Every CW-complex is a classifying space for proper bundles

by Ian J. Leary† and Brita E. A. Nucinkis‡

We prove that, up to homotopy equivalence, every connected CW-complex is the quotient of a contractible complex by a proper action of a discrete group, and that every CW-complex is the quotient of an aspherical complex by an action of a group of order two.

1. Introduction.

Throughout this note, G shall denote a discrete group. A G-CW-complex is a CW-complex upon which G acts by permuting the cells. A G-CW-complex is said to be proper if each cell stabiliser is finite, which is equivalent to the requirement that G should act properly discontinuously. Our main theorem may be stated as follows.

Theorem 1. For any connected CW-complex X, there exists a discrete group G_{X} and a contractible, proper G_{X}-CW-complex E_{X} such that E_{X}/G_{X} is homotopy-equivalent to X. Moreover, E_{X} has the property that for each finite H ≤ G_{X}, the fixed point subcomplex E_{X}^{H} is contractible.

The group G_{X} that we shall construct is not torsion-free (except in the case when X is 1-dimensional). However, it will transpire that G_{X} always contains a torsion-free subgroup K_{X} of index two. The subgroup K_{X} must act freely on E_{X}, and so E_{X}/K_{X} is aspherical. (Recall that a space Y is said to be aspherical if every map from an n-sphere to Y is homotopic to a constant map, for every n > 1.) Hence we obtain the following result.

Theorem 2. Let C_{2} denote the cyclic group of order two. For every CW-complex X, there is an aspherical C_{2}-CW-complex X̂ such that X̂/C_{2} is homotopy equivalent to X.

The statement of Theorem 1 hides the connection with the Kan-Thurston theorem and related results [11,3,12,13,9]. For any discrete group G there is, up to G-homotopy equivalence, a unique proper G-CW-complex E such that E^{H} is contractible for every finite H ≤ G. We write \underline{B}G for the quotient E/G, and call \underline{B}G a classifying space for proper Gbundles — see the appendix to [2] for an explanation of this nomenclature. The version of Theorem 1 that we prove is Theorem 1′, stated below. Note that Theorem 2 is a corollary of this statement, and that we have added a statement about finiteness conditions.

Theorem 1′. For any CW-complex X there exists a group G_{X} such that \underline{B}G_{X} is homotopy equivalent to X. The group G_{X} has a torsion-free subgroup K_{X} of index two. If X is finite, there is a finite model for BK_{X}.

The proof of this formulation of our theorem is similar to proofs of the theorem of D. M. Kan and W. P. Thurston mentioned above: for any connected X there is a group G and a map BG → X inducing an isomorphism on homology with any local coefficients coming from X [11]. Another similar result is the theorem of D. McDuff which says that for any connected X, there is a monoid M such that BM is homotopy equivalent to X

† Partially funded by the Nuffield Foundation and INTAS.

‡ Funded by EPSRC grant no. GR/L69398.

[13]. Other versions of these theorems appear in [3,12,9]. To prove these theorems one needs to know that the classifying space behaves well for free products of groups (resp. of monoids), and one needs a sufficiently large supply of groups with acyclic classifying spaces (resp. monoids with contractible classifying spaces).

To prove Theorem 1′, we check that \underline{B}(−) is well-behaved with respect to free products, and exhibit some groups G for which \underline{B}G is contractible. The result we need concerning free products has been proved independently by R. J. Platten, so we give only a sketch proof and refer the reader to [15,16] for a detailed proof (and a different application of this result).

In fact the arguments closest to those used in our proof of Theorem 1′ appear in the proof of the Kan-Thurston theorem given by G. Baumslag, E. Dyer and A. Heller [3], and the variation on that proof given by C. R. F. Maunder [12]. In this way we avoid having to construct a functorial embedding of groups into groups with contractible \underline{B}G’s. (Analogous embedding functors play a vital rôle in [11] and [9], and appear in [13].) In the next section we collect together various facts concerning \underline{B}G and generalizations, and in the third section we construct some groups for which \underline{B}G is contractible. In the fourth section we complete the proof of Theorem 1′. Finally in section 5 we give some examples of \underline{B}G that served as motivation for our work, although they are not needed in the proof of the main theorem.

The authors wish to thank Mike Davis and Guido Mislin for their encouragement with this work, and to thank Martin Dunwoody, Mary Jones and Graham Niblo for helpful discussions concerning G. Higman’s acyclic groups.

2. Classifying spaces for families of subgroups.

Some of the material in this section can be found in [8], but we have tried to make this section self-contained. Some of the material is covered in greater generality than is required in the sequel.

Let F = F(G) be a family of subgroups of G that is subgroup-closed and closed under conjugation. (The cases that are of most interest for our purposes are the case when F consists of all finite subgroups of G, and the case when F consists of just the trivial subgroup of G.) A G-CW-complex Y is said to be a model for E_{F}G if every cell stabiliser in Y is an element of F and for each H ∈ F, Y^{H} is contractible.

Suppose that Z is any G-CW-complex with all stabilisers in F, and that Y is a model for E_{F}G. An argument using obstruction theory shows that there is a unique G-homotopy class of equivariant maps from Z to Y. Hence any model for E_{F}G is a terminal object in the homotopy category of G-CW-complexes with stabilisers in F, and any two models for E_{F}G are G-homotopy equivalent. Conversely, if Y is a terminal object in the homotopy category of G-CW-complexes with stabilisers in F, the existence and uniqueness of a map G/H × S^{n} → Y for each n and each H ∈ F imply that each Y^{H} is contractible, and hence Y is a model for E_{F}G.

To establish the existence of E_{F}G, one may follow either Milnor’s or Segal’s construction of EG, the universal free G-CW-complex: for any set S, let E′_{S} be the infinite join of copies of S, topologized as in [14]. An argument due to Milnor shows that E′_{S} is contractible if S is non-empty. Alternatively, let E_{S} be the realisation of a semi-simplicial set with n-simplices the set S^{n+1}, and the obvious face maps. (Thus E_{S} is a CW-complex with one n-cell for every (n + 1)-tuple (s_{0}, …, s_{n}).) If S is empty, then so is E_{S}. If S is not empty, E_{S} may be identified with the nerve of a category equivalent to the category with one object and one morphism, and hence by an argument of Segal, E_{S} is contractible [17].

Now let Δ be a G-set such that the fixed-point subset Δ^{H} is non-empty if and only if H ∈ F, for example the union of the cosets of all subgroups in F. The action of G on Δ induces actions of G on E′_{Δ} and on E_{Δ}. It is easily verified that for any H ≤ G, (E′_{Δ})^{H} = E′_{(Δ^{H})}, and (E_{Δ})^{H} = E_{(Δ^{H})}, and hence that each of these spaces is contractible if H ∈ F and empty otherwise. It follows that E′_{Δ} and E_{Δ} are models for E_{F}G.

We refer to E_{F}G as a classifying space for G-CW-complexes with stabilisers in F. In the case when F is just the trivial group, E_{F}G is EG, the usual classifying space for free G-CW-complexes. In the case when F is the family of all finite subgroups of G, we write \underline{E}G instead of E_{F}G, and call \underline{E}G the classifying space for proper G-CW-complexes. There is a third model for \underline{E}G as well as Milnor’s model and Segal’s. This is the realisation of the poset of finite subsets of G. One could view this as the barycentric subdivision of the (possibly infinite) simplex with vertex set G.

The quotient E_{F}G/G will be denoted B_{F}G. The quotient EG/G is of course BG, the classifying space for principal G-bundles. The quotient \underline{E}G/G will be denoted \underline{B}G, and called the classifying space for proper G-bundles. The appendix to [2] explains the sense in which \underline{B}G deserves this name.

Since E_{F}G is well-defined up to G-homotopy equivalence, it follows that B_{F}G is welldefined up to (based) homotopy equivalence. It does not follow that any space homotopy equivalent to a model for B_{F}G is also a model for B_{F}G. For example, a group G can have a 0-dimensional model for \underline{E}G if and only if G is finite. Hence the groups for which a single point is a model for \underline{B}G are precisely the finite groups. On the other hand, we shall exhibit many infinite groups for which \underline{B}G is contractible.

Suppose that F (resp. F′) is a family of subgroups of G (resp. of G′) and f : G → G′ is a group homomorphism such that for every H ∈ F, f(H) is a member of F′. Then f gives rise to a unique equivariant map from E_{F}G to E_{F′}G′, and hence a unique based map from B_{F}G to B_{F′}G′. Now suppose that F is a class of groups that is closed under taking subgroups and homomorphic images. For each group G, a family of subgroups F = F(G) may be defined as the subgroups of G that are in F. In this way B_{F}G becomes a functor on the category of groups. Similarly, if F is only closed under taking subgroups, B_{F}G is a functor on the category of groups and injective homomorphisms.

We close this section by calculating the fundamental group of of B_{F}G for any G, and giving a description of B_{F}G in certain cases when G is a graph of groups.

Proposition 3. For any group G and family F, the fundamental group of B_{F}G is isomorphic to the quotient G/G′, where G′ is the subgroup generated by all subgroups of G that lie in F.

Proof. Let Δ be a G-set such that Δ^{H} is non-empty if and only if H ∈ F, and let E_{Δ} be Segal’s model for E_{F}G. All cell stabilisers in E_{Δ} are contained in G′, and so G/G′ acts freely on E_{Δ}/G′. Hence it will suffice to show that E_{Δ}/G′ is 1-connected.

Let δ_{0} be a vertex of E_{Δ}, and use the image of δ_{0} as a basepoint for E_{Δ}/G′. Any based cellular loop in E_{Δ}/G′ lifts to a path of the form

(δ_{0}, δ_{1}), (δ_{1}, δ_{2}), …, (δ_{n−2}, δ_{n−1}), (δ_{n−1}, δ_{n}),

where each δ_{i} ∈ Δ and for some g ∈ G′, δ_{n} = gδ_{0}. However, for each i, (δ_{i}, δ_{i+1}, δ_{i+2}) is a 2-cell in E_{Δ}, and affords a homotopy between the path (δ_{i}, δ_{i+1}), (δ_{i+1}, δ_{i+2}) and the path (δ_{i}, δ_{i+2}). Hence the path is homotopic, relative to its endpoints, to the single edge (δ_{0}, gδ_{0}). Consideration of the boundary of the 2-cell (δ_{0}, gδ_{0}, ghδ_{0}) shows that the function g ↦ (δ_{0}, gδ_{0}) induces a surjective homomorphism from G′ to π_{1}(E_{Δ}/G′). Now if g stabilises δ ∈ Δ, the 2-cell (δ_{0}, δ, gδ_{0}) has image in E_{Δ}/G equal to a disc with boundary (the image of) (δ_{0}, gδ_{0}). Hence a set of generators for G′ is mapped to 1 ∈ π_{1}(E_{Δ}/G′), and so E_{Δ}/G′ is 1-connected, as claimed.

Proposition 4. Let F be any class of groups that is subgroup closed and contains only finite groups, and let G = H_{1} ∗_{K} H_{2} be a group expressible as a free product with amalgamation. If X_{1}, X_{2} and Y are any models for B_{F}H_{1}, B_{F}H_{2} and B_{F}K respectively, then there is a model for B_{F}G consisting of

X_{1} ∐ Y × I ∐ X_{2}/ ∼,

Where ‘∼’ indicates that Y × {0} is identified with the image of the map Y → X_{1}, and similarly Y × {1} is identified with the image of the map Y → X_{2}.

As remarked in the introduction, we refer the reader to the work of R. J. Platten for a full proof [15,16], but provide the following: Sketch Proof. We prove the analogous result for models for E_{F}G. Let E_{1}, E_{2} and F be models for E_{F}H_{1}, E_{F}H_{2} and E_{F}K respectively, such that X_{i} = E_{i}/H_{i} and Y = F/K. Further, let f_{i} : F → E_{i} be the K-equivariant map induced by the inclusion of K in H_{i}.

The K-equivariant maps f_{i} give rise to G-equivariant maps from G×_{K} F to G×_{H_{i}} E_{i}. Using these maps as attaching maps, build a G-CW-complex E with stabilisers in F:

E = G ×_{H_{1}} E_{1} ∐ (G ×_{K} F) × I ∐ G ×_{H_{2}} E_{2} /∼ .

Let T be the G-tree corresponding to the free product decomposition for G, with two orbits of vertices of orbit types G/H_{i} and one orbit of edges of orbit type G/K (see [6]). There is a G-equivariant map p : E → T given by collapsing each E_{i} and F to a point.

For t a vertex of T, let S(t) denote its (open) star in T, and define an open subset O(t) of E by O(t) = p^{−1}(S(t)). The setwise stabiliser of O(t) is equal to Stab_{G}(t), the stabiliser of t, and is a conjugate of either H_{1} or H_{2}. Moreover, O(t) is a model for E_{F}Stab_{G}(t). If t and t′ are not adjacent in T, O(t) ∩ O(t′) is empty, and if t, t′ are the vertices of an edge e, O(t) ∩ O(t′) is a model for E_{F}Stab_{G}(e).

Now for any L ∈ F, the open cover of the fixed-point set E^{L} given by the (nonempty) O(t) ∩ E^{L} is a cover by contractible sets, whose intersections are either empty or contractible, such that the nerve of the cover is isomorphic to T^{L}. Since the fixed points for any action of a finite group on a tree are contractible (see [6]), it follows that E^{L} is contractible as required.

A similar statement may be proved for any graph of groups using the same technique. The reader may prefer the following version of the statement: for a graph of groups with fundamental group G, let C be the category whose objects are the vertex and edge groups of the graph, with morphisms the inclusions of the edge groups in the vertex groups. For F a class of finite groups as above, any choice of a model X_{H} for B_{F}H for each H ∈ C, together with maps between them, gives rise to a functor from C to the category of CW-complexes, and the complex

hocolim_{H∈C}B_{F}H

is a model for B_{F}G. The only cases that we shall require are those in which the graph of groups is modeled on a star-shaped tree, with one central vertex meeting every edge. These cases can be deduced directly from the special case proved in Proposition 4.

The condition that F should consist of finite groups is necessary, as may be seen from the following counterexample. Let F be the class of free abelian groups (or even the class of groups that are either infinite cyclic or trivial), and let G be the infinite dihedral group. Then G is a free product of two copies of C_{2}, the cyclic group of order two, and by Proposition 3, π_{1}(B_{F}G) is cyclic of order two. Hence B_{F}G is not homotopy equivalent to a wedge of two copies of B_{F}C_{2} = BC_{2}.

Proposition 4 should not be viewed as giving a quick proof of Whitehead’s theorem in the case when F is the class of trivial groups. The fact that BG can be built in this way is equivalent to the existence of the tree used in the proof of Proposition 4.

3. Some contractible groups.

Say that a group G is contractible if \underline{B}G is contractible, and as usual say that a group is acyclic if BG has the same integral homology as a point. For any finite group G, a single point will suffice as a model for \underline{E}G and hence also for \underline{B}G. Thus any finite group is contractible. By Proposition 4, it follows that the free product of any two finite groups is a contractible group. Provided that the two finite groups chosen are non-trivial, any such group contains an infinite cyclic subgroup. We shall also need a contractible group containing a (non-trivial) torsion-free acyclic group. This is provided by

Proposition 5. Let A be Higman’s acyclic group [10] given by the presentation

A = ⟨a, b, c, d: a^{b} = a^{2}, b^{c} = b^{2}, c^{d} = c^{2}, d^{a} = d^{2}⟩.

The presentation 2-complex for this presentation is a model for BA, and there is a contractible group containing A as a subgroup of index two.

Proof. It is well-known that A is an acyclic group of cohomological dimension two, and that the presentation 2-complex for the above presentation is a model for BA [3].

There is an automorphism t of A of order two, defined by a^{t} = c, b^{t} = d. Define a group C containing A as an index two subgroup as the split extension with kernel A and quotient the cyclic group of order two generated by t. This group C will be shown to be a contractible group.

Let E be the model for EA given as the universal cover of the presentation 2-complex given above. We claim that the action of A on E extends to an action of C in such a way that E becomes a model for \underline{E}C. The complex E consists of one free A-orbit of 0-cells, four free orbits of 1-cells, and four free orbits of 2-cells. Let v, e_{i} and f_{i} for 1 ≤ i ≤ 4 be orbit representatives, and write a_{1}, …, a_{4} instead of a, …, d for the generating set for A. With this notation, the boundaries of the cells of E may be defined by

δ(e_{i}) = a_{i}v − v, δ(f_{i}) = (1 + a_{i} − a_{i}^{2}a_{i+1}^{−1}a_{i}^{−1})e_{i} + (a_{i}^{2}a_{i+1}^{−1}a_{i}^{−1} − a_{i}^{2}a_{i+1}^{−1})e_{i+1},

where all indices are to be read modulo 4. It may be checked that the formulae

t · gv = g^{t}v,  t · ge_{i} = g^{t}e_{i+2},  t · gf_{i} = g^{t}f_{i+2},

where g ∈ A, define an action of the involution t on E, and that this together with the given action of A gives rise to an action of C on E. The 2-cells form two free C-orbits, as do the 1-cells, and the 0-cells form a single orbit of type C/⟨t⟩.

Since A has index two in C and acts freely on E, it follows that C acts properly on E. To verify that E is a model for \underline{E}C, it suffices to show that the fixed point set for any finite subgroup of C is contractible. Since the 1-cells and 2-cells are freely permuted by C, the fixed point set of any (non-trivial) finite subgroup consists of 0-cells. P. A. Smith’s theorem tells us that the fixed points for any action of a cyclic group of order two on a finite-dimensional contractible complex must be mod-2-acyclic [1]. Hence any non-trivial finite subgroup of C fixes a single point in E, and E is a model for \underline{E}C.

The model for \underline{B}C consisting of E/C has one 0-cell, two 1-cells and two 2-cells, and is the presentation complex for the presentation

⟨a, b: a^{b} = a^{2}, b^{a} = b^{2}⟩.

This is a presentation for the trivial group, and hence the 2-complex E/C is contractible as claimed.

We mention in passing that the above argument also shows that the involution t fixes no non-identity element of A.

The above construction can easily be generalized to construct a contractible group containing a non-trivial torsion-free acyclic subgroup of index p, for any prime p. Let A(n) be Higman’s group given by the n-generator, n-relator presentation

A(n) = ⟨a_{1}, …, a_{n}: a_{i}^{a_{i+1}} = a_{i}^{2}⟩,

where the indices should be taken modulo n. For n ≤ 3, A(n) is trivial, but for n ≥ 4, A(n) is a torsion-free acyclic group of cohomological dimension two, and the presentation 2-complex for the above presentation is a model for BA(n). The group A(n) has an automorphism of order n that cyclically permutes the generators and relators. Define G(n, m) for any m dividing n to be the group generated by A(n) and the n/mth power of the cyclic automorphism. It may be shown that for any p dividing n, \underline{B}G(n, p) is homotopy equivalent to BA(n/p). In particular, G(2p, p) is a contractible group containing a torsionfree acyclic group of index p, as is G(p, p) for p ≥ 5.

All the other contractible groups that we need come from the following analogue of theorem 6.1 of [3].

Proposition 6. Suppose that G is a subgroup of a contractible group H. There is a contractible group P containing H ∗_{G} H as a subgroup. If H has an index two torsion-free subgroup, P may be chosen to have an index two torsion-free subgroup, whose intersection with each copy of H is the given torsion-free subgroup of H.

Proof. Let A be Higman’s acyclic group and let C be the contractible group constructed in Proposition 5. Define P to be H ∗_{G} (G × A) ∗_{A} C. Then P contains H ∗_{G} (G × A) as a subgroup, and as in [3], H ∗_{G} H is a subgroup of this group, so is a subgroup of P. Let ϕ : H → C_{2} be a homomorphism with torsion-free kernel, and let ϕ′ : C → C_{2} be the homomorphism with kernel A. Define a homomorphism ϕ′′ : G×A → C_{2} as the composite of the projection onto G with ϕ|_{G}. These three homomorphisms from the vertex groups of P to C_{2} agree on the edge groups, and so define a homomorphism from P to C_{2} with torsion-free kernel.

It remains to show that P is contractible. It is easy to show that for any groups K_{1} and K_{2}, \underline{B}(K_{1} × K_{2}) = \underline{B}K_{1} × \underline{B}K_{2}. Since A is torsion-free, \underline{B}A = BA and so \underline{B}A is acyclic. It follows that the inclusion of G in G × A induces a homology isomorphism from \underline{B}G to \underline{B}(G×A). The fundamental group π_{1}(\underline{B}(G×A)) is isomorphic to π_{1}(\underline{B}G)×A. From Proposition 4, it follows that \underline{B}(G × A) ∗_{A} C is homotopy equivalent to \underline{B}(G × A) with a cone on the acyclic subspace \underline{B}A attached. The map f : \underline{B}G → \underline{B}(G × A) ∗_{A} C induces an isomorphism of fundamental groups, which are isomorphic to G/G′, the quotient of G by its subgroup generated by torsion. The universal covers are respectively \underline{E}G/G′ and \underline{E}G/G′ × BA with a free G/G′-orbit of cones on BA attached to a subspace G/G′ × BA, and so the map between universal covers induces an isomorphism of homology groups. It follows that f is a homotopy equivalence. By Proposition 4, it follows that \underline{B}P is homotopy equivalent to the mapping cylinder of the map from \underline{B}G to \underline{B}H, and so is contractible since \underline{B}H is.

The direct analogue of the proof given in [3] would be to take P = H ∗_{G} (G × C), for some non-trivial contractible group C. For example, we could take C to be cyclic of order two. This P is a contractible group containing H ∗_{G} H, but since any contractible group contains torsion, this P would contain larger torsion subgroups than H did. This simpler construction suffices to prove a weaker version of Theorem 1′, in which the minimal index of a torsion-free subgroup of G_{X} depends on the dimension of X.

4. Proofs of the main theorems.

Proof of Theorem 2. It clearly suffices to prove Theorem 2 under the additional assumption that X be connected. As remarked in the introduction, this is a corollary of Theorem 1′, since we may take X̂ = BK_{X} = \underline{B}K_{X} with the action of G_{X}/K_{X} ≅ C_{2}.

Proof of Theorem 1′. By the simplicial approximation theorem, we may assume that X is a simplicial complex. Pick a basepoint x_{0} for X, pick a maximal tree T in X, and wellorder the vertices of X in such a way that the shortest path in T from any vertex to x_{0} is a descending sequence. (In particular, x_{0} should be the first element of the ordering.) For any subcomplex Y of X, this gives rise to a way to base Y, without changing its homotopy type, by adding to Y the shortest path in T from the least vertex of Y to x_{0}.

We shall define a pair of functors from the category of based connected subcomplexes of X and inclusions to the category of groups and monomorphisms, Y ↦ (H_{Y} , G_{Y}), with G_{Y} ≤ H_{Y}, a model for \underline{B}H_{Y} containing a model for \underline{B}G_{Y} as a subcomplex, and a map of pairs from (\underline{B}H_{Y} , \underline{B}G_{Y}) to (CY, Y) that is a homotopy equivalence. (Here CY denotes the cone on Y.) We shall also define a natural (for inclusions of subcomplexes) map ϕ : H_{Y} → C_{2}, with torsion-free kernel. For the remainder of the proof, suppose that Y is a based subcomplex of X.

In the case when Y is 1-dimensional, take G_{Y} = π_{1}(Y), and take Y as the chosen model for \underline{B}G_{Y}. The definition of H_{Y} is slightly more complicated. Fix an element g of infinite order in an infinite dihedral group D. Let H_{Y} be a free product of copies of D indexed by the edges of Y. By Proposition 4, H_{Y} is a contractible group. Define a homomorphism from G_{Y} to H_{Y} as follows. An element of G_{Y} can be represented by a based loop in Y, denoted e_{1}^{ϵ_{1}} ⋯ e_{n}^{ϵ_{n}}, where e_{i} is an edge of Y, and ϵ_{i} = ±1 depending whether the edge is traversed with or against its orientation. Send the corresponding element of G_{Y} to the word g_{1}^{ϵ_{1}} ⋯ g_{n}^{ϵ_{n}}, where g_{i} denotes the copy of g in the copy of D corresponding to the edge e_{i}. This defines a homomorphism from G_{Y} to H_{Y} having the required naturality properties. By applying Proposition 4 to the free product decomposition H_{Y} = G_{Y} ∗_{G_{Y}} H_{Y}, obtain a model for \underline{B}H that is the mapping cylinder of a morphism from Y to a contractible complex. The morphism sending this contractible complex to a point induces the required map of pairs (\underline{B}H_{Y} , \underline{B}G_{Y}) → (CY, Y). There is a unique homomorphism ϕ : H_{Y} → C_{2} with torsion-free kernel, which takes every element of order two to the non-identity element of C_{2}.

Now suppose that the construction has been completed for every based subcomplex of X of dimension at most n − 1, and let Y be n-dimensional. Let Z be the (n − 1)-skeleton of Y, and for each n-simplex σ of Y, let T (σ) denote the based complex consisting of the boundary of σ and the shortest path in T from the least vertex of the boundary of σ to x_{0}.

The group G_{Y} is constructed as a star-shaped graph of groups, with central vertex group G_{Z}, and for each σ, an edge group G_{T (σ)} joining the central vertex to a vertex with group H_{T (σ)}. By Proposition 5, there is a contractible group P_{T (σ)} containing H_{T (σ)} ∗_{G_{T (σ)}} H_{T (σ)} as a subgroup. Fix such a group P_{T (σ)} depending only on σ and not on Y. In this way one obtains a map of pairs of groups (H, G) → (P, H), where the two copies of H have intersection equal to G.

Now define the group H_{Y} to be a star-shaped graph of groups, with central vertex group H_{Z}, and for each σ an edge group H_{T (σ)} joining the central vertex to a vertex with group P_{T (σ)}. This graph of groups contains G_{Y} as a subgroup, where the inclusion is given by the ‘graph of inclusions’. The homomorphisms ϕ : H_{Z} → C_{2}, ϕ : H_{T (σ)} → C_{2} and ϕ : P_{T (σ)} → C_{2} already constructed are compatible, and so give rise to a homomorphism ϕ : H_{Y} → C_{2} with torsion-free kernel.

Build a model for \underline{B}G_{Y} from the models for the edge and vertex groups that have already been constructed, using Proposition 4. Make a map from this model for \underline{B}G_{Y} to Y as follows. Take the map already constructed on \underline{B}G_{Z}, for each n-simplex σ map \underline{B}H_{T (σ)} to the centroid of σ, and define the map on \underline{B}G_{T (σ)} ×I using a homotopy (within T (σ)∪σ) between the identity map of T (σ) and the constant map to the centroid of σ.

To build a model for \underline{B}H_{Y}, work in stages. Let J_{Y} be a star-shaped graph of groups with centre vertex group H_{Z}, and edge groups G_{T (σ)} joining the central vertex to other vertices with groups H_{T (σ)}. Now H_{Y} may be constructed as a star-shaped graph of groups with centre vertex group J_{Y}, edge groups H_{T (σ)} ∗_{G_{T (σ)}} H_{T (σ)} and outer vertex groups P_{T (σ)}. Applying Proposition 4 to the graph of groups decomposition for J_{Y} gives a model for \underline{B}J_{Y} containing the given model for \underline{B}G_{Y}. A map from this model to CZ ∪ Y may be constructed as above: take the given map from \underline{B}H_{Z} to CZ, map the whole of \underline{B}H_{T (σ)} to the centroid of σ, and use a homotopy from the identity map on T (σ) to the constant map to the centroid of σ to define the map on \underline{B}G_{T (σ)} × I.

Now apply Proposition 4 again to obtain a model for \underline{B}H_{Y} containing the above model for \underline{B}J_{Y}. Make a map from this model to CY as in the two previous cases. Take the given map from \underline{B}J_{Y} to CZ ∪ Y, and for each σ map \underline{B}P_{T (σ)} to the centroid of the cone on σ. Use a homotopy from the identity map on CT (σ) ∪ σ to the constant map to the centroid of Cσ to define the map on \underline{B}(H_{T (σ)} ∗_{G_{T (σ)}} H_{T (σ)}) × I.

Finally, we leave it as an exercise for the reader to check that when X is finite the inductive process used above may be used to build a finite model for BK_{X}.

We use the prime 2 to ease the notation, but by varying the construction slightly we could obtain G_{X} having a torsion-free subgroup of index p, for any prime p. Most of the changes that need to be made are trivial. For example, every occurrence of C_{2} (resp. D) in the above proof should be replaced by C_{p} (resp. C_{p} ∗ C_{p}). The p-analogue of Proposition 6 works as before, given the existence of a contractible group having a torsion-free acyclic group of index p. Such a group can be constructed as in the remarks below the proof of Proposition 5.

If every occurrence of the group C_{2} and the group D in the proof is replaced by a copy of Higman’s acyclic group [10] (or any other acyclic group containing an element of infinite order), the word ‘contractible’ is replaced by the word ‘acyclic’ and every occurrence of \underline{B}(−) is replaced by B(−), the proof becomes a proof of the Kan-Thurston theorem, modelled on Maunder’s version [12]. We have felt obliged to be more pedantic about basepoints than Maunder, however, for the following reason. To determine whether a space X is a BG for some G, it suffices to show that X is aspherical. On the other hand, before proving Theorem 1′, one has no way of determining whether a space is a \underline{B}G. One way to avoid these worries would be to work throughout with groupoids instead of groups.

5. Motivating examples and questions.

One corollary of Theorem 1′ is the group-theoretic statement given below. The direct proof of Corollary 7 was found before the proof of Theorem 1′, and provided motivating evidence.

Corollary 7. Let G be any group. There is a group G̃ and a surjection ϕ : G̃ → G such that the kernel of ϕ is the subgroup of G̃ generated by all torsion elements.

Proof. Let X be any connected complex with fundamental group G, and let G̃ = G_{X}. The result follows from Proposition 3.

Direct proof. Let ψ : F → G be a surjection from a free group to G, with kernel N. Thus N is a free group. Let H be a free product of copies of C_{2} that contains a subgroup isomorphic to N. (Provided that F is countable, H = C_{2} ∗ C_{2} ∗ C_{2} will suffice.) Now take G̃ = F ∗_{N} H, with ϕ induced by ψ.

Our construction of the groups G_{X} is not as explicit as McDuff’s construction of monoids in [13]. One class of spaces X for which we were able to find a more explicit construction is the class of suspensions. The examples are based on right-angled Coxeter groups. They were known to us quite early during this project and provided useful motivation.

A right-angled Coxeter group is a group generated by a given set of elements of order two (the ‘Coxeter generators’), subject only to the relations that certain pairs of the generators commute. For more information concerning these groups, see [5,7]. A rightangled Coxeter group is either an elementary abelian 2-group, or is a free product with amalgamation of right-angled Coxeter groups in which each factor has fewer generators. It follows from Proposition 4 that \underline{B}G is contractible for every right-angled Coxeter group G. There is another way to see this fact, using a simplicial complex that was introduced by M. Davis, which turns out to be a model for \underline{E}G [5].

Let G be a right-angled Coxeter group. A Coxeter subgroup of G is by definition a subgroup generated by a subset of the Coxeter generators for G. Davis’ complex Σ = Σ(G) is the realisation of the poset whose elements are the cosets of the finite Coxeter subgroups of G. The action of G on the cosets induces an action on Σ such that all stabilisers are finite. For a proof that Σ is a model for \underline{E}G, and another application of this fact, see [4]. The Coxeter presentation for G may be encoded as a simplicial complex, K = K(G), whose vertices are the Coxeter generators for G and whose simplices are sets of commuting generators. Any flag complex (for example, the barycentric subdivision of any simplicial complex) arises in this way for some G. For any right-angled Coxeter group, G, there is a surjection from G to C_{2} that sends each Coxeter generator to the non-zero element of C_{2}. The kernel of this homomorphism consists of those elements of G expressible as words of even length in the Coxeter generators.

Proposition 8. Let G be the right-angled Coxeter group corresponding to a flag complex K, and let H be the index two subgroup of G consisting of even length words in the Coxeter generators. The model for \underline{B}H constructed as Σ/H is isomorphic to the suspension of the barycentric subdivision of K.

Proof. It is easily checked that the model for \underline{B}G given by Σ/G is isomorphic to the realisation of the poset of subsets of the Coxeter generators that generate finite subgroups, and hence is isomorphic to the cone on K′.

Since H does not contain any non-trivial Coxeter subgroup of G, the G-orbits in the poset used to construct Σ consist of single H-orbits, except for the orbit consisting of the cosets of the trivial subgroup, which splits into two H-orbits. The realisation of the poset of H-orbits of cosets is the suspension of K′, as claimed.

It would be interesting to have a proof of Theorem 1′ along the lines of the original proof of the Kan-Thurston theorem. The missing ingredient is a functorial embedding of groups into contractible groups. Given such a functor, Kan and Thurston’s proof would translate verbatim into a proof of a version of Theorem 1′ (which would not necessarily give a bound on the index of a torsion-free subgroup).

Mike Davis asked the following question, which our methods are unable to address. Suppose that X is a manifold. Is there a choice for G_{X} so that \underline{E}G_{X} is a manifold upon which G_{X} acts as homeomorphisms, in such a way that \underline{B}G_{X} is homeomorphic to X?

6. References.

[1] C. Allday and V. Puppe, Cohomological Methods in Transformation Groups, Cambridge Studies in Advanced Mathematics 32, Cambridge Univ. Press (1993).

[2] P. Baum and A. Connes, Chern character for discrete groups, in A Fête of Topology: papers dedicated to Itiro Tamura, eds. Y. Matsumoto, T. Mizutani and S. Morita, Academic Press (1988).

[3] G. Baumslag, E. Dyer and A. Heller, The topology of discrete groups, Journal of Pure and Applied Algebra 16 (1980) 1–47.

[4] N. P. Brady, I. J. Leary and B. E. A. Nucinkis, On algebraic and geometric dimensions for groups with torsion, in preparation.

[5] M. W. Davis, Groups generated by reflections, Ann. of Math. 117 (1983) 293–324.

[6] W. Dicks and M. J. Dunwoody, Groups Acting on Graphs, Cambridge Studies in Advanced Mathematics 17, Cambridge Univ. Press (1989).

[7] W. Dicks and I. J. Leary, On subgroups of Coxeter groups, Geometry and Cohomology in Group Theory, London Mathematical Society Lecture Note Series 252, Cambridge Univ. Press (1998).

[8] T. tom Dieck, Transformation Groups, Studies in Math. 8 de Gruyter (1987).

[9] Z. Fiedorowicz, Classifying spaces of topological monoids and categories, Amer. J. Math 106 (1984) 301–350.

[10] G. Higman, A finitely generated infinite simple group, J. London Math. Soc. 24 (1951) 61–64.

[11] D. M. Kan and W. P. Thurston, Every connected space has the homology of a K(π, 1), Topology 15 (1976), 253–258.

[12] C. R. F. Maunder, A short proof of a theorem of Kan and Thurston, Bull. London Math. Soc. 13 (1981), 325–327.

[13] D. McDuff, On the classifying spaces of discrete monoids, Topology 18 (1979), 313–320.

[14] J. W. Milnor, Construction of universal bundles I, Ann. of Math. 63 (1956), 272–284.

[15] R. J. Platten, PhD thesis, Queen Mary and Westfield, London (1999).

[16] R. J. Platten, On the minimal dimension of a classifying space for proper bundles, in preparation.

[17] G. Segal, Classifying spaces and spectral sequences, Publ. I.H.E.S. 34 (1968), 105–112.

Authors’ address:

Faculty of Mathematical Studies, University of Southampton, Southampton, SO17 1BJ.