Classifying spaces for families of subgroups
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[edit] 1 Introduction
Given a discrete group and a family of subgroups
of
(see Definition 2.2 below), there is a
-CW complex,
, that classifies
-CW complexes with isotropy contained in
. That is, the isotropy subgroups of
are contained in
and for every
-CW complex
with isotropy in
, there is a
-equivariant map
that is unique up to
-equivariant homotopy. There can be many models for
, but the universal property implies that they are all
-homotopy equivalent. For this reason
is known as the classifying space (or universal space) of
for the family
. A useful characterization of
is given below (Corollary 2.5).
Classifying spaces for families of subgroups play an important role in the classification of manifolds with a given fundamental group . The Farrell-Jones Conjecture relates the
- and
-theory of groups rings
to certain equivariant homology theories evaluated at
, the classifying space of
for the family of virtually cyclic subgroups. Similarly, the Baum-Connes Conjecture relates the topological
-theory of the reduced
-algebra of
to an appropriate equivariant homology theory evaluated at
, the classifying space for proper
-actions. (In the case of a discrete group, this means that the isotropy is contained in the family of finite subgroups, i.e., all of the stabilizer subgroups are finite.) From the point of view of computations it is also important to find nice models for these spaces, and in particular for
, since in many cases calculations can be reduced to working with the family of finite subgroups.
[edit] 2 Construction and examples
Definition (-CW complex) 2.1.
A
-CW complex
is a
-space with a
-invariant filtration of spaces

such that:
- a set
is closed if and only if
is closed for every
;
- for each
,
is obtained from
by attaching
-dimensional
-cells. That is, there exists a
-pushout
whereis a collection of subgroups of
.
Examples
- The real line with the translation action of
is a
-CW complex with one equivariant
-cell (the integers) and one equivariant
-cell (the orbit of the interval
).
-
with the antipodal
-action is a
-CW complex with one equivariant
-cell for every
. (Notice that as a CW complex
has two
-cells for every
.)
-
is a
-CW complex with one equivariant
-cell in each dimension. The filtration is
, with the antipodal action of
on each
.
- Let
be the infinite dihedral group. It acts on
, where
acts by reflection through zero and
acts by translation by 1. Thus,
is a
-CW complex with two equivariant
-cells (the orbits of 0 and
) and one equivariant
-cell (the orbit of the interval
).
Definition (Family of subgroups) 2.2.
A family of subgroups of a discrete group
is a collection of subgroups of
that is closed under conjugation and taking subgroups.
Examples of families:
- The family containing only the trivial subgroup.
- The family of all subgroups.
- The family of finite subgroups,
.
- The family of virtually cyclic subgroups,
.
Definition (Classifying space for a family of subgroups) 2.3.
Let be a family of subgroups of a discrete group
. The classifying space of
for
is a
-CW complex
whose isotropy groups are contained in
and has the property that for any
-CW complex
with isotropy in
, there is a
-equivariant map
that is unique up to
-homotopy.
An important fact that follows from the Generalized Whitehead Theorem ([Lück2005, Theorem 1.6]) is:
Theorem 2.4.
Let be a
-equivariant map of
-CW complexes. Then
is a
-homotopy equivalence if and only if for every subgroup
of
, the induced map on fixed sets
is a weak homotopy equivalence (i.e., induces an isomorphism on homotopy groups).
This gives us a way to determine whether or not a given -CW complex is a model for
.
Corollary 2.5.
A -CW complex
is a model for
if and only if
is weakly contractible for every
and is empty otherwise. In particular,
is contractible.
Examples
-
with the
-CW structure described above is a model for the classifying space of
for the trivial family.
-
with the
-CW structure described above is not a model for the classifying space of
for the trivial family, because
is not contractible.
-
is a model for the classifying space of
for the trivial family, since it is weakly contractible and hence contractible.
-
with the
-CW structure described above is a model for the classifying space of
for the family of finite subgroups. Notice that the nontrivial finite subgroups of
are of the form
, where
, and for each
,
fixes
. All other points have trivial stabilizers.
So far the existence question has been ignored. However, given a discrete group and a family of subgroups
of
, a model for
always exists. As explained in [Lück2005, Theorem 1.9], Corollary 2.5 implies that a model for
can be constructed by attaching equivariant
-cells,
to kill each of the homotopy groups of the
-fixed point sets, for each
in
. This process is illustrated in the
-example. Clearly, this procedure for constructing classifying spaces will typically produce very large models for
and is therefore usually not a useful model to work with.
More examples of classifying spaces:
- The classifying space of
for the trivial family (i.e., for free actions) is just
, the universal cover of the classifying space
. (Recall that
is a CW complex whose fundamental group is
and whose higher homotopy groups are all zero; it is unique up to homotopy.)
- For any group, a point is a model for the classifying space for the family of all subgroups.
is also known as the classifying space (or universal space) for proper
-actions, and is commonly written as
. There are typically "nice" models for
, as the
-example above and the next few examples show.
- Let
be a discrete subgroup of a Lie group
with finitely many path components. If
is a maximal compact subgroup of
, then
is a finite dimensional model for
. ([Lück2005])
- Let
be a word hyperbolic group in the sense of Gromov. Then the Rips complex,
, is a finite model for
( i.e., there are only finitely many
-cells), provided
is sufficiently large. ([Meintrup&Schick2002])
- If
is a
-CW complex with the structure of a CAT(0)-space with respect to which
acts by isometries, then
is a model for
. ([Bridson&Haefliger1999])
- Models for
are hard to construct, but are needed in the formulation of the Farrell-Jones Conjecture.
Classical sources on this topic are: [Bredon1967] [Bredon1972] [Tom Dieck1987]
[edit] 3 References
- [Bredon1967] G. E. Bredon, Equivariant cohomology theories, Lecture Notes in Mathematics, No. 34, Springer-Verlag, Berlin, 1967. MR0214062 (35 #4914) Zbl 0162.27202
- [Bredon1972] G. E. Bredon, Introduction to compact transformation groups, Academic Press, New York, 1972. MR0413144 (54 #1265) Zbl 0484.57001
- [Bridson&Haefliger1999] M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Springer-Verlag, Berlin, 1999. MR1744486 (2000k:53038) Zbl 0988.53001
- [Lück2005] W. Lück, Survey on classifying spaces for families of subgroups, Infinite groups: geometric, combinatorial and dynamical aspects, Progr. Math., 248, Birkhäuser, Basel (2005), 269–322. MR2195456 (2006m:55036) Zbl 1117.55013
- [Meintrup&Schick2002] D. Meintrup and T. Schick, A model for the universal space for proper actions of a hyperbolic group, New York J. Math. 8 (2002), 1–7 (electronic). MR1887695 (2003b:57002) Zbl 0990.20027
- [Tom Dieck1987] T. tom Dieck, Transformation groups, Walter de Gruyter & Co., Berlin, 1987. MR889050 (89c:57048) Zbl 0646.00011