Mathematics – Group Theory
Scientific paper
2001-02-13
Geom. Topol. 5(2001) 7-74
Mathematics
Group Theory
Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper2.abs.html
Scientific paper
Associated to any finite flag complex L there is a right-angled Coxeter group W_L and a cubical complex \Sigma_L on which W_L acts properly and cocompactly. Its two most salient features are that (1) the link of each vertex of \Sigma_L is L and (2) \Sigma_L is contractible. It follows that if L is a triangulation of S^{n-1}, then \Sigma_L is a contractible n-manifold. We describe a program for proving the Singer Conjecture (on the vanishing of the reduced L^2-homology except in the middle dimension) in the case of \Sigma_L where L is a triangulation of S^{n-1}. The program succeeds when n < 5. This implies the Charney-Davis Conjecture on flag triangulations of S^3. It also implies the following special case of the Hopf-Chern Conjecture: every closed 4-manifold with a nonpositively curved, piecewise Euclidean, cubical structure has nonnegative Euler characteristic. Our methods suggest the following generalization of the Singer Conjecture. Conjecture: If a discrete group G acts properly on a contractible n-manifold, then its L^2-Betti numbers b_i^{(2)} (G)$ vanish for i>n/2.
Davis Michael W.
Okun Boris
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