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'' | + | * 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. |
* If $X$ is a $G$-CW complex with the structure of a CAT(0)-space with respect to which $G$ acts by isometries, then $X=\underbar{E}G$. | * If $X$ is a $G$-CW complex with the structure of a CAT(0)-space with respect to which $G$ acts by isometries, then $X=\underbar{E}G$. | ||
* 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''. |
Revision as of 16:47, 7 June 2010
Contents |
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, for every -CW complex , there is a -equivariant map that is unique up to -equivariant homotopy. This universal property implies that is unique up to -homotopy. For this reason is known as the classifying space (or universal space) of for the family .
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, .
An important fact that follows from the Generalized Whitehead Theorem is:
Theorem 2.2. Let be a -equivariant map of -CW complexes. Then is a -equivariant homotopy equivalence if and only if for every subgroup of , is a weak homotopy equivalence (i.e., induces an isomorphism on homotopy groups).
Theorem 2.2 implies that any two models for are -equivariantly homotopy equivalent.
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 , is -equivariantly homotopy equivalent to a point.
- 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 also appear in 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 .
- 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
This page has not been refereed. The information given here might be incomplete or provisional. |