Half dimensional collapse of ends of manifolds of nonpositive curvature

Grigori Avramidi, T. Tam Nguyen Phan

Published 2016-08-07Version 1

We study the topology of ends of noncompact, complete Riemannian $n$-manifolds $M$ with bounded nonpositive sectional curvature and finite volume. It is known that if such a manifold is negatively curved or does not contain arbitrarily small geodesic loops, then it is tame in the sense that it is homeomorphic to the interior of a compact manifold $\overline{M}$ with boundary $\partial\overline{M}$ since the thin part $M_{<\epsilon}$, i.e. the end of $M$, is topologically a product of a closed manifold with a ray. Let $\widetilde{M}_{<\epsilon}$ be a lift of $M_{<\epsilon}$ in the universal cover $\widetilde{M}$. We show that in this case, any finite polyhedron $P$ in $\widetilde{M}_{<\epsilon}$ can be homotoped within $\widetilde{M}_{<\epsilon}$ to factor through a polyhedron $Q$ of dimension less than $ \lfloor n/2\rfloor$. A corollary of this is that the homology of $\widetilde{M}_{<\epsilon}$ vanishes in dimension greater or equal to $ \lfloor n/2\rfloor$. Another corollary is that when $M$ has dimension less than $6$ each component of the boundary $\partial \overline{M}$ is aspherical. A third corollary is that any complex homotopy equivalent to $M$ has dimension greater or equal to $\lceil n/2\rceil$. These bounds are sharp by examples such as products of noncompact hyperbolic surfaces.