A metric Kan-Thurston theorem

Abstract

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 from T(X) to X with the following properties:

t is equivariant for tau; t is a homology isomorphism; the induced map from the quotient space T(X)/tau to X is a homotopy equivalence; the induced map from the tau-fixed point set in T(X) 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 of some other group. From this we obtain extensions of a theorem of Quillen on the spectrum of a (Borel) equivariant cohomology ring and of a result of Block concerning assembly conjectures. Another corollary of our 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, notably in 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.

[A version of this paper was submitted in September 2010. This is a revised version I made in April 2011 (improvements to some material in the appendices).]

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ORCID for Ian J. Leary: orcid.org/0000-0001-8300-4979

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