Using simplicial volume to count maximally broken Morse trajectories

TL;DR

This paper establishes a lower bound on the number of maximally broken Morse trajectories in hyperbolic manifolds, linking Morse theory with simplicial volume and hyperbolic geometry.

Contribution

It introduces a novel connection between Morse trajectory counts and simplicial volume in hyperbolic manifolds, combining Morse theory with Gromov's lemmas.

Findings

Number of n-part broken trajectories ≥ hyperbolic volume

Uses Gromov's lemmas on simplicial volume of stratified spaces

Bridges Morse theory with hyperbolic geometry

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.