## Journal of Differential Geometry

- J. Differential Geom.
- Volume 93, Number 1 (2013), 133-174.

### Resonance for loop homology of spheres

Nancy Hingston and Hans-Bert Rademacher

#### 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.

#### Article information

**Source**

J. Differential Geom., Volume 93, Number 1 (2013), 133-174.

**Dates**

First available in Project Euclid: 2 January 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.jdg/1357141508

**Digital Object Identifier**

doi:10.4310/jdg/1357141508

**Mathematical Reviews number (MathSciNet)**

MR3019513

**Zentralblatt MATH identifier**

1285.53031

#### Citation

Hingston, Nancy; Rademacher, Hans-Bert. Resonance for loop homology of spheres. J. Differential Geom. 93 (2013), no. 1, 133--174. doi:10.4310/jdg/1357141508. https://projecteuclid.org/euclid.jdg/1357141508