A metric Kan-Thurston theorem

Abstract

For every simplicial complex X, we construct a locally CAT(0) cubical complex TX, a cellular isometric involution τ on TX and a map tX:TX→X with the following properties: tXτ=tX; tX is a homology isomorphism; the induced map from the quotient space TX/〈τ〉 to X is a homotopy equivalence; the induced map from the fixed point space T X τ 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, E ¯ G ˜ of some other group G∼. From this we obtain extensions of a theorem of Quillen on the spectrum of a (Borel) equivariant cohomology ring and 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.