Classifying spaces for families of subgroups
(→Construction and examples) |
(→Construction and examples) |
||
Line 38: | Line 38: | ||
{{endthm}} | {{endthm}} | ||
− | Theorem \ref{thm:GCW} implies that any two models for $E_\mathcal{ | + | Theorem \ref{thm:GCW} implies that any two models for $E_\mathcal{F}G$ are $G$-equivariantly homotopy equivalent. |
Line 45: | Line 45: | ||
* A model for $E_\mathcal{TR}\mathbb{Z}$ is $\mathbb{R}$ on which $\mathbb{Z}$ acts by translation. | * A model for $E_\mathcal{TR}\mathbb{Z}$ is $\mathbb{R}$ on which $\mathbb{Z}$ acts by translation. | ||
More generally, $E_\mathcal{TR}G=EG$, 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 all zero. It is unique up to homotopy.) | More generally, $E_\mathcal{TR}G=EG$, 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 all zero. It is unique up to homotopy.) | ||
− | * For any $G$, $E_\mathcal{ALL} | + | * For any $G$, $E_\mathcal{ALL}G$ is $G$-equivariantly homotopy equivalent to a point. |
</wikitex> | </wikitex> |
Revision as of 15:54, 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, .
An important fact that follows from the Generalized Whitehead Theorem is:
Theorem 2.2. Let be a -equivariant map of -CW complexes. Then is a -equivariant homotopy equivalence if and only if for every subgroup of , is a weak homotopy equivalence (i.e., induces an isomorphism on homotopy groups).
Theorem 2.2 implies that any two models for are -equivariantly homotopy equivalent.
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 , is -equivariantly homotopy equivalent to a point.
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. |