Classifying spaces for families of subgroups

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1 Introduction

Given a discrete group $G$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}G$ and a family of subgroups $\mathcal{F}$$\mathcal{F}$ of $G$$G$ (see Definition 2.2 below), there is a $G$$G$-CW complex, $E_\mathcal{F}G$$E_\mathcal{F}G$, that classifies $G$$G$-CW complexes with isotropy contained in $\mathcal{F}$$\mathcal{F}$. That is, the isotropy subgroups of $E_\mathcal{F}G$$E_\mathcal{F}G$ are contained in $\mathcal{F}$$\mathcal{F}$ and for every $G$$G$-CW complex $X$$X$ with isotropy in $\mathcal{F}$$\mathcal{F}$, there is a $G$$G$-equivariant map $X \to E_\mathcal{F}G$$X \to E_\mathcal{F}G$ that is unique up to $G$$G$-equivariant homotopy. There can be many models for $E_\mathcal{F}G$$E_\mathcal{F}G$, but the universal property implies that they are all $G$$G$-homotopy equivalent. For this reason $E_\mathcal{F}G$$E_\mathcal{F}G$ is known as the classifying space (or universal space) of $G$$G$ for the family $\mathcal{F}$$\mathcal{F}$. A useful characterization of $E_\mathcal{F}G$$E_\mathcal{F}G$ 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 $G$$G$. The Farrell-Jones Conjecture relates the $K$$K$- and $L$$L$-theory of groups rings $RG$$RG$ to certain equivariant homology theories evaluated at $E_\mathcal{VCYC}G$$E_\mathcal{VCYC}G$, the classifying space of $G$$G$ for the family of virtually cyclic subgroups. Similarly, the Baum-Connes Conjecture relates the topological $K$$K$-theory of the reduced $C^*$$C^*$-algebra of $G$$G$ to an appropriate equivariant homology theory evaluated at $\underbar{E}G$$\underbar{E}G$, the classifying space for proper $G$$G$-actions. (In the case of a discrete group, this means that the isotropy is contained in the family of finite subgroups, i.e., all of the stabilizer subgroups are finite.) From the point of view of computations it is also important to find nice models for these spaces, and in particular for $\underbar{E}G$$\underbar{E}G$, since in many cases calculations can be reduced to working with the family of finite subgroups.

2 Construction and examples

Definition ($G$$G$-CW complex) 2.1. A $G$$G$-CW complex $X$$X$ is a $G$$G$-space with a $G$$G$-invariant filtration of spaces

$\displaystyle \emptyset=X_{-1} \subseteq X_0 \subseteq X_1 \subseteq \cdots \subseteq X_n \subseteq \cdots \subseteq \bigcup_{n\geq 0} X_n = X$ $$\displaystyle \emptyset=X_{-1} \subseteq X_0 \subseteq X_1 \subseteq \cdots \subseteq X_n \subseteq \cdots \subseteq \bigcup_{n\geq 0} X_n = X$$

such that:

1. a set $C \subseteq X$$C \subseteq X$ is closed if and only if $C \cap X_n$$C \cap X_n$ is closed for every $n\geq 0$$n\geq 0$;
2. for each $n\geq 0$$n\geq 0$, $X_n$$X_n$ is obtained from $X_{n-1}$$X_{n-1}$ by attaching $n$$n$-dimensional $G$$G$-cells. That is, there exists a $G$$G$-pushout
   $\displaystyle \xymatrix{ \coprod_{i \in I_n}G/H_i \times S^{n-1} \ar@{^{(}->}[d] \ar[r] & X_{n-1} \ar@{^{(}->}[d] \\ \coprod_{i \in I_n}G/H_i \times D^{n} \ar[r] & X }$
   where $\{ H_i\;|\; i\in I_n \}$$\{ H_i\;|\; i\in I_n \}$ is a collection of subgroups of $G$$G$.

Examples

1. The real line with the translation action of $\mathbb{Z}$$\mathbb{Z}$ is a $\mathbb{Z}$$\mathbb{Z}$-CW complex with one equivariant $0$$0$-cell (the integers) and one equivariant $1$$1$-cell (the orbit of the interval $[0,1]$$[0,1]$).
2. $S^n$$S^n$ with the antipodal $\mathbb{Z}/2$$\mathbb{Z}/2$-action is a $\mathbb{Z}/2$$\mathbb{Z}/2$-CW complex with one equivariant $k$$k$-cell for every $k\leq n$$k\leq n$. (Notice that as a CW complex $S^n$$S^n$ has two $k$$k$-cells for every $k\leq n$$k\leq n$.)
3. $S^\infty$$S^\infty$ is a $\mathbb{Z}/2$$\mathbb{Z}/2$-CW complex with one equivariant $n$$n$-cell in each dimension. The filtration is $\emptyset \subseteq S^0 \subseteq S^1 \subseteq \cdots \subseteq S^n \subseteq \cdots \subseteq \bigcup_{n\geq 0} S^n = S^\infty$$\emptyset \subseteq S^0 \subseteq S^1 \subseteq \cdots \subseteq S^n \subseteq \cdots \subseteq \bigcup_{n\geq 0} S^n = S^\infty$, with the antipodal action of $\mathbb{Z}/2$$\mathbb{Z}/2$ on each $S^n$$S^n$.
4. Let $D_\infty=\langle a,b \;|\; a^2=1, aba^{-1}=b^{-1} \rangle$$D_\infty=\langle a,b \;|\; a^2=1, aba^{-1}=b^{-1} \rangle$ be the infinite dihedral group. It acts on $\mathbb{R}$$\mathbb{R}$, where $a$$a$ acts by reflection through zero and $b$$b$ acts by translation by 1. Thus, $\mathbb{R}$$\mathbb{R}$ is a $D_\infty$$D_\infty$-CW complex with two equivariant $0$$0$-cells (the orbits of 0 and $\frac{1}{2}$$\frac{1}{2}$) and one equivariant $1$$1$-cell (the orbit of the interval $[0,\frac{1}{2}]$$[0,\frac{1}{2}]$).

Definition (Family of subgroups) 2.2. A family $\mathcal{F}$$\mathcal{F}$ of subgroups of a discrete group $G$$G$ is a collection of subgroups of $G$$G$ 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, $\mathcal{FIN}$$\mathcal{FIN}$.
• The family of virtually cyclic subgroups, $\mathcal{VCYC}$$\mathcal{VCYC}$.

Definition (Classifying space for a family of subgroups) 2.3. Let $\mathcal{F}$$\mathcal{F}$ be a family of subgroups of a discrete group $G$$G$. The classifying space of $G$$G$ for $\mathcal{F}$$\mathcal{F}$ is a $G$$G$-CW complex $E_\mathcal{F}G$$E_\mathcal{F}G$ whose isotropy groups are contained in $\mathcal{F}$$\mathcal{F}$ and has the property that for any $G$$G$-CW complex $X$$X$ with isotropy in $\mathcal{F}$$\mathcal{F}$, there is a $G$$G$-equivariant map $X \to E_\mathcal{F}G$$X \to E_\mathcal{F}G$ that is unique up to $G$$G$-homotopy.

An important fact that follows from the Generalized Whitehead Theorem ([Lück2005, Theorem 1.6]) is:

Theorem 2.4. Let $f: X \to Y$$f: X \to Y$ be a $G$$G$-equivariant map of $G$$G$-CW complexes. Then $f$$f$ is a $G$$G$-homotopy equivalence if and only if for every subgroup $H$$H$ of $G$$G$, the induced map on fixed sets $f^H: X^H \to Y^H$$f^H: X^H \to Y^H$ 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 $G$$G$-CW complex is a model for $E_\mathcal{F}G$$E_\mathcal{F}G$.

Corollary 2.5. A $G$$G$-CW complex $X$$X$ is a model for $E_\mathcal{F}G$$E_\mathcal{F}G$ if and only if $X^H$$X^H$ is weakly contractible for every $H\in \mathcal{F}$$H\in \mathcal{F}$ and is empty otherwise. In particular, $E_\mathcal{F}G$$E_\mathcal{F}G$ is contractible.

Examples

1. $\mathbb{R}$$\mathbb{R}$ with the $\mathbb{Z}$$\mathbb{Z}$-CW structure described above is a model for the classifying space of $\mathbb{Z}$$\mathbb{Z}$ for the trivial family.
2. $S^n$$S^n$ with the $\mathbb{Z}/2$$\mathbb{Z}/2$-CW structure described above is not a model for the classifying space of $\mathbb{Z}/2$$\mathbb{Z}/2$ for the trivial family, because $S^n$$S^n$ is not contractible.
3. $S^\infty$$S^\infty$ is a model for the classifying space of $\mathbb{Z}/2$$\mathbb{Z}/2$ for the trivial family, since it is weakly contractible and hence contractible.
4. $\mathbb{R}$$\mathbb{R}$ with the $D_\infty$$D_\infty$-CW structure described above is a model for the classifying space of $D_\infty$$D_\infty$ for the family of finite subgroups. Notice that the nontrivial finite subgroups of $D_\infty$$D_\infty$ are of the form $\langle ab^i \rangle$$\langle ab^i \rangle$, where $i\in \mathbb{Z}$$i\in \mathbb{Z}$, and for each $i$$i$, $\langle ab^i \rangle$$\langle ab^i \rangle$ fixes $-\frac{i}{2}\in \mathbb{R}$$-\frac{i}{2}\in \mathbb{R}$. All other points have trivial stabilizers.

So far the existence question has been ignored. However, given a discrete group $G$$G$ and a family of subgroups $\mathcal{F}$$\mathcal{F}$ of $G$$G$, a model for $E_\mathcal{F}G$$E_\mathcal{F}G$ always exists. As explained in [Lück2005, Theorem 1.9], Corollary 2.5 implies that a model for $E_\mathcal{F}G$$E_\mathcal{F}G$ can be constructed by attaching equivariant $n$$n$-cells, $G/H\times D^n$$G/H\times D^n$ to kill each of the homotopy groups of the $H$$H$-fixed point sets, for each $H$$H$ in $\mathcal{F}$$\mathcal{F}$. This process is illustrated in the $S^\infty$$S^\infty$-example. Clearly, this procedure for constructing classifying spaces will typically produce very large models for $E_\mathcal{F}G$$E_\mathcal{F}G$ and is therefore usually not a useful model to work with.

More examples of classifying spaces:

• The classifying space of $G$$G$ for the trivial family (i.e., for free actions) is just $EG$$EG$, the universal cover of the classifying space $BG=G\backslash EG$$BG=G\backslash EG$. (Recall that $BG$$BG$ is a CW complex whose fundamental group is $G$$G$ and whose higher homotopy groups are all zero; it is unique up to homotopy.)
• For any group, a point is a model for the classifying space for the family of all subgroups.
• $E_\mathcal{FIN}G$$E_\mathcal{FIN}G$ is also known as the classifying space (or universal space) for proper $G$$G$-actions, and is commonly written as $\underbar{E}G$$\underbar{E}G$. There are typically "nice" models for $\underbar{E}G$$\underbar{E}G$, as the $D_\infty$$D_\infty$-example above and the next few examples show.
• Let $G$$G$ be a discrete subgroup of a Lie group $L$$L$ with finitely many path components. If $K$$K$ is a maximal compact subgroup of $L$$L$, then $G/K$$G/K$ is a finite dimensional model for $\underbar{E}G$$\underbar{E}G$. ([Lück2005])
• Let $G$$G$ be a word hyperbolic group in the sense of Gromov. Then the Rips complex, $P_d(G)$$P_d(G)$, is a finite model for $\underbar{E}G$$\underbar{E}G$ ( i.e., there are only finitely many $G$$G$-cells), provided $d$$d$ is sufficiently large. ([Meintrup&Schick2002])
• If $X$$X$ is a $G$$G$-CW complex with the structure of a CAT(0)-space with respect to which $G$$G$ acts by isometries, then $X$$X$ is a model for $\underbar{E}G$$\underbar{E}G$. ([Bridson&Haefliger1999])
• Models for $E_\mathcal{VCYC}G$$E_\mathcal{VCYC}G$ 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]

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