Spherical complexities, with applications to closed geodesics
Geometric Topology
2021-05-12 v3 Algebraic Topology
Differential Geometry
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.
Cite
@article{arxiv.1911.03948,
title = {Spherical complexities, with applications to closed geodesics},
author = {Stephan Mescher},
journal= {arXiv preprint arXiv:1911.03948},
year = {2021}
}
Comments
35 pages, revised version, fixed typos and a mistake in Theorem 6.5 pointed out by D. Kotschick. To appear in Algebraic & Geometric Topology