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, 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'' (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}. | ||
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+ | 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 \ref{ex:classifying} below.) From the point of view of computations, it is also important to find nice models for these spaces, and in particular for $\underbar{E}G$. | ||
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{{endthm}} | {{endthm}} | ||
− | Examples of families | + | {{beginthm|Examples of families}} |
* The family containing only the trivial subgroup, $\mathcal{TR}$. | * The family containing only the trivial subgroup, $\mathcal{TR}$. | ||
* The family of all subgroups, $\mathcal{ALL}$. | * The family of all subgroups, $\mathcal{ALL}$. | ||
* The family of finite subgroups, $\mathcal{FIN}$. | * The family of finite subgroups, $\mathcal{FIN}$. | ||
* The family of virtually cyclic subgroups, $\mathcal{VCYC}$. | * The family of virtually cyclic subgroups, $\mathcal{VCYC}$. | ||
+ | {{endthm}} | ||
{{beginthm|Definition}} | {{beginthm|Definition}} | ||
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{{endthm}} | {{endthm}} | ||
− | Examples of classifying spaces: | + | {{beginthm|Examples of classifying spaces}}\label{ex:classifying} |
* A model for $E_\mathcal{TR}\mathbb{Z}$ is $\mathbb{R}$ on which $\mathbb{Z}$ acts by translation. | * 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.) | 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.) | ||
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* 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 $\underbar{E}G$. | * 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''. | * Models for $E_\mathcal{VCYC}G$ are hard to construct, but are needed in the formulation of the ''Farrell-Jones Conjecture''. | ||
+ | {{endthm}} | ||
For more on classifying spaces, a good source is \cite{Lück2005}. | For more on classifying spaces, a good source is \cite{Lück2005}. |
Revision as of 10:54, 8 June 2010
1 Introduction
Given a discrete group and a family of subgroups (see Definition 2.1 below) of , there is a -CW complex, , that classifies -CW complexes with isotropy contained in . That is, the isotropy subgroups of are contained in and for every -CW complex , there is a -equivariant map that is unique up to -equivariant homotopy. There can be many models for , but the universal property implies that they are all -homotopy equivalent. For this reason is known as the classifying space (or universal space) of for the family . A useful characterization of is given below in Corollary 2.5.
Classifying spaces for families of subgroups play an important role in the classification of manifolds with a given fundamental group . The Farrell-Jones Conjecture relates the - and -theory of groups rings to certain equivariant homology theories evaluated at , the classifying space of for the family of virtually cyclic subgroups. Similarly, the Baum-Connes Conjecture relates the topological -theory of the reduced -algebra of to an appropriate equivariant homology theory evaluated at , the classifying space for proper -actions. (See 2.6 below.) From the point of view of computations, it is also important to find nice models for these spaces, and in particular for .
2 Construction and examples
Definition 2.1. A family of subgroups of a discrete group is a collection of subgroups of that is closed under conjugation and taking subgroups.
Examples of families 2.2.
- The family containing only the trivial subgroup, .
- The family of all subgroups, .
- The family of finite subgroups, .
- The family of virtually cyclic subgroups, .
Definition 2.3. Let be a family of subgroups of a discrete group . The classifying space of for is a -CW complex whose isotropy groups are contained in and has the property that for any -CW complex with isotropy in , there is a -equivariant map that is unique up to -homotopy.
An important fact that follows from the Generalized Whitehead Theorem ([Bredon1967]) is:
Theorem 2.4. Let be a -equivariant map of -CW complexes. Then is a -homotopy equivalence if and only if for every subgroup of , is a weak homotopy equivalence (i.e., induces an isomorphism on homotopy groups).
Theorem 2.4 gives us a way to determine whether or not a given -CW complex is a model for .
Corollary 2.5. A -CW complex is a model for if and only if is weakly contractible for every and is empty otherwise.
Examples of classifying spaces 2.6.
- A model for is on which acts by translation.
More generally, , the universal cover of the classifying space . (Recall that is a CW complex whose fundamental group is and whose higher homotopy groups are all zero. It is unique up to homotopy.)
- For any , a point is a model for .
- is also known as the classifying space (or universal space) for proper -actions, and is commonly written as . There are typically "nice" models for , as the next several examples show. They appear in the statement of the Baum-Connes Conjecture.
- Let be the infinite dihedral group. A model for is , where acts by reflection through zero and acts by translation by 1. (Notice that the nontrivial finite subgroups of are of the form , where , and for each , fixes .)
- Let be a discrete subgroup of a Lie group with finitely many path components. If is a maximal compact subgroup of , then is a finite dimensional model for .
- Let be a word hyperbolic group in the sense of Gromov. Then the Rips complex, , is a finite model for ( i.e., there are only finitely many -cells), provided is sufficiently large.
- If is a -CW complex with the structure of a CAT(0)-space with respect to which acts by isometries, then is a model for .
- Models for are hard to construct, but are needed in the formulation of the Farrell-Jones Conjecture.
For more on classifying spaces, a good source is [Lück2005].
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
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