Classifying spaces for families of subgroups
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Given a discrete group $G$ and a ''family of subgroups'' (see Definition \ref{def:family} below) $\mathcal{F}$ of $G$, there is a $G$-CW complex, $E_\mathcal{F}G$, that classifies $G$-CW complexes with isotropy contained in $\mathcal{F}$. That is, for every $G$-CW complex $X$, there is a $G$-equivariant map $X \to E_\mathcal{F}G$ that is unique up to $G$-equivariant homotopy. This universal property implies that $E_\mathcal{F}G$ is unique up to $G$-homotopy. For this reason $E_\mathcal{F}G$ is known as the classifying space (or universal space) of $G$ for the family $\mathcal{F}$. | Given a discrete group $G$ and a ''family of subgroups'' (see Definition \ref{def:family} below) $\mathcal{F}$ of $G$, there is a $G$-CW complex, $E_\mathcal{F}G$, that classifies $G$-CW complexes with isotropy contained in $\mathcal{F}$. That is, for every $G$-CW complex $X$, there is a $G$-equivariant map $X \to E_\mathcal{F}G$ that is unique up to $G$-equivariant homotopy. This universal property implies that $E_\mathcal{F}G$ is unique up to $G$-homotopy. For this reason $E_\mathcal{F}G$ is known as the classifying space (or universal space) of $G$ for the family $\mathcal{F}$. | ||
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Revision as of 14:48, 7 June 2010
Contents |
1 Introduction
Given a discrete group and a family of subgroups (see Definition \ref{def:family} 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
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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. |