Ends of finite volume, nonpositively curved manifolds

TL;DR

This paper investigates the topology of finite volume, nonpositively curved manifolds, proving new homological properties of their boundaries and implications for their fundamental groups, extending classical results in geometric topology.

Contribution

It establishes vanishing homology and cohomology results for the boundaries of such manifolds, and shows that their fundamental groups are freely indecomposable, extending previous work to higher dimensions.

Findings

Vanishing (n-2)-homology of boundary covers for n≥3

Boundary components are aspherical when n=4

Fundamental groups are freely indecomposable

Abstract

We study complete, finite volume $n$-manifolds $M$ of bounded nonpositive sectional curvature. A classical theorem of Gromov says that if such $M$ has negative curvature then it is homeomorphic to the interior of a compact manifold-with-boundary, and we denote this boundary $\partial M$. If $n\geq 3$, we prove that the universal cover of the boundary $\widetilde{\partial M}$ and also the $\pi_1M$-cover of the boundary $\partial\widetilde M$ have vanishing $(n-2)$-dimensional homology. For $n=4$ the first of these recovers a result of Nguyen Phan saying that each component of the boundary $\partial M$ is aspherical. For any $n\geq 3$, the second of these implies the vanishing of the first group cohomology group with group ring coefficients $H^1(B\pi_1M;\mathbb Z\pi_1M)=0$. A consequence is that $\pi_1M$ is freely indecomposable. These results extend to manifolds $M$ of bounded nonpositive curvature if we assume that $M$ is homeomorphic to the interior of a compact manifold with boundary. Our approach is a form of "homological collapse" for ends of finite volume manifolds of bounded nonpositive curvature. This paper is very much influenced by earlier, yet still unpublished work of Nguyen Phan.