Classifying spaces for families of subgroups
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* 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 $D_\infty=\langle a,b \;|\; a^2=1, aba^{-1}=b^{-1} \rangle$ 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 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. | + | * 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 $\underbar{E}G$. | + | * 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''. | * Models for $E_\mathcal{VCYC}G$ are hard to construct, but are needed in the formulation of the ''Farrell-Jones Conjecture''. | ||
Revision as of 14:08, 8 June 2010
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1 Introduction
Given a discrete group and a family of subgroups of (see Definition 2.1 below), 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 (Corollary 2.4).
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 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 , since in many cases calculations can be reduced to working with the family of finite subgroups.
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
- The family containing only the trivial subgroup.
- The family of all subgroups.
- The family of finite subgroups, .
- The family of virtually cyclic subgroups, .
Definition 2.2. 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.
Such a space always exists.
An important fact that follows from the Generalized Whitehead Theorem ([Lück2005, Theorem 1.6]) is:
Theorem 2.3. 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 2.3 gives us a way to determine whether or not a given -CW complex is a model for .
Corollary 2.4. A -CW complex is a model for if and only if is weakly contractible for every and is empty otherwise.
Examples of classifying spaces
- A model for the classifying space of for the trivial family is on which acts by translation.
- More generally, the classifying space of for the trivial family (i.e., for free actions) is just , 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 group, a point is a model for the classifying space for the family of all subgroups.
- 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
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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. ([Meintrup&Schick2002])
- 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 . ([Bridson&Haefliger1999])
- Models for 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
- [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