Classifying spaces for families of subgroups

Revision as of 16:04, 7 June 2010

1 Introduction

Given a discrete group $<!-- 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 $E_\mathcal{F}G$ 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 $G$, $E_\mathcal{ALL}G$ 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) $\mathcal{F}$ 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 $\mathcal{F}$.

2 Construction and examples

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:

Theorem 2.2 implies that any two models for $E_\mathcal{F}G$ 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 $G$, $E_\mathcal{ALL}G$ is $G$-equivariantly homotopy equivalent to a point.
• Let $D_\infty=\langle a,b \;|\; a^2=1, aba^{-1}=b^{-1} \rangle$ be the infinite dihedral group. A model for $E_\mathcal{FIN}D_\infty$ is $\mathbb{R}$, where $a$ acts by reflection through zero and $b$ acts by translation by 1. The nontrivial finite subgroups of $D_\infty$ are of the form $\langle ab^i \rangle$, where $i\in \mathbb{Z}$. For each $i$, $\langle ab^i \rangle$ fixes $-i/2\in \mathbb{R}$.

3 Invariants

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4 Classification/Characterization

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5 Further discussion

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6 References