Resonance for loop homology of spheres

Abstract

A Riemannian or Finsler metric on a compact manifold $M$ gives rise to a length function on the free loop space $\Lambda M$, whose critical points are the closed geodesics in the given metric. If $X$ is a homology class on $\Lambda M$, the “minimax” critical level $\mathsf{cr}(X)$ is a critical value. Let $M$ be a sphere of dimension $\gt 2$, and fix a metric $g$ and a coefficient field $G$. We prove that the limit as $\deg(X)$ goes to infinity of $\mathsf{cr}(X)/ \deg(X)$ exists. We call this limit $\overline\alpha = \overline\alpha(M, g,G)$ the global mean frequency of $M$. As a consequence we derive resonance statements for closed geodesics on spheres; in particular either all homology on $\Lambda$ of sufficiently high degreee lies hanging on closed geodesics of mean frequency (length/average index) $\overline\alpha$, or there is a sequence of infinitely many closed geodesics whose mean frequencies converge to $\overline\alpha$. The proof uses the Chas-Sullivan product and results of Goresky-Hingston.