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Topology of ends of nonpositively curved manifolds

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NPCW05 - Group actions and cohomology in non-positive curvature

Co-author: Tam Nguyen Phan (Binghamton University)

The structure of ends of a finite volume, nonpositively curved, locally symmetric manifold M is very well understood. By Borel-Serre, the thin part of the universal cover of such a manifold is homotopy equivalent to a rational Tits building. This is a simplicial complex built out of the algebra of the locally symmetric space which turns out to have dimension <M/2. In this talk, I will explain aspects of the locally symmetric situation that are true for more general finite volume nonpositively curved manifolds satisfying a mild tameness assumption (there are no arbitrarily small closed geodesic loops). The main result is that the homology of the thin part of the universal cover vanishes in dimension greater or equal to dim M/2. One application is that any complex X homotopy eqiuvalent to M has dimension >= dim M/2. Another application is that the group cohomology with group ring coefficients of the fundamental group of M vanishes in low dimensions (<dim M/2). 

This talk is part of the Isaac Newton Institute Seminar Series series.

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