Classifying spaces for families of subgroups
Contents |
1 Introduction
Given a discrete group and a family of subgroups (see Definition 2.1 below)
of
, 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
, 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 in Corollary 2.4.
2 Construction and examples
Definition (Family of Subgroups) 2.1.
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 2.2.
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 is:
Theorem 2.3.
Let be a
-equivariant map of
-CW complexes. Then
is a
-homotopy equivalence if and only if for every subgroup
of
,
is a weak homotopy equivalence (i.e., induces an isomorphism on homotopy groups).
Theorem 2.3 gives us a way to determine whether or not a given -CW complex is a model for
.
Corollary 2.4.
A -CW complex
is a model for
if and only if
is weakly contractible for every
and is empty otherwise.
Examples of classifying spaces:
- A model for
is
on which
acts by translation.
More generally, , 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
, a point is a model for
.
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 next several examples show. They appear in the statement of the Baum-Connes Conjecture.
- Let
be the infinite dihedral group. A model for
is
, where
acts by reflection through zero and
acts by translation by 1. (Notice that the nontrivial finite subgroups of
are of the form
, where
, and for each
,
fixes
.)
- 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
.
- 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.
- 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
.
- Models for
are hard to construct, but are needed in the formulation of the Farrell-Jones Conjecture.
3 Invariants
...
4 Classification/Characterization
...
5 Further discussion
...
6 References
This page has not been refereed. The information given here might be incomplete or provisional. |