other hand, if x 2 CWT \ n-1 then x 2 CwWT0 with rank(T 0)
n - 1 so x 2 CWT \ wCWT0 which, by equation (4.2), equals to either empty or
w0CWT\T0 @CWT .
(b) In general, the intersection of two different Coxeter cells of rank n and
m is either empty or another Coxeter cell of rank strictly less than min{n, m}
_ see equation (4.2).
(c) The first part is obvious: wCWT = CwWT = CWT if w 2 WT .
Suppose now that x 2 CWT and wx = x. We want w 2 WT . Identify sd(CWT ) =
|WT S|. Suppose that x belongs to a cell corresponding to the chain WT0 _ . ._.
WTk, Tk T . If w fixes x, it has to fix at least one of the vertices of the c*
*ell,
i.e., there is an i such that wWTi= WTi so that w 2 WTi WT . }
Secondly, we observe that the copies of a Coxeter cell in admit an inter-
pretation as induced spaces. If H is a subgroup of G and X is an H-space, the
associated induced G-space is
IndGHX = G xH X ,
the orbit space for the H-action h_._(g,_x)_=_(gh-1,_hx). The (left) G-action on
the induced space is given by g . (k, x)= (gk, x). This definition carries on to
pairs of H-spaces.
13
Proposition 5.4. For each spherical T S, there is a W -homeomorphism
[
CwWT ~=W W xWT CWT .
wWT2W=WT
_______
Proof.If x 2 CwWT = w . CWT , write x = wx0 and send x to (w, x0)2 W xWT
CWT . This is well-defined by equation (4.2): x = wx0 = w0y0 then there is
w002 WT such that w = w0w00. This defines a continuous W -map, bijective,
with continuous inverse
______
(w, x)7! w . x 2 CwWT . }
Consequently, we may write n, n 1, as a union of induced spaces
[
n ~= W xWT CWT .
|T| n
Next, we will need the following easy consequence of a relative Mayer-
Vietoris sequence.
Proposition 5.5. Let X be a G-CW-complex and Y, A1, . .,.An G-subcomple-
xes such that X = A1 [ . .[.An and Ai\ Aj Y for all i 6= j. Write Hn(-)
for Bredon homology with some fixed coefficients and with respect to a family
F Iso(X) or, more generally, any G-homology theory. Then
Mn
Hn(X, Y ) ~= Hn(Ai, Ai\ Y ) .
i=1
Proof.By induction on n. For n = 1 it is a tautology. For n > 1 call A = A1,
B = A2 [ . .[.An, C = A \ Y and D = B \ Y . The relative Mayer-Vietoris of
the CW-pairs (A, C) and (B, D) is
. .!.Hi(A\B, C\D) ! Hi(A, C) Hi(B, D) ! Hi(A[B, C[D) ! . . . (5.1)
Observe that
A [ B = X
C [ D = (A [ B) \ Y = Y
A \ B Y
C \ D = A \ B \ Y = A \ B
Therefore Hi(A \ B, C \ D) = 0 for all i and the sequence (5.1)gives isomor-
phisms
Hi(X, Y ) ~=Hi(A, C) Hi(B, D)
Now apply induction to X0 = B, Y 0= B \ Y and the result follows. }
Finally, recall that if h?*is an equivariant homology theory (see, for insta*
*nce,
[13, x1]), it has an induction structure. Hence, for a pair of H-spaces (X, A),
hGn(IndGH(X, A)) ~=hHn(X, A) .
Bredon homology has an induction structure [15]; in particular,
i j
HFin(W)nIndWWT(CWT , @CWT ) ~=HFin(WT)n(CWT , @CWT ).
14
Proof.(Theorem 5.2) Write Hn(-) for Bredon homology of proper G-spaces
with coefficients in the representation ring. Let T1, . .,.Tm be the spherical
subsets of generators up to rank n. Define
Ai= W xWTi CWTi 0 i m .
S
Then n = Ai and Ai \ Aj n-1 for all i 6= j, by Lemma 5.3(b). By
Proposition 5.5
Mn
Hn( n, n-1) ~= Hn(Ai, Ai\ n-1) .
i=1
If rank(Ti) < n then Ai n-1 and the corresponding term is zero. If
rank(Ti) = n then Ai \ n-1 = @Ai (Lemma 5.3(a)) and, by the induction
structure,
Hn(Ai, @Ai) ~=Hn(CWTi, @CWTi) . }
Corollary 5.6. Write Hi(-) for Bredon homology with respect to finite sub-
groups and some fixed Bredon coefficient system. For each n 1, there is a
long exact sequence
M
. .!.HFini( n-1) ! HFini( n) ! HFini(CWT , @CWT )! . . .
raT2Snk(T)=n
Remark 5. Theorem 5.2 also holds for any (proper) equivariant homology the-
ory, in particular, for equivariant K-homology.
6 Relative Bredon homology of some Coxeter
cells
As an application of Theorem 5.2, we compute the relative Bredon homology
of ( n, n-1), with coefficients in the representation ring, for the first cases
n = 0, 1, 2. To do so, we need the Bredon homology of (CWT , @CWT ) when
T S spherical with rank(T ) = 0, 1, 2.
_____________
| rank(T ) = 0|T = ?, WT = {1} and CW is a point. So the homology is
|____________| T
RC({1}) ~=Z at degree 0 and vanishes elsewhere.
_____________
| rank(T ) = 1|T = {si}, WT is cyclic of order two and CW is an interval. For
|____________| T
the relative homology, we consider one 0-cell and one 1-cell, with stabilizers
cyclic order two and trivial respectively (see Figure 4). The associated Bredon
chain complex
0 -! RC({1}) -Ind!RC (C2) -! 0
1 7! (1, 1)
gives ae
Hi(CWT , @CWT ; R)= Z0 ii=60=.0 (6.1)
15
Figure 4: Coxeter cell of rank 1
_____________
| rank(T ) = 2|T = {si, sj}, WT is dihedral of order 2n (n = mij 6= 1) and
|____________|
CWT is a 2n-gon. The orbit space is a sector of the polygon (see Figure 5) and
the relative Bredon chain complex is
Figure 5: Coxeter cell of rank 2
0 -! RC ({1})-d2!RC (C2) RC (C2)-d1!RC (Dn) -! 0 ,
where the differentials are given by induction among representation rings. In-
ducing the trivial representation always yields the regular one, so d2 in matrix
form is
d2 1 1 1 1 ~ 1 0 0 0 ,
the right-hand side being its Smith normal form. For d1, recall the charac-
ter tables of the cyclic group of order two and dihedral group (with Coxeter
generators a and b):
__C2_|1____a__ _Dn__|____(ab)k_______b(ab)k__
ae1 1| 1 O1 | 1 1
ae2 1| -1 O2 | 1 -1
cO3| (-1)k (-1)k
cO4| (-1)k (-1)k+1
OEl |2 cos(2sslk=n) 0
16
where 0 k n - 1, l varies from 1 to n=2 (n even) or (n - 1)=2 (n odd) and
the hat b denotes a character which only appears when n is even. The induced
representations corresponding to the inclusions and ** a*
*re
X
Ind(ae1)= O1 + cO3+ OEl,
X
Ind(ae2)= O2 + cO4+ OEl .
Consequently, the differential d1 is written in matrix form as
0 1
1 0 b1 b0 1 . . .1
B 0 1 b0 b1 1 . . .1C
d1 B@ 1 0 b0 b1 1 . . .1CA,
0 1 b1 b0 1 . . .1
which reduces by elementary operations to
` ' ` '
_I3_|0_ _I2_|0_
0 |0 if n is even, 0 |0 if n is odd.
Therefore, the relative Bredon homology of a Coxeter cell of rank two and
n = mij6= 1 is
n even: H0 = Zc(Dn)-3= Zn=2, Hi= 0 8 i 1 ;
n odd: H0 = Zc(Dn)-2= Z(n-1)=2, H1 = Z , Hi= 0 8 i 2 .
Here we write c(H) for the number of conjugacy classes in a finite group H, and
Zn, or sometimes n . Z, for ni=1Z.
We can now use Theorem 5.2 to deduce the lower-rank relative Bredon ho-
mology of . We omit the family Finand coefficients R for clarity.
Proposition 6.1. Let (W, S) be a Coxeter system and its associated Coxeter
complex. Then
ae
Hi( 0, -1) = Z0 ii=60= 0.
ae
Hi( 1, 0) = |S|0. Zii=60= 0.
0 1
X X
H0( 2, 1)= 1_2@ mij+ (mij- 1)A . Z ,
even,i 0 and the split exact sequence
0 -! H0( n-1) -! H0( n) -! H0( n, n-1) -! 0 .
Using induction on n,
Hi( n) = Hi( 0) = 0 8 i > 0 ,
H0( n) = H0( n-1) H0( n, n-1) = (s(0) + . .+.s(n)). Z .
This gives the Bredon homology of (W, S) when n = |S|.
Theorem 7.1. The Bredon homology of a right-angled Coxeter group (W, S)
with respect to the family Fin(W ) and coefficients in the representation ring *
*is
ae
HFini(W ; R)= s0. Z ii=60= 0
where s = s(0) + . .+.s(|S|), the number of spherical subsets T S.
Finally, since the Bredon homology concentrates at degree 0, it coincides
with the equivariant K-homology (Proposition 2.1). Also, the Baum-Connes
conjecture holds for Coxeter groups, so we have the following corollary.
Corollary 7.2. If W is a right-angled Coxeter group, the equivariant K-homo-
logy of E_W coincides with its Bredon homology at degree 0 and 1 respectively
(given in the previous theorem). This corresponds to the topological K-theory of
the reduced C*-algebra of W , via the Baum-Connes assembly map.
8 Bredon homology of even Coxeter groups
A Coxeter system (W, S) is even if mijis even or infinite for all i, j. Right-a*
*ngled
Coxeter groups are then a particular case. Again, the only spherical subgroups
are direct products of cyclic (order two) and dihedral, since any irreducible f*
*inite
Coxeter group with more than two generators have at least one odd mij.
Suppose that WT = Dm1 x . .x.Dmr x (C2)k, with mi (1 i r) even
numbers and k 0. We know that the relative Bredon homology of the Coxeter
cells corresponding to dihedral Dmi and C2 concentrates at degree 0 and
H0 = (mi=2) . Z , respectivelyH0 = Z .
18
By the relative K"unneth formula we have
ae ffir
Hi(CWT , @CWT )= m1 . . ...mr02 . Z ii=60=.0
Alternatively, if T = {t1, . .,.tn} is a spherical subset of rank n,
Y
H0(CWT , @CWT )= mij=2 . Z , (8.1)
i**