Spherical complexities with applications to closed geodesics

Mescher, Stephan

doi:10.2140/agt.2021.21.1021

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Spherical complexities with applications to closed geodesics

Stephan Mescher

Algebraic & Geometric Topology 21 (2021) 1021–1074

DOI: 10.2140/agt.2021.21.1021

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

Volume 21, issue 2 (2021)