### Every CW-complex is a classifying space for proper bundles

### I. J. Leary and B. 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.

These results may be viewed as analogues of the Kan-Thurston theorem,
in which the universal free *G*-space has been replaced by the
universal proper *G*-space. The fact that our result concerns
homotopy type (whereas the Kan-Thurston theorem concerns homology)
is a reflection of the existence of `contractible groups', i.e.,
groups for which the quotient of the universal proper *G*-space
by *G* is contractible.

Topology 40 (2001) 539-550.

The Mathematical Review of this paper mis-states one of our results as
`every CW-complex is the quotient of an acyclic complex by an action
of a group of order two'. I don't know whether this result is true or
not, but it certainly can't be true that `every finite-dimensional
CW-complex is the quotient of a finite-dimensional acyclic complex by
an action of a group of order two', since any such space would have to
be mod-2 acyclic by Smith theory.