Classifying spaces for families of subgroups
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{{beginthm|Definition (Family of Subgroups)|}}\label{def:family} | {{beginthm|Definition (Family of Subgroups)|}}\label{def:family} | ||
− | A family of subgroups of a discrete group $G$ is a collection of subgroups | + | A family $\mathcal{F}$ of subgroups of a discrete group $G$ is a collection of subgroups of $G$ that is closed under conjugation and taking subgroups. |
{{endthm}} | {{endthm}} | ||
Revision as of 14:48, 7 June 2010
Contents |
1 Introduction
Given a discrete group and a family of subgroups (see Definition 1.1 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 .
Definition (Family of Subgroups) 1.1. A family of subgroups of a discrete group is a collection of subgroups of that is closed under conjugation and taking subgroups.
2 Construction and examples
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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. |