Classifying spaces for families of subgroups
(→Introduction) |
(→Construction and examples) |
||
Line 33: | Line 33: | ||
{{beginthm|Definition}} | {{beginthm|Definition}} | ||
− | Let $\mathcal{F}$ be a family of subgroups of a discrete group $G$. The classifying space of $G$ for $\mathcal{F}$ is a $G$-CW complex $E_\mathcal{F}G$ whose isotropy groups are contained in $\mathcal{F}$ and has the property that for any $G$-CW complex $X$ with isotropy in $\mathcal{F}$, there is a $G$-equivariant map $X \to E_\mathcal{F}G$ that is unique up to $G$- | + | Let $\mathcal{F}$ be a family of subgroups of a discrete group $G$. The classifying space of $G$ for $\mathcal{F}$ is a $G$-CW complex $E_\mathcal{F}G$ whose isotropy groups are contained in $\mathcal{F}$ and has the property that for any $G$-CW complex $X$ with isotropy in $\mathcal{F}$, there is a $G$-equivariant map $X \to E_\mathcal{F}G$ that is unique up to $G$-homotopy. |
{{endthm}} | {{endthm}} | ||
An important fact that follows from the Generalized Whitehead Theorem is: | An important fact that follows from the Generalized Whitehead Theorem is: | ||
{{beginthm|Theorem}}\label{thm:GCW} | {{beginthm|Theorem}}\label{thm:GCW} | ||
− | Let $f: X \to Y$ be a $G$-equivariant map of $G$-CW complexes. Then $f$ is a $G$- | + | Let $f: X \to Y$ be a $G$-equivariant map of $G$-CW complexes. Then $f$ is a $G$-homotopy equivalence if and only if for every subgroup $H$ of $G$, $f^H: X^H \to Y^H$ is a weak homotopy equivalence (i.e., induces an isomorphism on homotopy groups). |
{{endthm}} | {{endthm}} | ||
− | Theorem \ref{thm:GCW} | + | Theorem \ref{thm:GCW} gives us a way to determine whether or not a given $G$-CW complex is a model for $E_\mathcal{F}G$. |
{{beginthm|Corollary}}\label{def:alternate} | {{beginthm|Corollary}}\label{def:alternate} | ||
A $G$-CW complex $X$ is a model for $E_\mathcal{F}G$ if and only if $X^H$ is weakly contractible for every $H\in \mathcal{F}$ and is empty otherwise. | A $G$-CW complex $X$ is a model for $E_\mathcal{F}G$ if and only if $X^H$ is weakly contractible for every $H\in \mathcal{F}$ and is empty otherwise. | ||
Line 49: | Line 49: | ||
* A model for $E_\mathcal{TR}\mathbb{Z}$ is $\mathbb{R}$ on which $\mathbb{Z}$ acts by translation. | * A model for $E_\mathcal{TR}\mathbb{Z}$ is $\mathbb{R}$ on which $\mathbb{Z}$ acts by translation. | ||
More generally, $E_\mathcal{TR}G=EG$, 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, $E_\mathcal{TR}G=EG$, 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 $G$, $E_\mathcal{ALL}G$ | + | * For any $G$, a point is a model for $E_\mathcal{ALL}G$. |
− | *$E_\mathcal{FIN}G$ 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 | + | *$E_\mathcal{FIN}G$ 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''. |
* 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. | ||
− | * 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 | + | * 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 $\underbar{E}G$. |
* 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 17:32, 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, 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
Tex syntax error
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
...
4 Classification/Characterization
...
5 Further discussion
...
6 References
This page has not been refereed. The information given here might be incomplete or provisional. |