Using simplicial volume to count maximally broken Morse trajectories

Abstract

Given a closed Riemannian manifold of dimension $n$ and a Morse–Smale function, there are finitely many $n$–part broken trajectories of the negative gradient flow. We show that if the manifold admits a hyperbolic metric, then the number of $n$–part broken trajectories is always at least the hyperbolic volume. The proof combines known theorems in Morse theory with lemmas of Gromov about simplicial volumes of stratified spaces.