Every CW-complex is a classifying space for proper bundles

                     by Ian J. Learyy and Brita E. A. Nucinkisz
    We prove that, up to homotopy equivalence, every connected CW-complex is th*
 *e quotient of a con-
    tractible 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-c*
 *omplex
upon which G acts by permuting the cells. A G-CW-complex is said to be proper i*
 *f 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 GX and*
 * a
contractible, proper GX -CW-complex EX such that EX =GX is homotopy-equivalent *
 *to X.
Moreover, EX  has the property that for each finite H  GX , the fixed point sub*
 *complex
EHX is contractible.

    The group GX  that we shall construct is not torsion-free (except in the ca*
 *se when
X is 1-dimensional).  However, it will transpire that GX  always contains a tor*
 *sion-free
subgroup KX  of index two. The subgroup KX  must act freely on EX , and so EX =*
 *KX  is
aspherical. (Recall that a space Y is said to be aspherical if every map from a*
 *n n-sphere
to Y  is homotopic to a constant map, for every n > 1.)  Hence we obtain the fo*
 *llowing
result.

Theorem 2. Let C2 denote the cyclic group of order two.  For every CW-complex X,
there is an aspherical C2-CW-complex X^such that X^=C2 is homotopy equivalent t*
 *o X.

    The statement of Theorem 1 hides the connection with the Kan-Thurston theor*
 *em
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 EH is contractible for ev*
 *ery finite
H  G. We write B_G for the quotient E=G, and call B_G a classifying space for p*
 *roper G-
bundles _ see the appendix to [2] for an explanation of this nomenclature. The *
 *version of
Theorem 1 that we prove is Theorem 10, stated below. Note that Theorem 2 is a c*
 *orollary
of this statement, and that we have added a statement about finiteness conditio*
 *ns.

Theorem 10. For any CW-complex X there exists a group GX such that B_GX is homo-
topy equivalent to X. The group GX  has a torsion-free subgroup KX  of index tw*
 *o. If X
is finite, there is a finite model for BKX .

    The proof of this formulation of our theorem is similar to proofs of the th*
 *eorem of
D. M. Kan and W. P. Thurston mentioned above: for any connected X there is a gr*
 *oup
G and a map BG ! X inducing an isomorphism on homology with any local coefficie*
 *nts
coming from X [11]. Another similar result is the theorem of D. McDuff which sa*
 *ys that
for any connected X, there is a monoid M such that BM is homotopy equivalent to*
 * X
_________________________
  y Partially funded by the Nuffield Foundation and INTAS.
  z Funded by EPSRC grant no. GR/L69398.


                                        1

[13].  Other versions of these theorems appear in [3,12,9].  To prove these the*
 *orems one
needs to know that the classifying space behaves well for free products of grou*
 *ps (resp.
of monoids), and one needs a sufficiently large supply of groups with acyclic c*
 *lassifying
spaces (resp. monoids with contractible classifying spaces).
    To prove Theorem 10, we check that B_(-) is well-behaved with respect to fr*
 *ee products,
and exhibit some groups G for which B_G is contractible. The result we need con*
 *cerning
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 app*
 *lication of this
result).
    In fact the arguments closest to those used in our proof of Theorem 10 appe*
 *ar 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 contracti*
 *ble B_G's.
(Analogous embedding functors play a vital r^ole in [11] and [9], and appear in*
 * [13].)  In
the next section we collect together various facts concerning B_G and generaliz*
 *ations, and
in the third section we construct some groups for which B_G is contractible. In*
 * the fourth
section we complete the proof of Theorem 10. Finally in section 5 we give some *
 *examples
of 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 encourageme*
 *nt 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 t*
 *han is required
in the sequel.
    Let F = F(G) be a family of subgroups of G that is subgroup-closed and clos*
 *ed
under conjugation. (The cases that are of most interest for our purposes are th*
 *e case when
F consists of all finite subgroups of G, and the case when F consists of just t*
 *he trivial
subgroup of G.) A G-CW-complex Y is said to be a model for EF G if every cell s*
 *tabiliser
in Y is an element of F and for each H 2 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 EF G. An argument using obstruction theory shows that there is a unique G-h*
 *omotopy
class of equivariant maps from Z to Y . Hence any model for EF G is a terminal *
 *object in
the homotopy category of G-CW-complexes with stabilisers in F, and any two mode*
 *ls for
EF G are G-homotopy equivalent. Conversely, if Y is a terminal object in the ho*
 *motopy
category of G-CW-complexes with stabilisers in F, the existence and uniqueness *
 *of a map
G=H x Sn ! Y for each n and each H 2 F imply that each Y H is contractible, and*
 * hence
Y is a model for EF G.
    To establish the existence of EF G, one may follow either Milnor's or Segal*
 *'s con-
struction of EG, the universal free G-CW-complex: for any set S, let E0Sbe the *
 *infinite
join of copies of S, topologized as in [14]. An argument due to Milnor shows th*
 *at E0Sis
contractible if S is non-empty. Alternatively, let ES be the realisation of a s*
 *emi-simplicial


                                        2

set with n-simplices the set Sn+1 , and the obvious face maps. (Thus ES is a CW*
 *-complex
with one n-cell for every (n + 1)-tuple (s0; : :;:sn).) If S is empty, then so *
 *is ES. If S is
not empty, ES 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, ES is cont*
 *ractible
[17].
    Now let  be a G-set such that the fixed-point subset H  is non-empty if and*
 * only
if H 2 F, for example the union of the cosets of all subgroups in F.  The actio*
 *n of G
on  induces actions of G on E0  and on E  .  It is easily verified that for any*
 * H  G,
(E0 )H = E0(H ), and (E  )H = E(H ), and hence that each of these spaces is con*
 *tractible
if H 2 F and empty otherwise. It follows that E0 and E  are models for EF G.
    We refer to EF G as a classifying space for G-CW-complexes with stabilisers*
 * in F. In
the case when F is just the trivial group, EF G is EG, the usual classifying sp*
 *ace for free
G-CW-complexes. In the case when F is the family of all finite subgroups of G, *
 *we write
E_G instead of EF G, and call E_G the classifying space for proper G-CW-complex*
 *es. There
is a third model for E_G as well as Milnor's model and Segal's. This is the rea*
 *lisation of
the poset of finite subsets of G. One could view this as the barycentric subdiv*
 *ision of the
(possibly infinite) simplex with vertex set G.
    The quotient EF G=G will be denoted BF G. The quotient EG=G is of course BG*
 *, the
classifying space for principal G-bundles.  The quotient E_G=G will be denoted *
 *B_G, and
called the classifying space for proper G-bundles. The appendix to [2] explains*
 * the sense
in which B_G deserves this name.
    Since EF G is well-defined up to G-homotopy equivalence, it follows that BF*
 * G is well-
defined up to (based) homotopy equivalence. It does not follow that any space h*
 *omotopy
equivalent to a model for BF G is also a model for BF G.  For example, a group *
 *G can
have a 0-dimensional model for E_G if and only if G is finite. Hence the groups*
 * for which a
single point is a model for B_G are precisely the finite groups. On the other h*
 *and, we shall
exhibit many infinite groups for which B_G is contractible.
    Suppose that F (resp. F0) is a family of subgroups of G (resp. of G0) and f*
 * : G ! G0
is a group homomorphism such that for every H 2 F, f(H) is a member of F0. Then*
 * f
gives rise to a unique equivariant map from EF G to EF0G0, and hence a unique b*
 *ased map
from BF G to BF0G0. Now suppose that F is a class of groups that is closed unde*
 *r 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 BF G becomes a *
 *functor
on the category of groups. Similarly, if F is only closed under taking subgroup*
 *s, BF G is
a functor on the category of groups and injective homomorphisms.
    We close this section by calculating the fundamental group of of BF G for a*
 *ny G, and
giving a description of BF G in certain cases when G is a graph of groups.

Proposition 3. For any group G and family F, the fundamental group of BF G is i*
 *so-
morphic to the quotient G=G0, where G0 is the subgroup generated by all subgrou*
 *ps of G
that lie in F.

Proof.  Let  be a G-set such that H is non-empty if and only if H 2 F, and let *
 *E  be
Segal's model for EF G. All cell stabilisers in E  are contained in G0, and so *
 *G=G0 acts
freely on E  =G0. Hence it will suffice to show that E  =G0 is 1-connected.


                                        3

    Let ffi0 be a vertex of E  , and use the image of ffi0 as a basepoint for E*
 *  =G0.  Any
based cellular loop in E  =G0 lifts to a path of the form


                     (ffi0; ffi1); (ffi1; ffi2); : :;:(ffin-2; ffin-1); (ffin-1*
 *; ffin);


where each ffii 2  and for some g 2 G0, ffin = gffi0.  However, for each i, (ff*
 *ii; ffii+1; ffii+2)
is a 2-cell in E  , and affords a homotopy between the path (ffii; ffii+1); (ff*
 *ii+1; ffii+2) and
the path (ffii; ffii+2).  Hence the path is homotopic, relative to its endpoint*
 *s, to the single
edge (ffi0; gffi0). Consideration of the boundary of the 2-cell (ffi0; gffi0; g*
 *hffi0) shows that the
function g 7! (ffi0; gffi0) induces a surjective homomorphism from G0 to ss1(E *
 * =G0). Now if
g stabilises ffi 2 , the 2-cell (ffi0; ffi; gffi0) has image in E  =G equal to *
 *a disc with boundary
(the image of) (ffi0; gffi0).  Hence a set of generators for G0 is mapped to 1 *
 *2 ss1(E  =G0),
and so E  =G0 is 1-connected, as claimed.     |

Proposition 4. Let F be any class of groups that is subgroup closed and contain*
 *s only
finite groups, and let G = H1 *K H2 be a group expressible as a free product wi*
 *th amal-
gamation. If X1, X2 and Y are any models for BF H1, BF H2 and BF K respectively*
 *, then
there is a model for BF G consisting of

                                 `       `
                              X1   Y x I    X2= ~;


Where `~' indicates that Y x {0} is identified with the image of the map Y ! X1*
 *, and
similarly Y x {1} is identified with the image of the map Y ! X2.

    As remarked in the introduction, we refer the reader to the work of R. J. P*
 *latten for
a full proof [15,16], but provide the following:
Sketch Proof.  We prove the analogous result for models for EF G. Let E1, E2 an*
 *d F be
models for EF H1, EF H2 and EF K respectively, such that Xi = Ei=Hi and Y = F=K.
Further, let fi: F ! Ei be the K-equivariant map induced by the inclusion of K *
 *in Hi.
    The K-equivariant maps fi give rise to G-equivariant maps from GxK F to GxH*
 *i Ei.
Using these maps as attaching maps, build a G-CW-complex E with stabilisers in *
 *F:

                               `               `
                  E = G xH1 E1   (G xK F ) x I    G xH2 E2 =~ :


Let T be the G-tree corresponding to the free product decomposition for G, with*
 * two orbits
of vertices of orbit types G=Hi 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 Ei 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 StabG(t), *
 *the stabiliser
of t, and is a conjugate of either H1 or H2. Moreover, O(t) is a model for EF S*
 *tabG(t). If
t and t0are not adjacent in T , O(t) \ O(t0) is empty, and if t, t0are the vert*
 *ices of an edge
e, O(t) \ O(t0) is a model for EF StabG(e).
    Now for any L 2 F, the open cover of the fixed-point set EL  given by the (*
 *non-
empty) O(t) \ EL is a cover by contractible sets, whose intersections are eithe*
 *r 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 follo*
 *ws that EL is
contractible as required.     |


                                        4

    A similar statement may be proved for any graph of groups using the same te*
 *chnique.
The reader may prefer the following version of the statement: for a graph of gr*
 *oups with
fundamental group G, let C be the category whose objects are the vertex and edg*
 *e groups of
the graph, with morphisms the inclusions of the edge groups in the vertex group*
 *s. For F a
class of finite groups as above, any choice of a model XH for BF H for each H 2*
 * C, together
with maps between them, gives rise to a functor from C to the category of CW-co*
 *mplexes,
and the complex
                                 hocolimH2CBF H

is a model for BF G.  The only cases that we shall require are those in which t*
 *he graph
of groups is modeled on a star-shaped tree, with one central vertex meeting eve*
 *ry 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 b*
 *e seen
from the following counterexample. Let F be the class of free abelian groups (o*
 *r even the
class of groups that are either infinite cyclic or trivial), and let G be the i*
 *nfinite dihedral
group. Then G is a free product of two copies of C2, the cyclic group of order *
 *two, and by
Proposition 3, ss1(BF G) is cyclic of order two. Hence BF G is not homotopy equ*
 *ivalent to
a wedge of two copies of BF C2 = BC2.
    Proposition 4 should not be viewed as giving a quick proof of Whitehead's t*
 *heorem
in the case when F is the class of trivial groups. The fact that BG can be buil*
 *t 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 B_G is contractible, and as usual say tha*
 *t a group
is acyclic if BG has the same integral homology as a point.  For any finite gro*
 *up G, a
single point will suffice as a model for E_G and hence also for B_G. Thus any f*
 *inite 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-tr*
 *ivial, any
such group contains an infinite cyclic subgroup. We shall also need a contracti*
 *ble 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: ab = a2; bc = b2; cd = c2; da = d2>:


The presentation 2-complex for this presentation is a model for BA, and there i*
 *s a con-
tractible 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 at = c, bt = d.  D*
 *efine a
group C containing A as an index two subgroup as the split extension with kerne*
 *l A and
quotient the cyclic group of order two generated by t. This group C will be sho*
 *wn 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 su*
 *ch a way


                                        5

that E becomes a model for 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, ei and fi *
 *for 1  i  4 be
orbit representatives, and write a1; : :;:a4 instead of a; : :;:d for the gener*
 *ating set for A.
With this notation, the boundaries of the cells of E may be defined by


     ffi(ei) = aiv - v;     ffi(fi) = (1 + ai- a2ia-1i+1a-1i)ei+ (a2ia-1i+1a-1i*
 *- a2ia-1i+1)ei+1;


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


                 t . gv = gtv;     t . gei= gtei+2;     t . gfi= gtfi+2;


where g 2 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 pro*
 *perly on
E. To verify that E is a model for E_C, it suffices to show that the fixed poin*
 *t 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 orde*
 *r two on a
finite-dimensional contractible complex must be mod-2-acyclic [1]. Hence any no*
 *n-trivial
finite subgroup of C fixes a single point in E, and E is a model for E_C.
    The model for B_C consisting of E=C has one 0-cell, two 1-cells and two 2-c*
 *ells, and
is the presentation complex for the presentation


                              <a; b: ab = a2; ba = b2>:


This is a presentation for the trivial group, and hence the 2-complex E=C is co*
 *ntractible
as claimed.     |

    We mention in passing that the above argument also shows that the involutio*
 *n t fixes
no non-identity element of A.
    The above construction can easily be generalized to construct a contractibl*
 *e group
containing a non-trivial torsion-free acyclic subgroup of index p, for any prim*
 *e p. Let A(n)
be Higman's group given by the n-generator, n-relator presentation


                          A(n) = <a1; : :;:an: aai+1i= a2i>;


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 pr*
 *esentation
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 po*
 *wer of
the cyclic automorphism. It may be shown that for any p dividing n, B_G(n; p) i*
 *s homotopy
equivalent to BA(n=p). In particular, G(2p; p) is a contractible group containi*
 *ng a torsion-
free 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 anal*
 *ogue of
theorem 6.1 of [3].


                                        6

Proposition 6. Suppose that G is a subgroup of a contractible group H.  There i*
 *s a
contractible group P containing H *G H as a subgroup. If H has an index two tor*
 *sion-free
subgroup, P may be chosen to have an index two torsion-free subgroup, whose int*
 *ersection
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 con*
 *structed
in Proposition 5.  Define P to be H *G (G x A) *A C.  Then P contains H *G (G x*
 * A)
as a subgroup, and as in [3], H *G H is a subgroup of this group, so is a subgr*
 *oup of P .
Let OE : H ! C2 be a homomorphism with torsion-free kernel, and let OE0: C ! C2*
 * be the
homomorphism with kernel A. Define a homomorphism OE00: GxA ! C2 as the composi*
 *te
of the projection onto G with OE|G . These three homomorphisms from the vertex *
 *groups
of P to C2 agree on the edge groups, and so define a homomorphism from P to C2 *
 *with
torsion-free kernel.
    It remains to show that P is contractible. It is easy to show that for any *
 *groups K1
and K2, B_(K1 x K2) = B_K1 x B_K2.  Since A is torsion-free, B_A = BA and so B_*
 *A is
acyclic. It follows that the inclusion of G in G x A induces a homology isomorp*
 *hism from
B_G to B_(GxA). The fundamental group ss1(B_(GxA)) is isomorphic to ss1(B_G)xA.*
 * From
Proposition 4, it follows that B_(G x A) *A C is homotopy equivalent to B_(G x *
 *A) with a
cone on the acyclic subspace B_A attached. The map f : B_G ! B_(G x A) *A C ind*
 *uces
an isomorphism of fundamental groups, which are isomorphic to G=G0, the quotien*
 *t of G
by its subgroup generated by torsion.  The universal covers are respectively E_*
 *G=G0 and
E_G=G0x BA with a free G=G0-orbit of cones on BA attached to a subspace G=G0x B*
 *A,
and so the map between universal covers induces an isomorphism of homology grou*
 *ps. It
follows that f is a homotopy equivalence. By Proposition 4, it follows that B_P*
 * is homotopy
equivalent to the mapping cylinder of the map from B_G to B_H, and so is contra*
 *ctible since
B_H is.     |

    The direct analogue of the proof given in [3] would be to take P = H *G (G *
 *x C), for
some non-trivial contractible group C. For example, we could take C to be cycli*
 *c of order
two. This P is a contractible group containing H *G H, but since any contractib*
 *le group
contains torsion, this P would contain larger torsion subgroups than H did. Thi*
 *s simpler
construction suffices to prove a weaker version of Theorem 10, in which the min*
 *imal index
of a torsion-free subgroup of GX 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 additiona*
 *l assumption
that X be connected. As remarked in the introduction, this is a corollary of Th*
 *eorem 10,
since we may take X^= BKX = B_KX with the action of GX =KX ~= C2.     |


Proof of Theorem 10.  By the simplicial approximation theorem, we may assume th*
 *at X
is a simplicial complex. Pick a basepoint x0 for X, pick a maximal tree T in X,*
 * and well-
order the vertices of X in such a way that the shortest path in T from any vert*
 *ex to x0 is
a descending sequence. (In particular, x0 should be the first element of the or*
 *dering.) For
any subcomplex Y of X, this gives rise to a way to base Y , without changing it*
 *s homotopy
type, by adding to Y the shortest path in T from the least vertex of Y to x0.


                                        7

    We shall define a pair of functors from the category of based connected sub*
 *complexes
of X and inclusions to the category of groups and monomorphisms, Y 7! (HY ; GY *
 *), with
GY  HY , a model for B_HY containing a model for B_GY as a subcomplex, and a map
of pairs from (B_HY ; B_GY ) to (CY; Y ) that is a homotopy equivalence. (Here *
 *CY denotes
the cone on Y .)  We shall also define a natural (for inclusions of subcomplexe*
 *s) map
OE : HY ! C2, 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 GY = ss1(Y ), and take Y as the c*
 *hosen model
for B_GY . The definition of HY is slightly more complicated. Fix an element g *
 *of infinite
order in an infinite dihedral group D. Let HY be a free product of copies of D *
 *indexed by
the edges of Y . By Proposition 4, HY is a contractible group. Define a homomor*
 *phism
from GY to HY as follows. An element of GY can be represented by a based loop i*
 *n Y ,
denoted effl11.e.f.flnn, where ei is an edge of Y , and ffli = 1 depending whet*
 *her the edge is
traversed with or against its orientation. Send the corresponding element of GY*
 * to the word
gffl11.g.f.flnn, where gi denotes the copy of g in the copy of D corresponding *
 *to the edge ei.
This defines a homomorphism from GY to HY having the required naturality proper*
 *ties.
By applying Proposition 4 to the free product decomposition HY = GY *GY HY , ob*
 *tain
a model for B_H that is the mapping cylinder of a morphism from Y  to a contrac*
 *tible
complex. The morphism sending this contractible complex to a point induces the *
 *required
map of pairs (B_HY ; B_GY ) ! (CY; Y ).  There is a unique homomorphism OE : HY*
 * ! C2
with torsion-free kernel, which takes every element of order two to the non-ide*
 *ntity element
of C2.

    Now suppose that the construction has been completed for every based subcom*
 *plex 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 oe of Y , let T (oe) denote the based complex con*
 *sisting of the
boundary of oe and the shortest path in T from the least vertex of the boundary*
 * of oe to
x0.

    The group GY is constructed as a star-shaped graph of groups, with central *
 *vertex
group GZ , and for each oe, an edge group GT(oe)joining the central vertex to a*
 * vertex with
group HT(oe). By Proposition 5, there is a contractible group PT(oe)containing *
 *HT(oe)*GT(oe)
HT(oe)as a subgroup. Fix such a group PT(oe)depending only on oe 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 HY to be a star-shaped graph of groups, with central v*
 *ertex
group HZ , and for each oe an edge group HT(oe)joining the central vertex to a *
 *vertex with
group PT(oe). This graph of groups contains GY as a subgroup, where the inclusi*
 *on is given
by the `graph of inclusions'.  The homomorphisms OE : HZ ! C2, OE : HT(oe)! C2 *
 *and
OE : PT(oe)! C2 already constructed are compatible, and so give rise to a homom*
 *orphism
OE : HY ! C2 with torsion-free kernel.

    Build a model for B_GY  from the models for the edge and vertex groups that*
 * have
already been constructed, using Proposition 4. Make a map from this model for B*
 *_GY to Y
as follows. Take the map already constructed on B_GZ , for each n-simplex oe ma*
 *p B_HT(oe)
to the centroid of oe, and define the map on B_GT(oe)xI using a homotopy (withi*
 *n T (oe)[oe)
between the identity map of T (oe) and the constant map to the centroid of oe.


                                        8

    To build a model for B_HY , work in stages. Let JY be a star-shaped graph o*
 *f groups
with centre vertex group HZ , and edge groups GT(oe)joining the central vertex *
 *to other
vertices with groups HT(oe). Now HY may be constructed as a star-shaped graph o*
 *f groups
with centre vertex group JY , edge groups HT(oe)*GT(oe)HT(oe)and outer vertex g*
 *roups
PT(oe). Applying Proposition 4 to the graph of groups decomposition for JY give*
 *s a model
for B_JY containing the given model for B_GY . A map from this model to CZ [ Y *
 *may be
constructed as above: take the given map from B_HZ to CZ, map the whole of B_HT*
 *(oe)to
the centroid of oe, and use a homotopy from the identity map on T (oe) to the c*
 *onstant map
to the centroid of oe to define the map on B_GT(oe)x I.
    Now apply Proposition 4 again to obtain a model for B_HY containing the abo*
 *ve model
for B_JY . Make a map from this model to CY as in the two previous cases. Take *
 *the given
map from B_JY to CZ [ Y , and for each oe map B_PT(oe)to the centroid of the co*
 *ne on oe.
Use a homotopy from the identity map on CT (oe) [ oe to the constant map to the*
 * centroid
of Coe to define the map on B_(HT(oe)*GT(oe)HT(oe)) x 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 BKX .     |

    We use the prime 2 to ease the notation, but by varying the construction sl*
 *ightly we
could obtain GX  having a torsion-free subgroup of index p, for any prime p. Mo*
 *st of the
changes that need to be made are trivial. For example, every occurrence of C2 (*
 *resp. D) in
the above proof should be replaced by Cp (resp. Cp* Cp). The p-analogue of Prop*
 *osition 6
works as before, given the existence of a contractible group having a torsion-f*
 *ree acyclic
group of index p. Such a group can be constructed as in the remarks below the p*
 *roof of
Proposition 5.
    If every occurrence of the group C2 and the group D in the proof is replace*
 *d by a
copy of Higman's acyclic group [10] (or any other acyclic group containing an e*
 *lement of
infinite order), the word `contractible' is replaced by the word `acyclic' and *
 *every occurrence
of B_(-) is replaced by B(-), the proof becomes a proof of the Kan-Thurston the*
 *orem,
modelled on Maunder's version [12].  We have felt obliged to be more pedantic a*
 *bout
basepoints than Maunder, however, for the following reason.  To determine wheth*
 *er a
space X is a BG for some G, it suffices to show that X is aspherical. On the ot*
 *her hand,
before proving Theorem 10, one has no way of determining whether a space is a B*
 *_G. One
way to avoid these worries would be to work throughout with groupoids instead o*
 *f groups.

                    5. Motivating examples and questions.

One corollary of Theorem 10 is the group-theoretic statement given below.  The *
 *direct
proof of Corollary 7 was found before the proof of Theorem 10, and provided mot*
 *ivating
evidence.

Corollary 7. Let G be any group. There is a group "Gand a surjection OE : "G! G*
 * such
that the kernel of OE is the subgroup of "Ggenerated by all torsion elements.

Proof. Let X be any connected complex with fundamental group G, and let "G= GX *
 *. The
result follows from Proposition 3.     |

Direct proof.   Let   : F ! G be a surjection from a free group to G, with kern*
 *el N.
Thus N is a free group. Let H be a free product of copies of C2 that contains a*
 * subgroup


                                        9

isomorphic to N. (Provided that F is countable, H = C2 * C2 * C2 will suffice.)*
 * Now take
G"= F *N H, with OE induced by  .     |

    Our construction of the groups GX  is not as explicit as McDuff's construct*
 *ion of
monoids in [13].  One class of spaces X for which we were able to find a more e*
 *xplicit
construction is the class of suspensions.  The examples are based on right-angl*
 *ed Cox-
eter 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 element*
 *s of
order two (the `Coxeter generators'), subject only to the relations that certai*
 *n pairs of the
generators commute.  For more information concerning these groups, see [5,7].  *
 *A right-
angled Coxeter group is either an elementary abelian 2-group, or is a free prod*
 *uct with
amalgamation of right-angled Coxeter groups in which each factor has fewer gene*
 *rators.
It follows from Proposition 4 that B_G is contractible for every right-angled C*
 *oxeter group
G. There is another way to see this fact, using a simplicial complex that was i*
 *ntroduced
by M. Davis, which turns out to be a model for E_G [5].
    Let G be a right-angled Coxeter group. A Coxeter subgroup of G is by defini*
 *tion a
subgroup generated by a subset of the Coxeter generators for G. Davis' complex *
 * = (G)
is the realisation of the poset whose elements are the cosets of the finite Cox*
 *eter subgroups
of G. The action of G on the cosets induces an action on  such that all stabili*
 *sers are
finite.  For a proof that  is a model for 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 s*
 *implicial
complex) arises in this way for some G. For any right-angled Coxeter group, G, *
 *there is a
surjection from G to C2 that sends each Coxeter generator to the non-zero eleme*
 *nt of C2.
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 B_H constructed as =H is isomorphic to the suspension*
 * of the
barycentric subdivision of K.

Proof.   It is easily checked that the model for B_G given by =G is isomorphic *
 *to the
realisation of the poset of subsets of the Coxeter generators that generate fin*
 *ite subgroups,
and hence is isomorphic to the cone on K0.
    Since H does not contain any non-trivial Coxeter subgroup of G, the G-orbit*
 *s in the
poset used to construct  consist of single H-orbits, except for the orbit consi*
 *sting 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 K0, as claimed.     |

    It would be interesting to have a proof of Theorem 10 along the lines of th*
 *e original
proof of the Kan-Thurston theorem. The missing ingredient is a functorial embed*
 *ding 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 10 (which would not nec*
 *essarily
give a bound on the index of a torsion-free subgroup).


                                       10

    Mike Davis asked the following question, which our methods are unable to ad*
 *dress.
Suppose that X is a manifold. Is there a choice for GX  so that E_GX  is a mani*
 *fold upon
which GX acts as homeomorphisms, in such a way that B_GX is homeomorphic to X?


                                 6. References.

 [1]C. Allday and V. Puppe, Cohomological Methods in Transformation Groups, Cam-
    bridge Studies in Advanced Mathematics 32, Cambridge Univ. Press (1993).
 [2]P. Baum and A. Connes, Chern character for discrete groups, in A F^ete of T*
 *opology:
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 *ita,
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 [3]G. Baumslag, E. Dyer and A. Heller, The topology of discrete groups, Journa*
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 [4]N. P. Brady, I. J. Leary and B. E. A. Nucinkis, On algebraic and geometric *
 *dimensions
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 [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 Coho*
 *mology
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 *idge
    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, A*
 *mer. J.
    Math 106 (1984) 301-350.
[10]G. Higman, A finitely generated infinite simple group, J. London Math. Soc.*
 * 24 (1951)
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[11]D. M. Kan and W. P. Thurston, Every connected space has the homology of a K*
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[12]C. R. F. Maunder, A short proof of a theorem of Kan and Thurston, Bull. Lon*
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    Math. Soc. 13 (1981), 325-327.
[13]D. McDuff, On the classifying spaces of discrete monoids, Topology 18 (1979*
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[14]J. W. Milnor, Construction of universal bundles I, Ann. of Math. 63 (1956),*
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[15]R. J. Platten, PhD thesis, Queen Mary and Westfield, London (1999).
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Authors' address:

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



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