Classifying spaces for families of subgroups

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1 Introduction

Given a discrete group ${{Stub}}  == 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, the isotropy subgroups of $E_\mathcal{F}G$ are contained in $\mathcal{F}$ and 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. There can be many models for $E_\mathcal{F}G$, but the universal property implies that they are all $G$-homotopy equivalent. For this reason $E_\mathcal{F}G$ is known as the classifying space (or universal space) of $G$ for the family $\mathcal{F}$. A useful characterization of $E_\mathcal{F}G$ is given below in Corollary \ref{def:alternate}. Classifying spaces for families of subgroups play an important role in the classification of manifolds with a given fundamental group $G$. The Farrell-Jones Conjecture relates the $K$- and $L$-theory of groups rings $RG$ to certain equivariant homology theories evaluated at $E_\mathcal{VCYC}G$, the classifying space of $G$ for the family of virtually cyclic subgroups. Similarly, the Baum-Connes Conjecture relates the topological $K$-theory of the reduced $C^*$-algebra of $G$ to an appropriate equivariant homology theory evaluated at $\underbar{E}G$, the classifying space for proper $G$-actions. (See the examples below.) From the point of view of computations, it is also important to find nice models for these spaces, and in particular for $\underbar{E}G$. </wikitex> == Construction and examples == <wikitex>; {{beginthm|Definition}}\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. * The family of all subgroups. * The family of finite subgroups, $\mathcal{FIN}$. * The family of virtually cyclic subgroups, $\mathcal{VCYC}$. {{beginthm|Definition}} Let $\mathcal{F}$ be a family of subgroups of a discrete group $G$. The classifying space of $G$ for $\mathcal{F}$ is a $G$-CW complex $E_\mathcal{F}G$ whose isotropy groups are contained in $\mathcal{F}$ and has the property that for any $G$-CW complex $X$ with isotropy in $\mathcal{F}$, there is a $G$-equivariant map $X \to E_\mathcal{F}G$ that is unique up to $G$-homotopy. {{endthm}} Such a space always exists. An important fact that follows from the Generalized Whitehead Theorem (\cite{Lück2005|Theorem 1.6}) is: {{beginthm|Theorem}}\label{thm:GCW} Let $f: X \to Y$ be a $G$-equivariant map of $G$-CW complexes. Then $f$ is a $G$-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} gives us a way to determine whether or not a given $G$-CW complex is a model for $E_\mathcal{F}G$. {{beginthm|Corollary}}\label{def:alternate} A $G$-CW complex $X$ is a model for $E_\mathcal{F}G$ if and only if $X^H$ is weakly contractible for every $H\in \mathcal{F}$ and is empty otherwise. {{endthm}} '''Examples of classifying spaces''' * A model for the classifying space of $\mathbb{Z}$ for the trivial family is $\mathbb{R}$ on which $\mathbb{Z}$ acts by translation. More generally, the classifying space of $G$ for the trivial family (i.e., for free actions) is just $EG$, 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 group, a point is a model for the classifying space for the family of all subgroups. *$E_\mathcal{FIN}G$ is also known as the ''classifying space (or universal space) for proper $G$-actions'', and is commonly written as $\underbar{E}G$. There are typically "nice" models for $\underbar{E}G$, as the next several examples show. They appear in the statement of the ''Baum-Connes Conjecture''. * Let $D_\infty=\langle a,b \;|\; a^2=1, aba^{-1}=b^{-1} \rangle$ be the infinite dihedral group. A model for $\underbar{E}D_\infty$ is $\mathbb{R}$, where $a$ acts by reflection through zero and $b$ acts by translation by 1. (Notice that the nontrivial finite subgroups of $D_\infty$ are of the form $\langle ab^i \rangle$, where $i\in \mathbb{Z}$, and for each $i$, $\langle ab^i \rangle$ fixes $-i/2\in \mathbb{R}$.) * Let $G$ be a discrete subgroup of a Lie group $L$ with finitely many path components. If $K$ is a maximal compact subgroup of $L$, then $G/K$ is a finite dimensional model for $\underbar{E}G$. * Let $G$ be a ''word hyperbolic group'' in the sense of Gromov. Then the ''Rips complex'', $P_d(G)$, is a finite model for $\underbar{E}G$ ( i.e., there are only finitely many $G$-cells), provided $d$ is sufficiently large. * If $X$ is a $G$-CW complex with the structure of a CAT(0)-space with respect to which $G$ acts by isometries, then $X$ is a model for $\underbar{E}G$. * Models for $E_\mathcal{VCYC}G$ are hard to construct, but are needed in the formulation of the ''Farrell-Jones Conjecture''. Classical sources are: \cite{Bredon1967} \cite{Tom Dieck1987} </wikitex> == References == {{#RefList:}}  [[Category:Theory]]G$ and a family of subgroups $\mathcal{F}$ of $G$ (see Definition 2.1 below), there is a $G$ -CW complex, $E_\mathcal{F}G$ , that classifies $G$ -CW complexes with isotropy contained in $\mathcal{F}$ . That is, the isotropy subgroups of $E_\mathcal{F}G$ are contained in $\mathcal{F}$ and 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. There can be many models for $E_\mathcal{F}G$ , but the universal property implies that they are all $G$ -homotopy equivalent. For this reason $E_\mathcal{F}G$ is known as the classifying space (or universal space) of $G$ for the family $\mathcal{F}$ . A useful characterization of $E_\mathcal{F}G$ is given below in Corollary 2.4.

Classifying spaces for families of subgroups play an important role in the classification of manifolds with a given fundamental group $G$ . The Farrell-Jones Conjecture relates the $K$ - and $L$ -theory of groups rings $RG$ to certain equivariant homology theories evaluated at $E_\mathcal{VCYC}G$ , the classifying space of $G$ for the family of virtually cyclic subgroups. Similarly, the Baum-Connes Conjecture relates the topological $K$ -theory of the reduced $C^*$ -algebra of $G$ to an appropriate equivariant homology theory evaluated at $\underbar{E}G$ , the classifying space for proper $G$ -actions. (See the examples below.) From the point of view of computations, it is also important to find nice models for these spaces, and in particular for $\underbar{E}G$ .

2 Construction and examples

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.

Examples of families

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

Let $\mathcal{F}$ be a family of subgroups of a discrete group $G$ . The classifying space of $G$ for $\mathcal{F}$ is a $G$ -CW complex $E_\mathcal{F}G$ whose isotropy groups are contained in $\mathcal{F}$ and has the property that for any $G$ -CW complex $X$ with isotropy in $\mathcal{F}$ , there is a $G$ -equivariant map $X \to E_\mathcal{F}G$ that is unique up to $G$ -homotopy.

Such a space always exists.

An important fact that follows from the Generalized Whitehead Theorem ([Lück2005, Theorem 1.6]) is:

Let $f: X \to Y$ be a $G$ -equivariant map of $G$ -CW complexes. Then $f$ is a $G$ -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).

Theorem 2.3 gives us a way to determine whether or not a given $G$ -CW complex is a model for $E_\mathcal{F}G$ .

A $G$ -CW complex $X$ is a model for $E_\mathcal{F}G$ if and only if $X^H$ is weakly contractible for every $H\in \mathcal{F}$ and is empty otherwise.

Examples of classifying spaces

A model for the classifying space of $\mathbb{Z}$ for the trivial family is $\mathbb{R}$ on which $\mathbb{Z}$ acts by translation.

More generally, the classifying space of $G$ for the trivial family (i.e., for free actions) is just $EG$ , 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 group, a point is a model for the classifying space for the family of all subgroups.
$E_\mathcal{FIN}G$ is also known as the classifying space (or universal space) for proper $G$ -actions, and is commonly written as $\underbar{E}G$ . There are typically "nice" models for $\underbar{E}G$ , as the next several examples show. They appear in the statement of the Baum-Connes Conjecture.
Let $D_\infty=\langle a,b \;|\; a^2=1, aba^{-1}=b^{-1} \rangle$ be the infinite dihedral group. A model for $\underbar{E}D_\infty$ is $\mathbb{R}$ , where $a$ acts by reflection through zero and
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acts by translation by 1. (Notice that the nontrivial finite subgroups of $D_\infty$ are of the form $\langle ab^i \rangle$ , where $i\in \mathbb{Z}$ , and for each $i$ , $\langle ab^i \rangle$ fixes $-i/2\in \mathbb{R}$ .)
Let $G$ be a discrete subgroup of a Lie group $L$ with finitely many path components. If $K$ is a maximal compact subgroup of $L$ , then $G/K$ is a finite dimensional model for $\underbar{E}G$ .
Let $G$ be a word hyperbolic group in the sense of Gromov. Then the Rips complex, $P_d(G)$ , is a finite model for $\underbar{E}G$ ( i.e., there are only finitely many $G$ -cells), provided $d$ is sufficiently large.
If $X$ is a $G$ -CW complex with the structure of a CAT(0)-space with respect to which $G$ acts by isometries, then $X$ is a model for $\underbar{E}G$ .
Models for $E_\mathcal{VCYC}G$ are hard to construct, but are needed in the formulation of the Farrell-Jones Conjecture.

Classical sources are: [Bredon1967] [Tom Dieck1987]

3 References

[Bredon1967] G. E. Bredon, Equivariant cohomology theories, Lecture Notes in Mathematics, No. 34, Springer-Verlag, Berlin, 1967. MR0214062 (35 #4914) Zbl 0162.27202
[Lück2005] W. Lück, Survey on classifying spaces for families of subgroups, Infinite groups: geometric, combinatorial and dynamical aspects, Progr. Math., 248, Birkhäuser, Basel (2005), 269–322. MR2195456 (2006m:55036) Zbl 1117.55013
[Tom Dieck1987] T. tom Dieck, Transformation groups, Walter de Gruyter & Co., Berlin, 1987. MR889050 (89c:57048) Zbl 0646.00011

Classifying spaces for families of subgroups

Revision as of 13:48, 8 June 2010

1 Introduction

2 Construction and examples

3 References

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@@ Line 15: / Line 15: @@
 <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, the isotropy subgroups of $E_\mathcal{F}G$ are contained in $\mathcal{F}$ and 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. There can be many models for $E_\mathcal{F}G$, but the universal property implies that they are all $G$-homotopy equivalent. For this reason $E_\mathcal{F}G$ is known as the classifying space (or universal space) of $G$ for the family $\mathcal{F}$. A useful characterization of $E_\mathcal{F}G$ is given below in Corollary \ref{def:alternate}.
+Given a discrete group $G$ and a ''family of subgroups'' $\mathcal{F}$ of $G$ (see Definition \ref{def:family} below), there is a $G$-CW complex, $E_\mathcal{F}G$, that classifies $G$-CW complexes with isotropy contained in $\mathcal{F}$. That is, the isotropy subgroups of $E_\mathcal{F}G$ are contained in $\mathcal{F}$ and 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. There can be many models for $E_\mathcal{F}G$, but the universal property implies that they are all $G$-homotopy equivalent. For this reason $E_\mathcal{F}G$ is known as the classifying space (or universal space) of $G$ for the family $\mathcal{F}$. A useful characterization of $E_\mathcal{F}G$ is given below in Corollary \ref{def:alternate}.
 Classifying spaces for families of subgroups play an important role in the classification of manifolds with a given fundamental group $G$. The Farrell-Jones Conjecture relates the $K$- and $L$-theory of groups rings $RG$ to certain equivariant homology theories evaluated at $E_\mathcal{VCYC}G$, the classifying space of $G$ for the family of virtually cyclic subgroups. Similarly, the Baum-Connes Conjecture relates the topological $K$-theory of the reduced $C^*$-algebra of $G$ to  an appropriate equivariant homology theory evaluated at $\underbar{E}G$, the classifying space for proper $G$-actions. (See the examples below.) From the point of view of computations, it is also important to find nice models for these spaces, and in particular for $\underbar{E}G$.