Volume 20, issue 5 (2016)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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
References
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