Using simplicial volume to count maximally broken Morse trajectories

Alpert, Hannah

doi:10.2140/gt.2016.20.2997

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Using simplicial volume to count maximally broken Morse trajectories

Hannah Alpert

Geometry & Topology 20 (2016) 2997–3018

DOI: 10.2140/gt.2016.20.2997

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

Volume 20, issue 5 (2016)