Author: Ian J Leary 

Ian Leary
Affiliation: The Ohio State University 

For every simplicial complex $X$, we construct a locally CAT(0)
cubical complex $T_X$, a cellular isometric involution $\tau$ on 
$T_X$ and a map $t_X:T_X\rightarrow X$ with the following properties: 
$t_X\tau = t_X$; $t_X$ is a homology isomorphism; the induced map from
the quotient space $T_X/\langle\tau\rangle$ to $X$ is a homotopy
equivalence; the induced map from the fixed point space $T_X^\tau$ to
$X$ is a homology isomorphism.  The construction is functorial in
$X$. 

One corollary is an equivariant Kan-Thurston theorem: every connected
proper $G$-CW-complex has the same equivariant homology as the
classifying space for proper actions, $\ebar \tilg$, of some other
group $\tilg$.  Corollaries of this include an extension to Quillen's
theorem describing the spectrum of an equivariant cohomology ring and
a generalization of a result of J. Block.  Another corollary of the
main result is that there can be no algorithm to decide whether a
CAT(0) cubical group is generated by torsion.

In appendices we prove some foundational results concerning cubical
complexes, including the infinite dimensional case.  We characterize
the cubical complexes for which the natural metric is complete; we
establish Gromov's criterion for infinite-dimensional cubical
complexes; we show that every CAT(0) cube complex is cubical; we
deduce that the second cubical subdivision of any locally CAT(0) cube
complex is cubical.


Status: preprint