Loop Products and Closed Geodesics

TL;DR

This paper explores the relationship between the Chas-Sullivan product and the Morse index of closed geodesics on Riemannian manifolds, introducing new algebraic structures in loop space cohomology.

Contribution

It establishes a connection between the Chas-Sullivan product and Morse indices, and constructs nontrivial products in loop space cohomology.

Findings

Chas-Sullivan product relates to Morse index of closed geodesics

Constructed nontrivial products in cohomology of loop spaces

Enhanced understanding of algebraic structures in loop space topology

Abstract

We show the Chas-Sullivan product (on the homology of the free loop space of a Riemannian manifold) is related to the Morse index of its closed geodesics. We construct related products in the cohomology of the free loop space and of the based loop space, and show they are nontrivial.