Volume 5, issue 1 (2001)

Download this article
For printing
Recent Issues

Volume 22
Issue 7, 3761–4380
Issue 6, 3145–3760
Issue 5, 2511–3144
Issue 4, 1893–2510
Issue 3, 1267–1891
Issue 2, 645–1266
Issue 1, 1–644

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Ethics Statement
Author Index
To Appear
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Other MSP Journals
Vanishing theorems and conjectures for the $\ell^2$–homology of right-angled Coxeter groups

Michael W Davis and Boris Okun

Geometry & Topology 5 (2001) 7–74

arXiv: math.GR/0102104


Associated to any finite flag complex L there is a right-angled Coxeter group WL and a cubical complex ΣL on which WL acts properly and cocompactly. Its two most salient features are that (1) the link of each vertex of ΣL is L and (2) ΣL is contractible. It follows that if L is a triangulation of Sn1, then ΣL is a contractible n–manifold. We describe a program for proving the Singer Conjecture (on the vanishing of the reduced 2–homology except in the middle dimension) in the case of ΣL where L is a triangulation of Sn1. The program succeeds when n 4. This implies the Charney–Davis Conjecture on flag triangulations of S3. 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 2–Betti numbers bi(2)(G) vanish for i > n2.

Coxeter group, aspherical manifold, nonpositive curvature, $\ell^2$–homology, $\ell^2$–Betti numbers
Mathematical Subject Classification 2000
Primary: 58G12
Secondary: 20F55, 57S30, 20F32, 20J05
Forward citations
Received: 1 September 2000
Revised: 13 December 2000
Accepted: 31 January 2001
Published: 2 February 2001
Proposed: Walter Neumann
Seconded: Steve Ferry, Ralph Cohen
Michael W Davis
Department of Mathematics
The Ohio State University
Ohio 43210
Boris Okun
Department of Mathematics
Vanderbilt University
Tennessee 37240