L₂-homology of Coxeter groups and applications
by
Michael Davis
Ohio State University
Coauthors: Boris Okun

Given a closed aspherical manifold (or orbifold) Mⁿ, a well-known conjecture of I. Singer asserts that the L₂-homology of its universal cover vanishes if n is odd and is concentrated in the middle dimension if n is even.

This conjecture implies another well-known conjecture that the Euler characteristic of M^2k satisfies, (-1)^kX(M^2k) >= 0.

I will discuss a proof of Singer's Conjecture in the case where Mⁿ is constructed from a right-angled Coxeter group. Not only does this imply the Euler characteristic conjecture for such Coxeter groups, but also for any nonpositively curved cubical manifold. Our results also have some interesting applications to combinatorics. For example, if a graph without 3-cycles embeds in a surface of genus g, then its second L₂ Betti number must be <= g. New inequalities involving the number of simplices in any triangulation of a sphere as a flag complex also follow.

Date received: May 20, 1998