Volume 21, issue 2 (2021)

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Spherical complexities with applications to closed geodesics

Stephan Mescher

Algebraic & Geometric Topology 21 (2021) 1021–1074
Abstract

We construct and discuss new numerical homotopy invariants of topological spaces that are suitable for the study of functions on loop and sphere spaces. These invariants resemble the Lusternik–Schnirelmann category and provide lower bounds for the numbers of critical orbits of SO(n)–invariant functions on spaces of n–spheres in a manifold. Lower bounds on these invariants are derived using weights of cohomology classes. As an application, we prove new existence results for closed geodesics on Finsler manifolds of positive flag curvature satisfying a pinching condition.

Keywords
sectional category, Lusternik–Schnirelmann theory, closed geodesics, topological complexity
Mathematical Subject Classification 2010
Primary: 55S40, 58E05
Secondary: 58E10
References
Publication
Received: 8 December 2019
Revised: 28 April 2020
Accepted: 15 June 2020
Published: 25 April 2021
Authors
Stephan Mescher
Mathematisches Institut
Universität Leipzig
Leipzig
Germany