Classifying spaces for families of subgroups
(→Construction and examples) |
(→Construction and examples) |
||
Line 27: | Line 27: | ||
{{endthm}} | {{endthm}} | ||
− | Examples of families | + | Examples of families: |
* The family containing only the trivial subgroup, $\mathcal{TR}$. | * The family containing only the trivial subgroup, $\mathcal{TR}$. | ||
* The family of all subgroups, $\mathcal{ALL}$. | * The family of all subgroups, $\mathcal{ALL}$. | ||
Line 33: | Line 33: | ||
* The family of virtually cyclic subgroups, $\mathcal{VCYC}$. | * The family of virtually cyclic subgroups, $\mathcal{VCYC}$. | ||
+ | Examples of classifying spaces: | ||
+ | * A model for $E_\mathcal{TR}\mathbb{Z}$ is $\mathbb{R}$. More generally, $E_\mathcal{TR}G=EG$ is the universal cover of the classifying space $BG$. (Recall that $BG$ is a CW complex whose fundamental group is $G$ and whose higher homotopy groups are zero.) | ||
+ | * | ||
</wikitex> | </wikitex> |
Revision as of 15:37, 7 June 2010
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, for every -CW complex , there is a -equivariant map that is unique up to -equivariant homotopy. This universal property implies that is unique up to -homotopy. For this reason is known as the classifying space (or universal space) of for the family .
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, .
Examples of classifying spaces:
- A model for is . More generally, is the universal cover of the classifying space . (Recall that is a CW complex whose fundamental group is and whose higher homotopy groups are zero.)
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. |