Classifying spaces for families of subgroups
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* 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, $\mathcal{ALL}$. | ||
− | * The family of finite subgroups $\mathcal{FIN}$. | + | * The family of finite subgroups, $\mathcal{FIN}$. |
− | * The family of virtually cyclic subgroups $\mathcal{VCYC}$. | + | * The family of virtually cyclic subgroups, $\mathcal{VCYC}$. |
+ | |||
</wikitex> | </wikitex> |
Revision as of 14:59, 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 are:
- The family containing only the trivial subgroup, .
- The family of all subgroups, .
- The family of finite subgroups, .
- The family of virtually cyclic subgroups, .
3 Invariants
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4 Classification/Characterization
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5 Further discussion
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6 References
This page has not been refereed. The information given here might be incomplete or provisional. |