Classifying spaces for families of subgroups

1 Introduction

Given a discrete group $G$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> == Introduction == <wikitex>; 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}$. </wikitex> == Construction and examples == <wikitex>; {{beginthm|Definition (Family of Subgroups)|}}\label{def:family} 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}} Examples of families: * The family containing only the trivial subgroup, $\mathcal{TR}$. * The family of all subgroups, $\mathcal{ALL}$. * The family of finite subgroups, $\mathcal{FIN}$. * The family of virtually cyclic subgroups, $\mathcal{VCYC}$. An important fact that follows from the Generalized Whitehead Theorem is: {{beginthm|Theorem}}\label{thm:GCW} Let $f: X \to Y$ be a $G$-equivariant map of $G$-CW complexes. Then $f$ is a $G$-equivariant homotopy equivalence if and only if for every subgroup $H$ of $G$, $f^H: X^H \to Y^H$ is a weak homotopy equivalence (i.e., induces an isomorphism on homotopy groups). {{endthm}} Theorem \ref{thm:GCW} implies that any two models for $E_\mathcal{ALL}\mathbb{G}$ are $G$-equivariantly homotopy equivalent. Examples of classifying spaces: * 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.) * For any $G$, $E_\mathcal{ALL}\mathbb{G}$ is $G$-equivariantly homotopy equivalent to a point. </wikitex> == Invariants == <wikitex>; ... </wikitex> == Classification/Characterization == <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]] {{Stub}}G$ and a family of subgroups (see Definition 2.1 below) $\mathcal{F}$$\mathcal{F}$ of $G$$G$, there is a $G$$G$-CW complex, $E_\mathcal{F}G$$E_\mathcal{F}G$, that classifies $G$$G$-CW complexes with isotropy contained in $\mathcal{F}$$\mathcal{F}$. That is, for every $G$$G$-CW complex $X$$X$, there is a $G$$G$-equivariant map $X \to E_\mathcal{F}G$$X \to E_\mathcal{F}G$ that is unique up to $G$$G$-equivariant homotopy. This universal property implies that $E_\mathcal{F}G$$E_\mathcal{F}G$ is unique up to $G$$G$-homotopy. For this reason $E_\mathcal{F}G$$E_\mathcal{F}G$ is known as the classifying space (or universal space) of $G$$G$ for the family $\mathcal{F}$$\mathcal{F}$.

2 Construction and examples

Definition (Family of Subgroups) 2.1. A family $\mathcal{F}$$\mathcal{F}$ of subgroups of a discrete group $G$$G$ is a collection of subgroups of $G$$G$ that is closed under conjugation and taking subgroups.

Examples of families:

• The family containing only the trivial subgroup, $\mathcal{TR}$$\mathcal{TR}$.
• The family of all subgroups, $\mathcal{ALL}$$\mathcal{ALL}$.
• The family of finite subgroups, $\mathcal{FIN}$$\mathcal{FIN}$.
• The family of virtually cyclic subgroups, $\mathcal{VCYC}$$\mathcal{VCYC}$.

An important fact that follows from the Generalized Whitehead Theorem is:

Theorem 2.2. Let $f: X \to Y$$f: X \to Y$ be a $G$$G$-equivariant map of $G$$G$-CW complexes. Then $f$$f$ is a $G$$G$-equivariant homotopy equivalence if and only if for every subgroup $H$$H$ of $G$$G$, $f^H: X^H \to Y^H$$f^H: X^H \to Y^H$ is a weak homotopy equivalence (i.e., induces an isomorphism on homotopy groups).

Theorem 2.2 implies that any two models for $E_\mathcal{ALL}\mathbb{G}$$E_\mathcal{ALL}\mathbb{G}$ are $G$$G$-equivariantly homotopy equivalent.

Examples of classifying spaces:

• A model for $E_\mathcal{TR}\mathbb{Z}$$E_\mathcal{TR}\mathbb{Z}$ is $\mathbb{R}$$\mathbb{R}$ on which $\mathbb{Z}$$\mathbb{Z}$ acts by translation.

More generally, $E_\mathcal{TR}G=EG$$E_\mathcal{TR}G=EG$, the universal cover of the classifying space $BG$$BG$. (Recall that $BG$$BG$ is a CW complex whose fundamental group is $G$$G$ and whose higher homotopy groups are all zero. It is unique up to homotopy.)

• For any $G$$G$, $E_\mathcal{ALL}\mathbb{G}$$E_\mathcal{ALL}\mathbb{G}$ is $G$$G$-equivariantly homotopy equivalent to a point.

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.