A metric Kan-Thurston theorem

作者：Leary Ian J^*

来源：Journal of Topology, 2013, 6(1): 251-284.

DOI：10.1112/jtopol/jts035

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 -%26gt; 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/%26lt;tau %26gt; 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. %26lt;br%26gt;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) under bar(G) over tilde, of some other group (G) over tilde. 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. %26lt;br%26gt;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%26apos;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.

• 出版日期2013-3