#### Volume 20, issue 5 (2016)

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Using simplicial volume to count maximally broken Morse trajectories

### Hannah Alpert

Geometry & Topology 20 (2016) 2997–3018
##### 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.

##### Keywords
simplicial volume, Gromov norm, hyperbolic volume, Morse–Smale vector field, Morse broken trajectories
##### Mathematical Subject Classification 2000
Primary: 53C23
Secondary: 58E05, 57N80
##### Publication
Received: 17 June 2015
Revised: 12 November 2015
Published: 7 October 2016
Proposed: Yasha Eliashberg
Seconded: Peter Teichner, John Lott
##### Authors
 Hannah Alpert Department of Mathematics, Massachusetts Institute of Technology Cambridge, MA 02139 USA