A metric Kan-Thurston theorem

TL;DR

This paper constructs a functorial locally CAT(0) cubical complex for any simplicial complex, leading to new results in equivariant homology, group theory, and the properties of cubical complexes.

Contribution

It introduces a new functorial construction of locally CAT(0) cubical complexes that generalizes the Kan-Thurston theorem and extends several key results in algebraic topology and geometric group theory.

Findings

Constructs a locally CAT(0) cubical complex T_X for any simplicial complex X.

Establishes an equivariant Kan-Thurston theorem relating G-CW-complexes to classifying spaces.

Proves no algorithm can decide if a CAT(0) cubical group is generated by torsion.

Abstract

For every simplicial complex X, we construct a locally CAT(0) cubical complex T_X, a cellular isometric involution i on T_X and a map t_X from T_X to X with the following properties: t_Xi = t_X; t_X is a homology isomorphism; the induced map from the quotient of T_X by the involution i to X is a homotopy equivalence; the induced map from the fixed point subspace for i 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 an extension of Quillen's theorem on the spectrum of an equivariant cohomology ring and an extension 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, 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.