Volume 9, issue 2 (2009)

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The $\ell^2$–homology of even Coxeter groups

Timothy A Schroeder

Algebraic & Geometric Topology 9 (2009) 1089–1104
Abstract

Given a Coxeter system (W,S), there is an associated CW–complex, denoted Σ(W,S) (or simply Σ), on which W acts properly and cocompactly. This is the Davis complex. The nerve L of (W,S) is a finite simplicial complex. When L is a triangulation of S3, Σ is a contractible 4–manifold. We prove that when (W,S) is an even Coxeter system and L is a flag triangulation of S3, then the reduced 2–homology of Σ vanishes in all but the middle dimension.

Keywords
Coxeter group, $\ell ^2$-homology, Singer Conjecture, Davis complex, aspherical manifold
Mathematical Subject Classification 2000
Primary: 20F55
Secondary: 57S30, 20J05, 57T15, 58H10
References
Publication
Received: 28 August 2008
Revised: 22 April 2009
Accepted: 22 April 2009
Published: 26 May 2009
Authors
Timothy A Schroeder
Department of Mathematics and Statistics
Murray State University
Murray, KY 42071
USA