"The type of the classifying space for a family
of subgroups"
by
Wolfgang L"uck
Key words: Classifying space for a family of subgroups, finiteness-conditions
AMS-classification number: 55R35
Abstract: The classifying space E(; F) for a family F of subgroups of a group *
*is defined up to
-homotopy as a -CW -complex E(; F) such that E(; F)H is contractible if H belon*
*gs to F and
is empty otherwise. We investigate the question whether there is a finite-dimen*
*sional -CW -model,
a finite -CW -model or a -CW -model of finite type for E(; F) focusing on the c*
*ase where F is
the family of finite subgroups.
0. Introduction
Let be a group and F be a family of subgroups, i.e. a set of subgroups of*
* which is
closed under conjugation and taking subgroups.
Definition 0.1. A classifying space E(; F) for F is a -CW -complex such that E(*
*; F)H
is contractible for H 2 F and empty otherwise.
If F is the family FIN of finite subgroups, we abbreviate
__
E_ = E(; FIN ): |__|
The existence of E(; F) and proofs that for any -CW -complex X whose isotr*
*opy
groups belong to F there is precisely one -map up to -homotopy from X to E(; F)*
* and
thus that two such classifying spaces are -homotopy equivalent, are presented i*
*n [9],[10,
I.6]. A functorial "bar-type" construction is given in [8, section 7]. These cl*
*assifying spaces
occur in the Isomorphism-Conjectures in algebraic K- and L-theory [11, 1.6 on p*
*age 257] for
F the family of virtually cyclic groups and in the Baum-Connes-Conjecture [2, C*
*onjecture
3.15 on page 254] for F the family FIN of finite subgroups. The space E_ play*
*s also a role
in the extension of the Atiyah-Segal-Completion Theorem from finite to infinite*
* groups [16].
Sometimes E_ is also called the classifying space for proper actions. Notice t*
*hat E(; F)
for F the family consisting of one element, namely the trivial subgroup, is the*
* same as the
total space E of the universal principal -bundle E -! B. If is torsionfree, th*
*en E_
is the same as E. If is finite, then the one-point-space is a model for E_.
In this paper we are dealing with the question whether one can find a d-di*
*mensional
-CW -model, a finite -CW -model or a -CW -model of finite type for E_. Recall *
*that
a -CW -complex X is finite if and only if it consists of finitely many -equivar*
*iant cells,
or, equivalently, \X is compact. It is called of finite type if each skeleton i*
*s finite. More
1
information about -CW -complexes can be found in [10, II.1 and II.2] and [14, s*
*ection 1
and 2]. A survey of groups for which nice geometric -CW -models for E
__ exist c*
*an be
found in [2, Section 2]. These include for instance (i) word-hyperbolic groups *
* for which
the Rips complex yields a finite -CW -model for E_, (ii) discrete subgroups G*
* of a
Lie group G with finitely many path components for which G=K with the left -act*
*ion for
a maximal compact subgroup K G is a -CW -model for E_ and (iii) groups acting
cellularly (without inversion) on trees with finite isotropy groups.
In Section 1 we give necessary and sufficient conditions for the existence*
* of a -CW -
model of E_ with prescribed dimensions of the H-fixed point sets E_H in terms o*
*f the Borel
cohomology of the posets of non-trivial finite subgroups of the Weyl groups WH *
*of the finite
subgroups H (see Theorem 1.6). We also give a necessary algebraic condition B*
*(d) for
a non-negative integer d for the existence of a d-dimensional -CW -model for E_*
*, namely
that for each finite subgroup H a ZWH-module M whose restriction to ZK is pro*
*jective
for any finite subgroup K WH has a d-dimensional projective ZWH-resolution (No*
*tation
1.4 and Lemma 1.5). The length l(H) 2 {0; 1; : :}:of a finite group H is the s*
*upremum
over all p for which there is a nested sequence H0 H1 : : :Hp of subgroups Hi*
* of H
with Hi 6= Hi+1. If there is an upper bound l on the length l(H) of finite subg*
*roups H of
and B(d) is satisfied we will prove that there is a (max {3; d} + l(d + 1))-di*
*mensional -
CW -model for E_ (Theorem 1.10). Such a result has already been proven for cert*
*ain classes
of groups by Kropholler and Mislin [12, Theorem B], for an in l exponential dim*
*ension
estimate. In Section 2 we show that E_ has a dm-dimensional -CW -model if ther*
*e is a
subgroup of finite index d with m-dimensional -CW -model for E_ (Theorem 2.4*
*).
In Section 3 we show for an exact sequence 1 ! ! ! ss ! 1 such that there is *
*an upper
bound on the orders of finite subgroups of ss that E_ has a finite-dimensional *
*-CW -model
if E_ and E_ss respectively have a finite-dimensional -CW -model and a finite-d*
*imensional
ss-CW -model respectively (Theorem 3.1). We discuss to which extend a statement*
* like this
holds if we ask for finite models or models of finite type (Theorem 3.2 and The*
*orem 3.3).
In Section 4 we prove that E_ has a -CW -model of finite type if and only if th*
*ere is a
CW -model of finite type for BWH for all finite subgroups H and contains only
finitely many conjugacy classes of finite subgroups. (Theorem 4.2). In Sectio*
*n 5 we deal
with the question whether there is a finitely dominated or finite -CW -complex *
*model for
E (_____Theorem 5.1 and Remark 5.2). In Section 6 we consider the special case *
*of groups of
finite virtual cohomological dimension. Provided that contains a torsionfree g*
*roup of finite
index, satisfies B(d) if and only if has virtual cohomological dimension d (*
*Theorem
6.3). If l is an upper bound on the length l(H) of finite subgroups H of and *
*has virtual
cohomological dimension d, then we will prove that there is a (max {3; d} + l)*
*-dimensional
-CW -model for E_ (Theorem 6.4). Finally we discuss some open problems in Secti*
*on 7.
We will always work in the category of compactly generated spaces (see [19*
*] and [21,
I.4]). A -space or a Z-module respectively is always to be understood as a left*
* -space or
left Z-module respectively. The letter stands always for a discrete group. The*
* paper is
organized as follows:
2
0. Introduction
1 . Finite-dimensional classifying spaces
2. Coinduction
3. Short exact sequences
4. Classifying spaces of finite type
5. Finitely dominated and finite classifying spaces
6. Groups with finite virtual cohomological dimension
7. Problems
References
The author wants to thank Frank Connolly, Peter Kropholler and Guido Misli*
*n for
stimulating and fruitful discussions.
1. Finite-dimensional classifying spaces
In this section we deal with the question whether there are finite-dimensi*
*onal -CW -
models for E_.
Define the -poset
P() := {K | K finite; K 6= 1}: *
*(1.1)
An element fl 2 sends K to flKfl-1 and the poset-structure comes from inclusio*
*n of
subgroups. Denote by |P()| the geometric realization of the category given by *
*the poset
P(). This is a -CW -complex but in general not proper, i.e. it can have points *
*with infinite
isotropy groups.
Let NH be the normalizer and let WH := NH=H be the Weyl group of H . Noti*
*ce
for a -space X that XH inherits a WH-action. Denote by oe (X) the singular se*
*t of the
-space X, i.e. the set of points with non-trivial isotropy groups. Notice that *
*oe0 (X) may
differ from oe (X) for a subgroup 0 . Denote by CX the cone over X. Notice that*
* C;
is the one-point-space. The meaning of |P()| lies in the following result which*
* follows from
[7, Lemma 2.4].
Lemma 1.2. There is a -equivariant map
f : (E_; oe (E_)) -! (C|P()|; |P()|);
*
* __
which is a (non-equivariant) homotopy equivalence. *
* |__|
If H and K are subgroups of and H is finite, then =KH is a finite union o*
*f WH-orbits
of the shape WH=L for finite L WH. Now one easily checks
Lemma 1.3. The WH-space E_H is a WH-CW -model for E_WH. In particular, if E_ *
*has
a -CW -model which is finite, of finite type or d-dimensional respectively, the*
*n there is a_
WH-model for E_WH which is finite, of finite type or d-dimensional respectively*
*. |__|
3
Notation 1.4. Let d 0 be an integer. A group satisfies the condition b(d) or*
* b(< 1)
respectively if any Z-module M with the property that M restricted to ZK is pro*
*jective
for all finite subgroups K has a projective Z-resolution of dimension d or of*
* finite
dimension respectively. A group satisfies the condition B(d) if WH satisfies t*
*he condition
b(d) for any finite subgroup H .
The length l(H) 2 {0; 1; : :}:of a finite group H is the supremum over all*
* p for_which
there is a nested sequence H0 H1 : : :Hp of subgroups Hi of H with Hi6= Hi+1.*
* |__|
Lemma 1.5. Suppose that there is a d-dimensional -CW -complex X with finite i*
*sotropy
groups such that Hp(X; Z) = Hp(*; Z) for all p 0 holds. This assumption is for*
* instance
satisfied if there is a d-dimensional -CW -model for E_. Then satisfies condit*
*ion B(d).
Proof. Let H be finite. Then X=H satisfies Hp(X=H; Z) = Hp(*; Z) for all p 0
[4, III.5.4 on page 131]. Let C* be the cellular ZWH-chain complex of X=H. Th*
*is is a
d-dimensional resolution of the trivial ZWH-module Z and each chain module is a*
* sum of
ZWH-modules of the shape Z[WH=K] for some finite subgroup K WH. Let N be a
ZWH-module such that N is projective over ZK for any finite subgroup K WH. Then
C* Z N with the diagonal WH-operation is a d-dimensional projective ZWH-resolut*
*ion of
*
* __
N. *
* |__|
Theorem 1.6. Let be a group. Suppose that we have for any finite subgroup H *
* an
integer d(H) 3 such that d(H) d(K) for H K and d(H) = d(K) if H and K are
conjugated in . Consider the following statements:
1. There is a -CW -model E_ such that for any finite subgroup H
dim(E_H ) = d(H):
2. We have for any finite subgroup H and for any ZWH-module M
Hd(H)+1ZWH(EWH x (C|P(WH)|; |P(WH)|); M) = 0:
3. We have for any finite subgroup H that its Weyl group WH satisfies b(< 1*
*) and
that there is a subgroup (H) WH of finite index such that for any Z(H)-mo*
*dule
M
Hd(H)+1Z(H)(E(H) x (C|P(WH)|; |P(WH)|); M) = 0:
Then (1) implies both (2) and (3). If there is an upper bound on the lengt*
*h l(H) of the
finite subgroups H of , then these statements (1), (2) and (3) are equivalent.
In the case that has finite virtual cohomological dimension a similar res*
*ult is proven
in [7, Theorem III].
Example 1.7. Suppose that is torsionfree. Then Theorem 1.6 reduces to the we*
*ll-known
result [6, Theorem VIII.3.1 on page 190,Theorem VIII.7.1 on page 205] that the *
*following
assertions are equivalent for an integer d 3:
4
1. There is a d-dimensional CW -model for B;
2. has cohomological dimension d;
*
* __
3. has virtual cohomological dimension d. *
* |__|
Example 1.8. In this example we use the notation of Theorem 1.6. If WH is tor*
*sionfree,
then
Hd(H)+1ZWH(EWH x (C|P(WH)|; |P(WH)|); M) = Hd(H)+1ZWH(EWH; M);
Hd(H)+1Z(H)(E(H) x (C|P(WH)|; |P(WH)|); M) = Hd(H)+1Z(H)(E(H); M);
and the condition that Hd(H)+1ZWH(EWHx(C|P(WH)|; |P(WH)|); M) = 0 or Hd(H)+1Z(H*
*)(E(H)x
(C|P(WH)|; |P(WH)|); M) = 0 respectively holds for all ZWH-modules M or Z(H)-
modules M respectively is equivalent to the existence of a d(H)-dimensional mod*
*el for BWH
or B(H) respectively.
If WH contains a non-trivial normal finite subgroup L, then |P(WH)| is con*
*tractible
and
Hd(H)+1ZWH(EWH x (C|P(WH)|; |P(WH)|); M) = 0;
Hd(H)+1Z(H)(E(H) x (C|P(WH)|; |P(WH)|); M) = 0:
Namely, define maps of posets C; F : P(WH) -! P(WH) by C(H) = L and F (H) =
where the subgroup generated by H and L is finite since L is normal in W*
*H. Since
H F (H) and L F (H) holds for all finite subgroups H, there are natural trans*
*formations
between the functors idand F and the functors C and F of the category given by *
*the poset
P(WH) and hence the maps induced by these functors on the geometric realization*
*s are
homotopic. Hence the identity is homotopic to the constant map which is the map*
* induced
*
* __
by C. *
* |__|
The proof of Theorem 1.6 needs some preparation. The next lemma deals with*
* the ques-
tion whether a -CW -complex can be made contractible by attaching free cells of*
* bounded
dimension and is the key ingredient in the induction step in the proof of Theor*
*em 1.6.
Lemma 1.9. Let X be a -CW -complex and d 3 be an integer. Consider the follo*
*wing
statements.
1. There is a -CW -complex Y which is obtained from X by attaching free cells*
* x Dn
of dimension n d and is contractible.
2. There is a -CW -complex Y which is obtained from E x X by attaching free *
*cells
x Dn of dimension n d and is contractible.
3. Hn(X; Z) = 0 for n d and for any (d - 2)-connected -CW -complex Z which is
obtained from X by attaching free cells x Dn of dimension n d - 1 the Z-*
*module
Hd-1(Z; Z) is Z-projective.
5
4. Hn(X; Z) = 0 for n d and there is a (d - 2)-connected -CW -complex Z whic*
*h is
obtained from X by attaching free cells x Dn of dimension n d - 1 such t*
*hat the
Z-module Hd-1(Z; Z) is Z-projective.
5. The Z-chain complex C*(E x (CX; X)) is Z-chain homotopy equivalent to a d-
dimensional projective Z-chain complex.
6. Hn(X; Z) = 0 for n d + 1 and Hd+1Z(E x (CX; X); M) = 0 for any Z-module M.
7. Hn(X; Z) = 0 for n d+1, the group satisfies b(<1) and there is a subgrou*
*p 0
of finite index such that Hd+1Z0(E0 x (CX; X); M) = 0 holds for any Z0-mod*
*ule M.
8. There is a non-negative integer e such that Hn(X; Z) = 0 for n d - e and *
*the group
satisfies b(e).
Then we have
(1) () (2) () (3) () (4) () (5) () (6).
If X is finite-dimensional and all isotropy groups of X are finite then (1*
*) =) (7).
If XH is contractible for all non-trivial finite subgroups H , then (7) *
*=) (1) and
(8) =) (1).
Proof. (1) =)(2) Let Z X be the given extension. Then one constructs a pair of*
* -CW -
complexes (Z0; E x X) such that Z0is obtained from E x X by attaching free cell*
*s x Dn
of dimension n d and for the -space Z00defined by the -pushout
E x X ---! Z0
? ?
pr?y ?y__pr
X ---! Z00
there is a -homotopy equivalence of pairs relative X from (Z00; X) to (Z; X). T*
*hen Z0 is
contractible and the desired -CW -complex.
In fact there is a bijective correspondence between the cells x Dn of Z0-*
* (E x X)
and of Z - X for all n. The construction is based on the observation that for a*
*ny -map
f : A - ! B of -spaces which is a (non-equivariant) homotopy equivalence any -m*
*ap
x Sn - ! B lifts up to -homotopy to a -map x Sn - ! A and changing the -
homotopy type of the attaching maps does not change the -homotopy type of a -CW*
* -
complex. Details can be found in [14, page 288 - 289].
(2) =) (5) There is a -homotopy equivalence (f; id) : (Y; E x X) -! E x (CX; X)*
* since
both E x CX and Y are contractible.
(5) =)(3) Define a -CW -complex Y by the -pushout
6
E x X - --! E x CX
?? ??
pry y __pr
X - --! Y
for pr : E x X - ! X the projection. Since the inclusion of E x X into E x CX *
*is
a cofibration and pr is a (non-equivariant) homotopy equivalence, __pris a (non*
*-equivariant)
homotopy equivalence. Hence Y is contractible. Obviously C*(Y; X) and C*(E x (C*
*X; X))
are Z-chain isomorphic. Choose a -map f : Z - ! Y which induces the identity *
*on X
and a Z-chain homotopy equivalence g* : C*(Y; X) -! P* for a d-dimensional proj*
*ective
Z-chain complex P*. Let C* be the mapping cone of the Z-chain map g* O C*(f; i*
*d) :
C*(Z; X) -! P*. Since the relative dimension of (Z; X) is d - 1 and the dimensi*
*on of P* is
d, C* is a d-dimensional projective Z-chain complex. Since Z is (d - 2)-connect*
*ed and Y
is contractible, Hj(C*) is trivial for j d - 1 and is Hd-1(Z; Z) for j = d. He*
*nce we obtain
an exact sequence of Z-modules
0 -! Hd-1(Z; Z) -! Cd -! Cd-1 -! : :-:! C0 -! 0:
This implies that Hd-1(Z; Z) is a projective Z-module. We conclude Hj(X; Z) = *
*0 for
j d from the long homology sequence associated to E x (CX; X) and the fact that
Hj(E x (CX; X); Z) = 0 for j d + 1.
(3) =) (4) is obvious.
(4) =)(1) By the Eilenberg swindle there exists a free Z-module F such that F H*
*d-1(Z) is
free. Hence we can achieve by attaching free (d - 1)-dimensional cells x Dd-1 *
*with trivial
attaching maps to Z that Hd-1(Z) is free. The Hurewicz homomorphism ssd-1(Z) -*
* !
Hd-1(Z; Z) is bijective since Z is (d - 2)-connected and d 3. Now choose a Z-b*
*asis for
ssd-1(Z) and attach for each basis element which is given by a map g : Sd-1 -! *
*Z a free
-cell x Dd to Z by the attaching map x Sd-1 -! Z (fl; s) 7! fl . g(s). The *
*resulting
space Y is obtained from X by attaching free -cells x Dn of dimension n d. *
*It is
(d - 2)-connected and in particular simply-connected by construction. By inspe*
*cting the
long homology sequence of (Y; Z) one concludes that the homology Hj(Y ; Z) is t*
*rivial for
1 j d - 1 and the map induced by the inclusion Hd(Z; Z) -! Hd(Y ; Z) is surje*
*ctive.
Using the long exact sequences of the pair (Z; X) one concludes that Hd(Y ; Z) *
*is trivial.
Since Hj(Y ; Z) = Hj(X; Z) = 0 holds for j d + 1, the space Y must be contract*
*ible.
(5) () (6) If we would substitute in (6) the condition Hn(X; Z) = 0 for n d + *
*1 by the
sharper condition Hn(X; Z) = 0 for n d, (5) () (6) would follow from [14, Prop*
*osition
11.10 on page 221] or [20]. Hence it remains to show that Hd(X; Z) = 0 holds pr*
*ovided that
Hd+1Z(E x (CX; X); M) = 0 for any Z-module M. Let A be any abelian group. Equip
i]A := hom Z(Z; A) with the left Z-module structure given by (fl . f)(x) = f(x *
*. fl-1) for
fl 2 , f 2 i]A and x 2 Z. This is the coinduction of A with respect to the incl*
*usion of the
trivial group in . Since coinduction is the right adjoint of restriction, we get
Hd(X; A) ~= Hd+1Z(E x (CX; X); A)
~= Hd+1Z(E x (CX; X); i]A)
~= 0:
7
From the universal coefficient theorem we conclude hom Z(Hd(X; Z); A) = 0 for a*
*ny abelian
group A. This implies H
d(X; Z) = 0.
Now we show (1) =) (7) provided that X is finite-dimensional and the isotr*
*opy groups
of X are finite. If i : 0 -! is the inclusion and i] denotes coinduction with *
*i, we have
Hd+1Z0(E0 x (CX; X); M) = Hd+1Z(E x (CX; X); i]M)
for any Z0-module M. Since we already know (1) =) (6), it suffices to show that*
* satisfies
b(d) if (1) holds. Then C*(Y ) is a finite-dimensional Z-resolution of the triv*
*ial Z-module
Z and each Cn(Y ) is a direct sum of Z-modules of the shape Z=L for appropriate*
* finite
subgroups L . If N is a Z-module which is projective over ZH for all finite su*
*bgroups
H , then C*(X) Z N with the diagonal -action is a finite-dimensional projecti*
*ve
Z-resolution of N.
From now on suppose that XK is contractible for any non-trivial finite sub*
*group K .
Notice that then for any non-trivial finite subgroup H the singular set oeH (*
*X) of the
H-space X is contractible because of [7, Lemma 2.5 on page 22] because oeH (X) *
*is the union
of the sets XK for K H with K 6= 1 and XK1 \ XK2 = X, where i*
*s the
subgroup of H generated by K1 and K2.
(7) =) (3) If the -space X satisfies (7), the 0-space X satisfies (6) and hence*
* (3). Therefore
Hn(X; Z) = 0 for n d and Hd-1(Z; Z) is projective over Z0. Hence it remains to*
* show
that Hd-1(Z; Z) is projective over Z. Because of [7, Lemma 4.1 (a) on page 26] *
*it suffices to
show that the cohomological dimension of the Z-module Hd-1(Z; Z) is finite. As *
* satisfies
b(<1) by assumption it suffices to prove for any finite subgroup H that the Z*
*H-module
Hd-1(Z; Z) is projective. Since the singular set oeH (X) of the H-space X is co*
*ntractible and
agrees with the singular set oeH (Z) of the H-space Z we conclude Hj(Z; Z) = Hj*
*(Z; oeH (Z); Z)
for j 1. Since Z is (d - 2)-connected and the relative dimension of (Z; oeH (Z*
*)) is less or
equal to d - 1, we obtain an exact sequence of ZH-modules
0 -! Hd-1(Z; Z) -! Cd-1(Z; oeH (Z)) -! : :-:! C0(Z; oeH (Z)) -! 0:
Since each Cj(Z; oeH (Z)) is ZH-free, the ZH-module Hd-1(Z; Z) is projective.
(8) =) (4) Assume that the -CW -complex Z0 is obtained from X by attaching free*
* cells
x Dn of dimension n d - e - 1 and is (d - e - 2)-connected. By the argument a*
*bove
Hd-e-1(Z0; Z) is ZH-projective for any finite subgroup H . Since satisfies b*
*(e) by
assumption, the cohomological dimension of the Z-module Hd-e-1(Z0; Z) is less o*
*r equal to
e. Now attach free cells x Dn of dimension d - e n d - 1 to Z0 such that the*
* resulting
-CW -complex Z is (d-2)-connected. We conclude from the long exact homology seq*
*uences
associated to (Z; Z0), (Z; X) and (Z0; X) that Hj(Z; Z0; Z) = 0 for d - e + 1 *
*j d - 2,
Hd-1(Z; Z) = Hd-1(Z; Z0; Z) and Hd-e-1(Z0; Z) = Hd-e(Z; Z0; Z) hold. Hence we o*
*btain an
exact sequence
0 -! Hd-1(Z; Z) -! Cd-1(Z; Z0) -! : :-:! Cd-e(Z; Z0) -! Hd-e-1(Z0; Z) -! 0
This implies that Hd-1(Z; Z) is a projective Z-module. This finishes the proof *
*of Lemma
*
* __
1.9. *
* |__|
8
Proof. Now we are ready to give the proof of Theorem 1.6. First we show that (1*
*) implies
both (2) and (3). This follows directly from Lemma 1.2, Lemma 1.3 and the impli*
*cation (1)
=) (6) and (1) =) (7) of Lemma 1.9 applied to the singular set X = oeWH (E_WH) *
*of the
WH-space E_WH. It remains to show that (1) holds provided that there is an uppe*
*r bound
l on the length l(H) of finite subgroups H and (2) or (3) is true.
We construct inductively a nested sequence of -CW -complexes
; = X[l + 1] X[l] : : :X[0]
with the following properties. The -CW -complex X[n] is obtained from X[n + 1]*
* by at-
taching cells of the type =H x Dk for finite subgroups H with l(H) = n and k *
* d(H)
and X[n]H is contractible for any finite subgroup H with l(H) n. Then X[0] i*
*s the
desired model for E_. The induction begin n = l + 1 is obvious, the induction *
*step from
n + 1 to n done as follows.
Let H be a finite subgroup with l(H) = n. Since d(K) d(H) holds for all
finite subgroups K with H K, the dimension of X[n + 1]H is less or equal to *
*d(H).
In particular Hn(X[n + 1]H ; Z) = 0 for n d(H) + 1. Fix any ZWH-module M. Let *
*f :
X[n + 1] -! E_ be a -map. It induces a homotopy equivalence fK for all finite *
*subgroups
K with l(K) n + 1. Hence the WH-map oeWH (fH ) : oeWH (X[n + 1]H ) -! oeWH (*
*E_H )
induced by f is a (non-equivariant) homotopy equivalence by [7, Lemma 2.5 on pa*
*ge 22]
since oeWH (X[n + 1]H ) is the union of the spaces X[n + 1]K for all finite su*
*bgroup K
with H K; H 6= K \ NH and similiar for oeWH (E_H ) and any such group K satisf*
*ies
l(K) n + 1. We get from Lemma 1.2 and Lemma 1.3 an isomorphism
Hd(H)+1ZWH(EWH x (CoeWH (X[n + 1]H ); oeWH (X[n + 1]H )); M)
~= d(H)+1 H H
-! HZWH (EWH x (CoeWH (E_ ); oeWH (E_ )); M)
~= d(H)+1
-! HZWH (EWH x (CoeWH (E_WH); oeWH (E_WH)); M)
~= d(H)+1
-! HZWH (EWH x (C|P(WH)|; |P(WH)|); M)
= 0:
Since X[n + 1]H is obtained from oeWH (X[n + 1]H ) by attaching free WH-cells W*
*H x Dk for
k d(H), we get for any ZWH-module M an isomorphism
Hd(H)+1ZWH(EWH x (CX[n + 1]H ; X[n + 1]H ); M)
~= d(H)+1 H H
-! HZWH (EWH x (CoeWH (X[n + 1] ); oeWH (X[n + 1] )); M) *
*= 0:
Similiarly we get Hd(H)+1Z(H)(E(H)x(CX[n+1]H ; X[n+1]H ); M) = 0 for any Z(H)-m*
*odule
M. Now the implications (6) =) (1) and (7) =) (1) of Lemma 1.9 yield the existe*
*nce of
a contractible WH-CW -complex Y (H) which is obtained from X[n + 1]H by attachi*
*ng free
cells WH x Dm of dimension m d(H). Now define X[n] as the -push out
` H i `
(H) xNH X[n?+ 1] - --! (H) xNH?Y [H]
p?y ?y
X[n + 1] - --! X[n]
9
where (H) runs through the conjugacy classes of finite subgroups H with l(H) *
*= n,
i is the obvious inclusion and p sends (fl; x) 2 x H
NH X[n + 1] to flx 2 X[n +*
* 1]. Since
X[n]H = Y (H) and X[n]H = X[n + 1]H respectively for any finite subgroup H w*
*ith
l(H) = n and l(H) > n respectively, one easily verifies that X[n] has the desir*
*ed properties.
*
* __
This finishes the proof of Theorem 1.6. *
* |__|
The next result has already been proven by Kropholler and Mislin [12, Theo*
*rem B] for
certain classes of groups and in l exponential dimension estimate (see also Rem*
*ark 1.12).
We will give an in l-linear one.
Theorem 1.10. Let be a group and let l 0 and d 0 be integers such that the*
* length
l(H) of any finite subgroup H is bounded by l and satisfies B(d). Then ther*
*e is a
-CW -model for E_ such that for any finite subgroup H
dim(E_H ) = max{3; d} + (l - l(H))(d + 1)
holds. In particular E_ has dimension max {3; d} + l(d + 1).
Proof. The proof is a variation of the proof of Theorem 1.6 now using in the in*
*duction step
the implication (8) =) (1) proven in Lemma 1.9 instead of the implications (6) *
*=) (1) and
*
* __
(7) =) (1). *
* |__|
Example 1.11. Next we give an example of a group for which there is a 1-dimen*
*sional
-CW -model for E_ but no upper bound l on the length l(H) of finite subgroups. *
*Namely,
let be Z[1=p]=Z for a fixed prime p, where Z[1=p] Q consists of those rationa*
*l numbers
r for which there is a positive integer k satisfying pk . r 2 Z. Let n be th*
*e unique
subgroup of order pn in . Notice that n is cyclic of order pn and is the colim*
*it of the
nested sequence 1 2 : :.:Moreover, there are no other finite subgroups in th*
*an the
n-s.
`
Define an oriented graph T as follows. The set of vertices is n1 =n. W*
*e join
the vertex fl1n1 and the vertex fl2n2 by an oriented edge if and only if n2 = n*
*1 + 1 and
fl1n2 = fl2n2 2 =n2 holds. There is an obvious -action on T induced by the cano*
*nical
-action on =n. Obviously T is a 1-dimensional -CW -complex. It remains to show *
*that
it is a -CW -model for E_. The isotropy group of a vertex fln is n. The isotrop*
*y group
of any point on the oriented edge from fl1n1 to fl2n2 is n1. Hence T K is empty*
* for infinite
K . It remains to show that T n is a tree for n 0. Obviously T n is connected*
* since it
is the subgraph given by the vertices gk for k n and for any g 2 there is an *
*m 1 with
gm = 1m 2 =m . Let w be a loop in T n with base point gk given by a sequence of
edges joining the vertices g1n1, g2n2, : :,:grnr such that g1n1 = grnr = gk. We*
* show
by induction over r that the path is homotopic relative end points to the const*
*ant path at
the vertex gk. The induction begin r = 0 is trivial, the induction step from r *
*- 1 to r 1
done as follows.
Suppose that there is an integer i with 1 < i < r such that ni = ni-1- 1 =*
* ni+1- 1.
Then we have gi-1ni-1= gi+1ni+1. Since there is at most one edge joining two ve*
*rtices, we
can shorten the path without changing its homotopy type relative endpoints by o*
*mitting the
portion from gi-1ni-1to gi+1ni+1. It remains to consider the case where such an*
* integer
i does not exist. Then there must be an integer i such that n1 < n2 < : : :< n*
*i and
10
ni > ni+1 > : :>:nr holds. This implies gi-1ni-1= gni-1and gi+1ni+1= gni+1and
hence g
i-1ni-1= gi+1ni+1and we can shorten the path again. Hence T is a 1-dimen*
*sional
-CW -model for E_.
Notice that a tree T on which a group acts cellularly (without inversion)*
* with finite_
isotropy groups is always a -CW -model for E_ (see [18, I.6.1]). *
* |__|
Remark 1.12. In [12] the class HFIN of hierarchically decomposable groups a*
*nd the Z-
module B(; Z) of bounded functions from to Z is investigated and related to th*
*e question
of the existence of a finite-dimensional -CW -model for E_. By definition HFIN*
* is the
smallest class of group which contains all finite subgroups and which contains *
* if there is
a finite-dimensional contractible -CW -complex whose isotropy groups belong to *
*the class.
Define the following classes of groups where hdim means the homological dimensi*
*on:
A := { | hdimZ (B(; Z)) < 1};
B := { | satisfiesb(d)};
C := { | satisfiesB(d)};
D := { | there is a finite-dimensional-CW -complex with finite isotropy gr*
*oups
and Hp(X; Z) = Hp(*; Z) forp 0};
E := { | there is a finite-dimensional-CW -model forE_};
L := { | there is an upper boundl on the lengthl(H) of finite subgroupsH *
*}:
Then
A \ HFIN = B \ HFIN = C \ HFIN
follows directly from [12, Proposition 6.1, Lemma 7.3] and the fact that B(; Z)*
* is projective
over ZH for any finite subgroup H [13]. We conclude
A \ HFIN \ L = B \ HFIN \ L = C \ L = D \ L = E \ L
*
* __
from Lemma 1.5 and Theorem 1.10. *
* |__|
2. Coinduction
In this section we study coinduction and what it does to a classifying space fo*
*r a family.
Definition 2.1. Let i : be an inclusion of groups. Define the coinduction wi*
*th i of
a -space X to be the -space
i]X := map (; X);
where map (; X) is the space of -equivariant maps of -spaces. The -action is *
*given
by (flf)(fl0) := f(fl0fl) for f : ! X and fl; fl0 2 . Given a family F of su*
*bgroups of ,
define the family i]F of subgroups of by the set of subgroups H for which fl*
*-1Hfl \__
belongs to F for all fl 2 . *
* |__|
Notice that i] is the right adjoint of the restriction functor which sends*
* a -space Y to
the -space i*Y , namely, there is a natural homeomorphism
~=
map (i*Y; X) -! map (Y; i]X): *
*(2.2)
11
The next result reduces for torsionfree to Serre's Theorem that there is *
*a finite-
dimensional model for B if contains a subgroup of finite index with finite-di*
*mensional
B ([6, Theorem VIII.3.1. on page 190, Corollary VIII.7.2 on page 205], [17]).
Theorem 2.3. Let i : be an inclusion of groups and let F be a family of su*
*bgroups
of . Then
1. If X is a -CW -complex of dimension d and the index [ : ] is finite, then *
*i]X is
-homotopy equivalent to a d . [ : ]-dimensional -CW -complex;
2. i]E(; F)H is empty for H =2i]F and is contractible for H 2 i]F.
Proof. 1.) Let l be the index [ : ]. Fix elements fl1, fl2, : :,:fll in such*
* that \ =
{flk | k = 1; 2; : :;:l}. Evaluation at flk defines a homeomorphism
~=Yl
ev: i]X -! X:
k=1
Q l
Let G be the semi-direct product of l = k=1 and the symmetric group Sl in l-l*
*etters
with respect to the action of Sl on l given by oe . (ffik) = (ffioe-1(k)). The*
* group G acts in
Q l
the obvious way on the target k=1X of ev. Let s : -! Sk be the group homomor*
*phism
satisfying fls(fl)(k)fl = flk 2 \ for all fl 2 and k 2 {1; : :;:l}. Then we o*
*btain an
embedding of group i j
i : -! G fl 7! flkflfl-1s(fl)-1(k). s(fl)
k
such that ev is a -homeomorphism with the given -actionQon the source and the r*
*estricted
-action on the target. Hence it suffices to show that lk=1X is G-homotopy eq*
*uivalent
to a dl-dimensional G-CW -complex Y . Recall that a G-CW -structure is the sam*
*e as a
CW -structure such that the G-action permutes the cells and for any g 2QG and c*
*ell e with
g . e = e multiplication with g induces the identity on e. Notice that lk=1X *
*has an obvious
l-CW -structure. However, the G-action does respect the skeleta and permutes t*
*he cells
but it happens that multiplication with an element g 2 G maps a cell to itselfQ*
*but not by
the identity so that this does not give the structure of a G-CW -complex on l*
*k=1X. This
problem can be solved by using simplicial complexes with equivariant simplicial*
* actions using
the fact that the second barycentric subdivision is an equivariant CW -complex *
*but then one
has to deal with product of simplicial complexes. We give another approach.
We next construct a nested sequence ofiG-CW -complexesj; = Y-1 Y0 : : :Y*
*ld
Q l
together with G-homotopy equivalence fn : k=1X -! Yn such that Yn is n-di*
*mensional
n
and fn is an extension of fn-1 for all n 0. Then fldis the desired -homotopy e*
*quivalence.
The begin n = -1 of the induction is trivial, the induction step from n - 1 to *
*n done as
follows.
For p 0 choose a -push out
` p-1
ff2Ip=Lffx?S ---! Xp-1?
? ?
y y
` p
ff2Ip=Lffx D ---! Xp
12
Define an index set
l
X
Jn := {(i_; ff_) | i_= (i1; i2; : :i:l); ik = 0; : :n:; ik = n; ff_= (ff1*
*; : :f:fl) 2 Ii1x : :x:Iil}:
k=1
Consider the space
!
a Yl
A = =Lffkx Dik :
(i_;ff_)2Jnk=1
There is an obvious Sl-action on Jn given by permutating the coordinates which *
*induces
an Sl-action on A. Together with the obvious l-action this yields a G-action on*
* A. Next
we show that (A; @A) carries the structure of a pair of G-CW -complexes with di*
*mension n
where @A is the topological boundary of A. Fix (i_; ff_) 2 Jn.Q Let S0l Sl be *
*the isotropy
group of (i_; ff_) under the Sl-action on Jn. Then S0lacts on lk=1Dik by per*
*muting the
Q l Q l
coordinates. Equip the pair ( k=1Dik; @ k=1Dik) with a S0l-CW -complex struc*
*ture. This
induces a G-CW -structure on the pair
0 1
a Yl a Yl
=LffkxDik; @ @ =Lffkx DikA*
* :
(i0___;ff0_)2Jn;Sl(i0_;ff0_)=Sl(i_;ff_)2Sl\Jnk=1(i0_;ff0_)2Jn;Sl(i0_;ff0_)=Sl(i*
*_;ff_)2Sl\Jnk=1
Since (A; @A) is the disjoint union of such pairs, it carries a G-CW -structure*
*. We have the
G-push out
q i Q l j
@A ---! k=1X
? ? n-1
? ?
y y
iQ j
A ---! lk=1X
n
By the equivariant cellular approximation theorem [10, II.2] we can choose a G-*
*homotopy h
from fn-1 O q to a cellular G-map c : @A -! Yn-1. Define Yn by the G-push out
@A - -c-! Yn-1
? ?
? ?
y y
A - --! Yn
Since c is cellular, (A; @A) is a pair of n-dimensional G-CW -complexes and Yn-*
*1 an (n - 1)-
dimensional G-CW -complex, Yn is an n-dimensional G-CW -complex. Using [14, Le*
*mma
2.13 onipage 38] and the G-homotopy h one constructs the desired G-homotopy equ*
*ivalence
Q l j
fn : k=1X -! Yn. This finishes the proof of Theorem 2.3.1.
n
2.) Let Z be any (non-equivariant) space and let H be a subgroup. We have t*
*he
isomorphism of -sets
al ~
=(flkHfl-1k\ ) =-!i*=H ffi(flkHfl-1k\ ) 7! ffiflkH:
k=1
13
We get from 2.2 the following composition of homeomorphisms
H *
map Z; (i]E(; F)) ~= map (=H x Z; i]E(; F)) ~= map (i =H x Z; E(; F)) *
* ~=
l !
a Yl
map =(flkHfl-1k\ ) x Z; E(; F) ~= map =(flkHfl-1k\ ) x Z; E(; F)
k=1 k=1
Y l i j
~= map Z; E(; F)flkHfl-1k\ :
k=1
Notice that H belongs to i]F if and only if flkHfl-1k\ belongs to F for k = 1;*
* : :l:. Hence
for any space Z the space map Z; (i]E(; F))H is empty if H does not belong to*
* i]F and
*
* __
is path-connected otherwise. This finishes the proof of Theorem 2.3. *
* |__|
Theorem 2.3 implies
Theorem 2.4. Let i : be an inclusion of a subgroup of finite index [ : ] a*
*nd F
be a family of subgroups of . Suppose that E(; F) has a -CW -model of dimension*
* d.__
Then E(; i]F) has a -CW -model of dimension d . [ : ]. *
* |__|
If we take in Theorem 2.4 the family F as the family of finite subgroups o*
*f or
virtually cyclic subgroup of respectively, then i]F is the family of finite su*
*bgroups of or
virtually cyclic subgroup of respectively.
3. Short exact sequences
Let 1 ! ! ! ss ! 1 be an exact sequence of groups. In this section we want to
investigate whether finiteness conditions about the type of a classifying space*
* for FIN for
and ss carry over to the one of .
p
Theorem 3.1. Let 1 ! -i! -! ss ! 1 be an exact sequence of groups. Suppose *
*that
there exists a positive integer d which is an upper bound on the orders of fini*
*te subgroups
of ss. Suppose that E_ has a k-dimensional -CW -model and E_ss has a m-dimensi*
*onal
ss-CW -model. Then E_ has a (dk + m)-dimensional -CW -model.
Proof. Let E_ss be a m-dimensional ss-CW -model for the classifying space of ss*
* for the family
of finite subgroups. Let p*E_ss be the -space obtained from E_ss by restriction*
* with p. We
will construct for each -1 n m a (dk + n)-dimensional -CW -complex Xn and a -*
*map
fn : Xn -! p*E_ssn to the n-skeleton of p*E_ss such that Xn-1 is a -CW -subcomp*
*lex of Xn,
fn restricted to Xn-1 is fn-1, for each finite subgroup H the map fHn: XHn- !*
* p*E_ssHnis
a homotopy equivalence and all isotropy groups of Xn are finite. Then Xm will b*
*e the desired
(dk + m)-dimensional -CW -model for E_. The begin n = -1 is given by the identi*
*ty on
the empty set, the induction step from n - 1 to n 0 is done as follows.
Choose a -pushout
14
` n-1 `i2Iqi *
i2I=Lix?S ----! p E_ssn-1?
? ?
y y
` n *
i2I=Lix D - --! p E_ssn
There is a canonical projection pri: xLiE_Li -! =Li: Fix a map of sets s : =Li*
* -!
whose composition with the projection -! =Liis the identity on =Li. Given an L*
*i-space
Y , there is a natural homeomorphism
a -1 ~= H
Y s(w) Hs(w)-!( xLiY )
w2=Li;s(w)-1Hs(w)Li
-1Hs(w) H
which sends y 2 Y s(w) to (s(w); y). Hence pri is a homotopy equivalence fo*
*r all finite
subgroups H and the isotropy groups of xLiE_Li are finite. Since fHn-1is a h*
*omotopy
equivalence for all finite subgroups H , there is a cellular -map
ri: ( xLiE_Li) x Sn-1 -! Xn-1
together with a -homotopy
h : ( xLiE_Li) x Sn-1 x [0; 1] -! p*E_ssn-1
from fn-1 O ri to qiO (prix id) [14, Proposition 2.3 on page 35]. Define the fo*
*llowing three
-spaces by the -push outs
` n-1 `i2IqiO(prix id)*
i2I( xLiE_Li)?x S ----------! p E_ssn-1?
? ?
y y
` n 0
i2I( xLiE_Li) x D - --! Xn
and
` n-1 ` i2Ifn-1Ori*
i2I( xLiE_Li)?x S -------! p E_ssn-1?
? ?
y y
` n 00
i2I( xLiE_Li) x D - --! Xn
and
` n-1 `i2Iri
i2I( xLiE_Li)?x S ----! Xn-1?
? ?
y y
` n
i2I( xLiE_Li) x D ---! Xn
` We obtain a`-map f0n: X0n-! p*E_ssn by the -pushout property and the -maps
00 00 00 0
i2Iprix idDn, i2Iprix idSn-1and idp*E_ssn-1. Let fn : Xn :`Xn -! Xn be *
*the -
homotopy equivalence relative p*E_ssn-1 induced by the -homotopy i2Ihi. Let f*
*000n: Xn -!
15
` `
X00nbe the -map induced by the -maps i2Iid n, i2Iid n-1and *
*fn-1
xLiE_Li)xD xLiE_Li)xS
and the -pushout property. We define the -map fn : Xn -! p*E_ssn by the composi*
*tion
f0nO f00nO f000n. Since (f0n)H and (f000n)H are homotopy equivalences for finit*
*e H as they are
push outs of homotopy equivalences [14, Lemma 2.13 on page 38] and f00nis a -ho*
*motopy
equivalence, fHn is a homotopy equivalence for all finite H .
We conclude that the resulting -CW -complex Xn has only finite isotropy gr*
*oups and
fH is a homotopy equivalence for each finite H . Notice that each cell =Li x*
* Dn
of p*E_ss satisfies Li = p-1(Hi) for some finite subgroup Hi ss and hence Li c*
*ontains
as subgroup of finite index |Hi|. Theorem 2.4 implies that that E_Li can be ch*
*oosen as a
*
* __
m|Hi|-dimensional Li-CW -complex. Hence the dimension of Xn is at most dk + n. *
* |__|
p
Theorem 3.2. Let 1 ! -i! -! ss ! 1 be an exact sequence of groups. Suppose *
*for
any finite subgroup ss0 ss and any extension 1 ! ! 0 ! ss0 ! 1 that E_0 has a
finite 0-CW -model or a 0-CW -model of finite type respectively and suppose tha*
*t E_ss has
a finite ss-CW -model or a ss-CW -model of finite type respectively. Then E_ h*
*as a finite
-CW -model or a -CW -model of finite type respectively.
Proof. The proof is exactly the same as the one of Theorem 3.1 except for the v*
*ery last step.
Namely, we have to know that E_Li has a finite Li-CW -model or a Li-CW -model o*
*f finite
*
* __
type respectively and this follows from the assumptions. *
* |__|
Notice that the assumption about E_0in Theorem 3.2 would follow from the a*
*ssump-
tion that E_ has a finite -CW -model or a -CW -model of finite type respectivel*
*y if
Problem 7.2 has an affirmative answer. If ss is torsionfree, this assumption a*
*bout E_0 re-
duces to the assumption that there is a finite -CW -model or a -CW -model of fi*
*nite type
for E_. If is word-hyperbolic, then its Rips complex yields a finite -CW -mod*
*el for
E. _____Moreover a group is word-hyperbolic if it contains a word-hyperbolic su*
*bgroup of finite
index, because the property word-hyperbolic is a quasi-isometry invariant. It i*
*s not hard to
check that any virtually poly-cyclic group has a finite -CW -model for E_. He*
*nce we
conclude from Theorem 3.2
p
Theorem 3.3. Let 1 ! -i! -!ss ! 1 be an exact sequence of groups. Suppose th*
*at is
word-hyperbolic or virtually poly-cyclic. Suppose that E_ss has a finite ss-CW *
*-model or a ss-
CW -model of finite type respectively. Then E_ has a finite -CW -model or a -CW*
* -model __
of finite type respectively. *
* |__|
4. Classifying spaces of finite type
In this section we deal with the question whether there are -CW -models of*
* finite type
for E_.
Lemma 4.1. If there is a -CW -complex Y of finite type which has only finite*
* isotropy
groups and is (non-equivariantly) contractible, then E has a -CW -model of fini*
*te type.
16
Proof. The idea is to replace a cell =Hix Dn in Y by the -space xHi EHix Dn. T*
*he
construction is an obvious modification of the construction in the proof of The*
*orem 3.1 using
the fact that EHican be choosen as a Hi-CW -complex of finite type. Namely, we *
*construct
for n -1 a -CW -complex Xn and a -map fn : Xn -! Yn to the n-skeleton of Y such
that Xn is obtained from Xn-1 by a -push out of the shape
` n-1
i2I( xHi EHi)?x S ---! Xn-1?
? ?
y y
` n
i2I( xHi EHi) x D ---! Xn
and fn extends fn-1 and is a non-equivariant homotopy equivalence. Then the co*
*limit
*
* __
X = colimn!1 Xn is the desired -CW -model for E. *
* |__|
Theorem 4.2. The following statements are equivalent for the group .
1. There is a -CW -model for E_ of finite type;
2. There are only finitely many conjugacy classes of finite subgroups of and*
* for any
finite subgroup H there is a CW -model for BWH of finite type;
3. There are only finitely many conjugacy classes of finite subgroups of and*
* for any
finite subgroup H the Weyl group WH is finitely presented and is of type*
* F P1 ,
i.e. there is a projective ZWH-resolution of finite type of the trivial ZW*
*H-module Z.
Proof. 1.) =) 2.) Let E_ be a -CW -model of finite type. Let =H1, : :,:=Hn b*
*e the
finitely many equivariant 0-cells . Then any other equivariant -cell =K x Dn in*
* E_ must
have the property that K is subconjugated to one of the Hi-s because the existe*
*nce of a
-map from =K to =L is equivalent to K being subconjugated to L. Since each Hi *
*is
finite and has only finitely many distinct subgroups, we conclude that there ar*
*e only finitely
many conjugacy classes of finite subgroups in . Now apply Lemma 1.3 and Lemma 4*
*.1.
2.) =) 1.) Let (H1), : :,:(Hr) be the conjugacy classes of finite subgroups wit*
*h a numeration
such that (Hi) is subconjugated to (Hj) only if i j. We construct inductively*
* -CW -
complexes X0, X1, : :,:Xr such that X0 is empty, Xn is obtained from Xn-1 by at*
*taching
cells =Hn x Dm for m 0, Xn is of finite type and XHnnis contractible. Notice t*
*hat then
Xr is a -CW -model of finite type for E_ because XHin= XHin-1for i < n.
In the induction step from n - 1 to n it suffices to construct an extensio*
*n of WHn-
CW -complexes XHnn-1i-!Z such that Z is contractible, obtained from XHnn-1by at*
*taching free
WHn-cells and of finite type. Then one defines Xn as the -pushout
idxNHni
xNHn XHnn-1 ------! xNHn Z
? ?
j?y ?y
Xn-1 ---! Xn
17
where j maps (fl; x) to flx. Notice that XHnn= ZHn . Since each isotropy group *
*of the WHn-
space XHn
n-1 is finite, we can construct a free WHn-CW -complex Y together with a*
* WHn-map
h : Y -! XHnn-1which is a (non-equivariant) homotopy equivalence by substituti*
*ng each
cell WHn=L x Dm by WHn xL EL x Dm . This construction is an easy modification o*
*f the
construction in the proof of Theorem 3.1. Since L is finite and hence EL can be*
* choosen as a
free L-CW -complex of finite type, Y is of finite type. Let f : Y - ! EWHn be t*
*he classifying
WHn-map. We can choose EWHn of finite type by assumption. Let i : Y -! cyl(f*
*) be
the inclusion into the mapping cylinder of f. Notice that cyl(f) is a WHn-CW -*
*model for
EWHn of finite type. Define Z by the WHn-pushout
Y --i-! cyl(f)
? ?
h?y ?yg
XHnn-1---! Z
where i is the inclusion. As h is a (non-equivariant) homotopy equivalence, g *
*is a (non-
equivariant) homotopy equivalence. Hence Z is contractible and of finite type.
2.) () 3.) is a variation of the proof of [6, Theorem VIII.7.1 on page 205*
*] or follows
*
* __
from [14, Proposition 14.9 on page 182]. *
* |__|
5. Finitely dominated and finite classifying spaces
In this section we deal with the question whether there are finitely domin*
*ated or finite
-CW -models for E_. Recall that a -CW -complex X is called finitely dominated i*
*f there
are a finite -CW -complex Y and -maps r : Y - ! X and i : X -! Y such that r O *
*i is
-homotopic to the identity.
Theorem 5.1. The following statements are equivalent for the group .
1. There is a finitely dominated -CW -model for E_;
2. There are only finitely many conjugacy classes of finite subgroups of and*
* for any
finite subgroup H the Weyl group WH is finitely presented, is of type F *
*P1 and
satisfies condition b(d) for some d 0.
Proof. A finitely dominated -CW -complex is -homotopy equivalent to a -CW -comp*
*lex
of finite orbit type [14, Proposition 2.12 on page 38]. A -CW -complex X of fin*
*ite orbit type
is finitely dominated if and only if it is -homotopy equivalent to both a -CW -*
*complex
Y of finite type and to a finite-dimensional -CW -complex Z. The argument in th*
*e proof
of [14, Proposition 14.9.a on page 282] applies to the general case. Hence the *
*claim follows
*
* __
from Lemma 1.5, Theorem 1.10 and Theorem 4.2. *
* |__|
If one knows that E_ is finitely dominated, then there is the equivariant *
*finiteness
obstruction
eo(E_) 2 eK0(Z Or(; FIN )) ~={(H)|H2FIN }eK0(ZWH)
18
whose vanishing is a necessary and sufficient condition for E_ being -homotopy *
*equivalent
to a finite -CW -complex ([14, Theorem 14.6 on page 278], [14, Theorem 10.34 on*
* page
196]).
Remark 5.2. If is a word-hyperbolic group, its Rips complex yields a finite *
*-CW -
model for E_. If is a discrete cocompact subgroup of a Lie group G with finit*
*ely many
components, then G=K with the left -action for K G a maximal compact subgroup *
*is
a finite -CW -model for E_ [1]. If contains Zn as subgroup of finite index, th*
*ere exists
an epimorphism of to a crystallographic group with finite kernel (as pointed o*
*ut to us by
Frank Connolly) and hence an n-dimensional -CW -model for E_ with Rn as underly*
*ing
space. Any virtually poly-cyclic group has a finite -CW -model for E_. If is *
*1 *0 2
for finite groups i, then there is a finite 1-dimensional -CW -model for E_ [18*
*, Theorem 7
*
* __
in I.4.1 on page 32]. *
* |__|
6. Groups with finite virtual cohomological dimension
In this section we investigate the condition b(d) and B(d) of Notation 1.4*
* and explain
how our results specialize in the case of a group of finite virtual cohomologic*
*al dimension.
Lemma 6.1. If satisfies b(d) or B(d) respectively, then any subgroup of sa*
*tisfies b(d)
or B(d) respectively.
Proof. We begin with b(d). Let M be a Z-module which is projective over ZH for *
*all finite
subgroups H . For a subgroup K the double coset formula gives an isomorphism
of ZK-modules
resKind M ~= Kfl2K\= indKflfl-1\Kresc(fl)M; *
*(6.2)
where c(fl) : flfl-1 \ K -! maps ffi to fl-1ffifl. If K is finite, each subgro*
*up \ fl-1Kfl
of is finite and hence the Zflfl-1 \ K-module resc(fl)M is projective. We conc*
*lude from
6.2 that the ZK-module resKind M is projective for all finite subgroups K . S*
*ince
satisfies b(d) by assumption the Z-module ind M has a projective resolution of *
*dimension
d. If one applies 6.2 to K = , then one concludes that the Z-module M is a dir*
*ect
summand in res ind M. Hence M has a projective Z-resolution of dimension d. *
*This
finishes the proof for b(d). The claim for B(d) follows since the Weyl group W *
* H of H in
*
*__
is a subgroup of the Weyl group W H of H in for any subgroup H . |*
*__|
Recall that a group has virtual cohomological dimension d if and only if*
* it contains
a torsionfree subgroup of finite index such that the trivial Z-module Z has a *
*projective
Z-resolution of dimension d.
Theorem 6.3. Let be a group which contains a torsionfree subgroup of finite*
* index.
Then the following assertions are equivalent:
1. satisfies B(d);
2. satisfies b(d);
19
3. has virtual cohomological dimension d.
Proof. 1.) =) 2.) is obvious.
2.) =) 3.) The subgroup satisfies b(d) by Lemma 6.1. Since is torsionfree thi*
*s shows
that the virtual cohomological dimension of is less or equal to d.
3.) =) 1.) Suppose that the virtual cohomological dimension of is d. Next we *
*show
that then satisfies b(d). Let M be a Z-module such that resHM is projective fo*
*r all finite
subgroups H . Then M has a d-dimensional projective Z-resolution by [6, Theor*
*em
VI.8.12 on page 152 and Proposition X.5.2 and Theorem X.5.3 on page 287]. Hence*
* satisfies
b(d). Next we show that satisfies B(d). Let H be a finite subgroup. Then \ *
*N H
is a subgroup of and hence has virtual cohomological dimension d. Since \ H i*
*s trivial,
\ N H is a subgroup in W H of finite index. Hence W H satisfies b(d) by the as*
*sertion
for b(d) we have just proven above. Therefore satisfies B(d). This finishes *
*the proof of
*
* __
Theorem 6.3. *
* |__|
We rediscover from Theorem 5.1 and Theorem 6.3 the result of [7, Theorem I*
* on page
18] that a group with finite virtual cohomological dimension has a finitely do*
*minated -
CW -model for E_ if and only if has only finitely many conjugacy classes of fi*
*nite subgroups
and for each finite subgroup H its Weyl group WH is finitely presented and of*
* type
F P1 .
Next we improve Theorem 1.10 in the case of groups with finite virtual coh*
*omological
dimension. Notice that for such a group there is an upper bound on the length l*
*(H) of finite
subgroups H .
Theorem 6.4. Let be a group with virtual cohomological dimension d. Let l *
*0 be an
integer such that the length l(H) of any finite subgroup H is bounded by l. T*
*hen there
is a -CW -model for E_ such that for any finite subgroup H
dim (E_H ) = max{3; d} + l - l(H)
holds. In particular E_ has dimension max {3; d} + l.
Proof. We want to use the implication (3) =) (1) of Theorem 1.6 where the subgr*
*oup
(H) WH is given by the image of \ NH under the projection NH -! WH. Because
of Theorem 6.3 the group satisfies B(d). Hence it remains to show
Hmax{3;d}+l-l(H)+1Z(H)(E(H) x (C|P(WH)|; |P(WH)|); M) = 0
for any Z(H)-module M. Notice that (H) is isomorphic to \ NH since is tor-
sionfree and H is finite and hence there is a d-dimensional model for E(H). The*
*refore it
suffices to show that |P(WH)| can be choosen to be (l - l(H) - 1)-dimensional b*
*ecause then
Hmax{3;d}+l-l(H)Z(H)(E(H) x |P(WH)|; M) = 0. This follows from
max {l(K) | K WH; K finite} l - l(H);
__
max {l(K) | K WH; K finite} = dim (|P(WH)|) + 1: |__|
20
7. Problems
In this section we discuss some open problems. We emphasize that we state most*
* of the
problems not because we have a strategy to prove them, but because we lack coun*
*terexam-
ples.
Problem 7.1. Which of the results in this paper carry over to a Lie group or *
*more general
to a (locally compact) topological group G and the classifying space E(G; K) fo*
*r the_family
K of compact subgroups? *
* |__|
Problem 7.2. Let be a subgroup of finite index. Suppose that E_ has a -CW*
* -
model of finite type or a finite -CW -model respectively. Does then E_ have a -*
*CW -model__
of finite type or a finite -CW -model respectively? *
* |__|
Remark 7.3. If Problem 7.2 has a positive answer, then the following is true *
*for an exact
sequence of groups 1 ! ! ! ss ! 1. If E_ has a finite -CW -model or a -CW -
model of finite type respectively and E_ss has a finite ss-CW -model or a ss-CW*
* -model of
finite type respectively, then E_ has a finite -CW -model or a -CW -model of fi*
*nite type
*
* __
respectively (cf. Theorem 3.2). *
* |__|
There are some cases where the answer to Problem 7.2 is positive, for exam*
*ple if is
word-hyperbolic or virtually poly-cyclic (see Theorem 3.3). Suppose for instanc*
*e that is
torsionfree. There is a normal subgroup 0 of finite index such that 0 holds. *
*If
B is of finite type or is finite respectively, then B0is of finite type or is f*
*inite respectively.
If ss is the finite group =0, then we have a fibration B -! B -! Bss. Since B0a*
*nd
Bss are of finite type, B is of finite type [15, Lemma 7.2]. We have shown that*
* B is of
finite type if is torsionfree and B is of finite type. If B is finite and is *
*torsionfree,
then B is finitely dominated since it is of finite type and by Theorem 2.4 fini*
*te-dimensional
[14, Proposition 14.9 on page 282]. In order to check that B is finite, one has*
* to compute
its finiteness obstruction eo(B) which takes values in fK0(Z). Notice that the*
*re is the
conjecture that fK0(Z) vanishes for torsionfree groups . These considerations s*
*how that
for torsionfree the answer to Problem 7.2 is positive if one asks for "finite *
*type" and very
likely to be positive if one asks for "finite".
There are some reasons to believe that the answer to Problem 7.2 is not al*
*ways positive
or that its proof is difficult. Consider the special case where OE : -! is an*
* automorphism
of a group with E_ of finite type such that OEn = id, the group is the semi-d*
*irect product
of and Z=n with respect to OE, and H is just Z=n. Let CH be the centralizer *
*of H in
. Since H is finite and hence has only finitely many automorphisms, CH has fini*
*te index
in NH. Then CH \ is the fixed point set Fix(OE) of OE and has finite index in *
*NH. Hence
because of Theorem 4.2 and [15, Lemma 7.2] a positive answer to Problem 7.2 wou*
*ld imply
for any periodic automorphism of a group with E_ of finite type that B Fix(OE)*
* has finite
type and in particular that Fix(OE) is finitely presented.
A positive answer to Problem 7.2 would imply also a positive answer to the*
* following
problem by Theorem 4.2.
Problem 7.4. If the group contains a subgroup of finite index which has a -*
*CW -
model of finite type for E_, does then contain only finitely many conjugacy cl*
*asses of
21
*
* __
finite subgroups? *
* |__|
To our knowledge Problem 7.4 is open even for torsionfree (see also [6, L*
*emma
IX.13.2 on page 267]). A stonger version would be the following problem which *
*may be
viewed as the 0-skeleton version of the conclusion of Remark 7.3.
p
Problem 7.5. Let 1 ! -i! -!ss ! 1 be an exact sequence such that and ss hav*
*e only
finitely many conjugacy classes of finite subgroups. Does then have only fini*
*tely many
*
* __
conjugacy classes of finite subgroups? *
* |__|
In view of Theorem 4.2 a positive answer to the next problem implies a pos*
*itive answer
to Problem 7.2 in the case of finite type provided that has only finitely many*
* conjugacy
classes of finite subgroups because B is of finite type if for some subgroup o*
*f finite index
B is of finite type [15, Lemma 7.2].
Problem 7.6. Let be a group such that B is of finite type. Is then BWH of fi*
*nite_type
for any finite subgroup H ? *
* |__|
An algebraic analogoue of Problem 7.6 would be the question whether for a *
*group
of type F P1 and any finite subgroup H the Weyl group WH is of type F P1 .
Remark 7.7. If Problem 7.6 has a positive answer, then Theorem 4.2 would impl*
*y that
there is a -CW -model of finite type for E_ if and only if is finitely present*
*ed and of type
*
* __
F P1 and there are only finitely many conjugacy classes of finite subgroups in*
* . |__|
p
Problem 7.8. Let 1 ! -i! -!ss ! 1 be an exact sequence of groups. Suppose th*
*at there
is a -CW -model of finite type for E_ and a -CW -model of finite type for E_. *
*Is then __
there a ss-CW -model of finite type for E_ss? *
* |__|
The answer to Problem 7.8 is yes if , and ss are torsionfree [15, Lemma 7*
*.2].
Since for instance the kernel of the obvious epimorphism from the free group of*
* rank 2 to its
abelianization is not even finitely generated, it does not make sense to ask a *
*similiar question
where one has information about the classifying spaces for and ss and wants to*
* conclude
something for the one of .
Problem 7.9. Suppose that there is a finite-dimensional -CW -complex or finit*
*e -CW -
complex or -CW -complex of finite type respectively whose isotropy groups are a*
*ll finite and
which is contractible. Is then there a finite-dimensional -CW -model or finite *
*-CW -model_
or -CW -model of finite type respectively for E_? *
* |__|
The answer to Problem 7.9 is positive in the case finite-dimensional if c*
*ontains a
normal torsionfree subgroup of finite index by Theorem 2.4. A positive answer t*
*o Problem
7.9 in the case finite-dimensional is given for groups of type F P1 in [12, Th*
*eorem A]. More
generally we get a positive answer to Problem 7.9 in the case finite-dimensiona*
*l from Lemma
1.5 and Theorem 1.10 provided that there is an upper bound l on the length l(H)*
* of the
finite subgroups H . Problem 7.9 in the case of finite type is equivalent to P*
*roblem 7.6
by Lemma 4.1 and Theorem 4.2 provided that there are only finitely many conjuga*
*cy classes
of finite subgroups of .
22
Recall that the condition B(d) is necessary for the existence of a d-dimen*
*sional -CW -
model for E
__ (see Lemma 1.5). This is not true for the condition appearing in T*
*heorem 1.6
and Theorem 1.10 that there is an upper bound l on the length l(H) of the finit*
*e subgroups
H (see Example 1.11). Therefore the question arises:
Problem 7.10. For which groups the following is true:
1. For d 3 the condition B(d) (or the condition b(d) alone) is equivalent to*
* the existence
of a d-dimensional -CW -model for E_;
2. The statements (1), (2) and (3) appearing in Theorem 1.6 are always equiva*
*lent (with-
out the assumption that there is an upper bound on the length l(H) of fini*
*te_subgroups
H ). *
*|__|
Because of Theorem 6.3 Problem 7.10 (1) reduces in the case that contains*
* a tor-
sionfree subgroup of finite index to the problem stated by Brown [5, page 32].
Problem 7.11. For which groups of virtual cohomological dimension d there ex*
*ists_a
d-dimensional -CW -model for E_. *
* |__|
More generally one may ask
Problem 7.12. Suppose that contains a subgroup of finite index such that th*
*ere_is_a d-
dimensional -CW -model for E_. Is then there a d-dimensional -CW -model for E_?*
* |__|
Remark 7.13. A positive answer to Problem 7.12 would imply for an extension o*
*f groups
1 ! ! ! ss ! 1 that E_ has a d + e-dimensional -CW -model if E_ has a d-
dimensional -CW -model and E_ss has a e-dimensional ss-CW -model. In this cont*
*ext the
*
* __
question is interesting whether satisfies B(d+e) if satisfies B(d) and ss sat*
*isfies B(e). |__|
Problem 7.14. Can one give nice algebraic conditions which ensure the vanishi*
*ng of the
finiteness obstruction eo(E_) provided that E_ is finitely dominated? Is there*
* at all an_
example where E_ is finitely dominated but does not admit a finite -CW -model? *
* |__|
Here is a suggestion for a solution of Problem 7.14. A (finite) permutatio*
*n Z-module
is a Z-module M which is isomorphic to a (finite) sum of Z-modules of the shape*
* Z=H
for some finite subgroup H .
Problem 7.15. Suppose that satisfies the following condition: There are onl*
*y finitely
many conjugacy classes of subgroups in and for each finite subgroup H the tr*
*ivial
ZWH-module Z has a finite-dimensional ZWH-resolution by finite ZWH-permutation *
*mod- __
ules. Is then there a finite -CW -model for E_? *
* |__|
Notice that the condition in Problem 7.15 is necessary for the existence o*
*f a finite
-CW -model for E_ by Lemma 1.3 and Theorem 5.1 since such a resolution of Z is *
*given
by the ZWH-chain complex of E_WH. Moreover, if the condition is satisfied, E_ i*
*s at least
finitely dominated by Theorem 5.1. Problem 7.15 has a positive answer if for e*
*ach finite
subgroup H and each exact ZWH-sequence 0 ! Q ! P1 ! P2 ! : : :! Pn ! 0
such that Q is finitely generated projective and Pi is a finite permutation ZWH*
*-module for
1 i n the ZWH-module Q is stably free, i.e. becomes free after taking a dire*
*ct sum
23
with a finitely generated free ZWH-module. The most optimistic version based on*
* Remark
7.7 and Problem 7.10 would be
Problem 7.16. For which groups are the following statements true for an integ*
*er d 3?
1. There is a d-dimensional -CW -model for E_ if and only satisfies b(d).
2. There is a -CW -model of finite type for E_ if and only if is finitely pr*
*esented, has
only finitely many conjugacy classes of finite subgroups and is of type F *
*P1 .
3. There is a d-dimensional finite -CW -model for E_ if and only if is finit*
*ely presented,
has only finitely many conjugacy classes of finite subgroups, is of type F*
* P1 and_satisfies
b(d). *
* |__|
References
[1] H. Abels, Parallizability of proper actions, global K-slices and maximal c*
*ompact sub-
groups,, Math. Ann. 212,(1974), 1 - 19
[2] Baum, P., Connes, A., and Higson, N.: "Classifying space for proper actio*
*ns
and K-theory of group C*-algebras", in: Doran, R.S. (ed.), C*-algebras, Co*
*ntemporary
Mathematics 167, 241-291, (1994)
[3] Borel, A. and Serre, J-P.: "Corners and arithmetic groups", Comment. Math.*
* Helv.
48, 436 - 491 (1974)
[4] Bredon, G.: "Introduction to compact transformation groups", Academic Pres*
*s (1972)
[5] Brown, K.S.: "Groups of virtually finite dimension", in "Homological group*
* theory",
editor: Wall, C.T.C., LMS Lecture Notes Series 36, Cambridge University Pr*
*ess, 27 -
70 (1979)
[6] Brown, K.S.: "Cohomology of groups", vol. 87 of Graduate texts in Mathema*
*tics,
Springer (1982)
[7] Connolly, F. and Kozniewski, T.: "Finiteness properties of classifying sp*
*aces of
proper -actions", Journal of Pure and Applied Algebra 41, 17-36 (1986)
[8] Davis, J. and L"uck, W.: "Spaces over a category and assembly maps in isom*
*orphism
conjectures in K-and L-theory", MPI preprint, to appear in K-Theory (1996)
[9] tom Dieck, T.: "Orbittypen und "aquivariante Homologie I", Arch. Math. 23,*
* 307 - 317
(1972).
[10] tom Dieck, T.: "Transformation groups", Studies in Math. 8, de Gruyter (19*
*87)
[11] Farrell, F.T. and Jones, L.E.: "Isomorphism conjectures in algebraic K-the*
*ory", J.
of the AMS 6, 249 - 298 (1993).
24
[12] Kropholler, P.H. and Mislin, G.: "Groups acting on finite dimensional spac*
*es with
finite stabilizers", Comment. Math. Helv. 73 (1998), 122-136.
[13] Kropholler, P.H. and Talelli, O: "On a property of fundamental groups of g*
*raphs of
finite groups", J. of Pure and Applied Algebra 74, 57 - 59 (1991)
[14] L"uck, W.: "Transformation groups and algebraic K-theory", vol. 1408 of Le*
*cture notes
in mathematics, Springer (1989)
[15] L"uck, W.: "Hilbert modules and modules over finite von Neumann algebras a*
*nd appli-
cations to L2-invariants", Math. Annalen 309, 247 - 285 (1997)
[16] L"uck, W. and Oliver, R., "The completion theorem in K-theory for proper a*
*ctions
of a discrete group". preprint (1998)
[17] Serre, J-P.: "Cohomologie des groupes discrets", Ann. of Math. Studies 70,*
* 77 - 169
(1971)
[18] Serre, J-P.: "Trees", Springer (1980)
[19] Steenrod, N.E.: "A convenient category of topological spaces", Mich. Math.*
* J. 14, 133
- 152 (1967)
[20] Wall, C.T.C.: "Finiteness conditions for CW -complexes II ", Proc. Roy. So*
*c. London
Ser. A 295, 129-139 (1966)
[21] Whitehead, G.W.: "Elements of homotopy theory", Graduate Texts in Mathemat*
*ics,
Springer (1978)
Address
Wolfgang L"uck
Fachbereich Mathematik und Informatik
Westf"alische Wilhelms-Universit"at M"unster
Einsteinstr. 62
48149 M"unster
Bundesrepublik Deutschland
email: lueck@math.uni-muenster.de
FAX: 0251 8338370
internet: http://wwwmath.uni-muenster.de/math/u/lueck/
Version of April 6, 1998
25