Classifying spaces for families of subgroups
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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 2.4.
2 Construction and examples
Definition (Family of Subgroups) 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.
An important fact that follows from the Generalized Whitehead Theorem 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 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.
3 Invariants
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4 Classification/Characterization
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5 Further discussion
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6 References
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