Classifying spaces for families of subgroups
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[edit] 1 Introduction
Given a discrete group and a family of subgroups of (see Definition 2.2 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 with isotropy in , 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 2.5).
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. (In the case of a discrete group, this means 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 , since in many cases calculations can be reduced to working with the family of finite subgroups.
[edit] 2 Construction and examples
Definition (-CW complex) 2.1. A -CW complex is a -space with a -invariant filtration of spaces
such that:
- a set is closed if and only if is closed for every ;
- for each , is obtained from by attaching -dimensional -cells. That is, there exists a -pushout where is a collection of subgroups of .
Examples
- The real line with the translation action of is a -CW complex with one equivariant -cell (the integers) and one equivariant -cell (the orbit of the interval ).
- with the antipodal -action is a -CW complex with one equivariant -cell for every . (Notice that as a CW complex has two -cells for every .)
- is a -CW complex with one equivariant -cell in each dimension. The filtration is , with the antipodal action of on each .
- Let be the infinite dihedral group. It acts on , where acts by reflection through zero and acts by translation by 1. Thus, is a -CW complex with two equivariant -cells (the orbits of 0 and ) and one equivariant -cell (the orbit of the interval ).
Definition (Family of subgroups) 2.2. 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 (Classifying space for a family of subgroups) 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 ([Lück2005, Theorem 1.6]) 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 , the induced map on fixed sets is a weak homotopy equivalence (i.e., induces an isomorphism on homotopy groups).
This 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. In particular, is contractible.
Examples
- with the -CW structure described above is a model for the classifying space of for the trivial family.
- with the -CW structure described above is not a model for the classifying space of for the trivial family, because is not contractible.
- is a model for the classifying space of for the trivial family, since it is weakly contractible and hence contractible.
- with the -CW structure described above is a model for the classifying space of for the family of finite subgroups. Notice that the nontrivial finite subgroups of are of the form , where , and for each , fixes . All other points have trivial stabilizers.
So far the existence question has been ignored. However, given a discrete group and a family of subgroups of , a model for always exists. As explained in [Lück2005, Theorem 1.9], Corollary 2.5 implies that a model for can be constructed by attaching equivariant -cells, to kill each of the homotopy groups of the -fixed point sets, for each in . This process is illustrated in the -example. Clearly, this procedure for constructing classifying spaces will typically produce very large models for and is therefore usually not a useful model to work with.
More examples of classifying spaces:
- 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 -example above and the next few examples show.
- 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 . ([Lück2005])
- 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 on this topic are: [Bredon1967] [Bredon1972] [Tom Dieck1987]
[edit] 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