Classifying spaces for families of subgroups

Revision as of 14:23, 8 June 2010

1 Introduction

Given a discrete group ${{Stub}} <!-- 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'' $\mathcal{F}$ 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 (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. (In the case of a discrete group, a proper action implies that the isotropy is contained in the family of finite subgroups, i.e., all of the stabilizer subgroups are finite.) From the point of view of computations it is also important to find nice models for these spaces, and in particular for $\underbar{E}G$, since in many cases calculations can be reduced to working with the family of finite subgroups. </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 $\underbar{E}G$ \cite{Lück2005}. * 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 \cite{Meintrup&Schick2002}. * 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 $\underbar{E}G$ \cite{Bridson&Haefliger1999}. * 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{Bredon1972} \cite{Tom Dieck1987} </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[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 (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. (In the case of a discrete group, a proper action implies that the isotropy is contained in the family of finite subgroups, i.e., all of the stabilizer subgroups are finite.) From the point of view of computations it is also important to find nice models for these spaces, and in particular for $\underbar{E}G$, since in many cases calculations can be reduced to working with the family of finite subgroups.

2 Construction and examples

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, $\mathcal{VCYC}$.

Such a space always exists.

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

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

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 $\underbar{E}G$ [Lück2005].
• 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 [Meintrup&Schick2002].
• 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 $\underbar{E}G$ [Bridson&Haefliger1999].
• 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] [Bredon1972] [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
• [Bredon1972] G. E. Bredon, Introduction to compact transformation groups, Academic Press, New York, 1972. MR0413144 (54 #1265) Zbl 0484.57001
• [Bridson&Haefliger1999] M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Springer-Verlag, Berlin, 1999. MR1744486 (2000k:53038) Zbl 0988.53001
• [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
• [Meintrup&Schick2002] D. Meintrup and T. Schick, A model for the universal space for proper actions of a hyperbolic group, New York J. Math. 8 (2002), 1–7 (electronic). MR1887695 (2003b:57002) Zbl 0990.20027
• [Tom Dieck1987] T. tom Dieck, Transformation groups, Walter de Gruyter & Co., Berlin, 1987. MR889050 (89c:57048) Zbl 0646.00011