Classifying spaces for families of subgroups
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'''Examples of classifying spaces''' | '''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. | * 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.) | + | * 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. | * 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''. | *$E_\mathcal{FIN}G$ 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''. |
Revision as of 13:57, 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 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.
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