Using simplicial volume to count maximally broken Morse trajectories

Alpert, Hannah¹

¹ Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 USA

Geometry & topology, Tome 20 (2016) no. 5, pp. 2997-3018.

Résumé

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.

Détail

DOI : 10.2140/gt.2016.20.2997

Keywords: simplicial volume, Gromov norm, hyperbolic volume, Morse–Smale vector field, Morse broken trajectories

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