A Morse theoretic description of string topology

TL;DR

This paper develops a Morse theoretic framework for understanding string topology operations on a closed manifold, connecting gradient flows on loop spaces with topological algebraic structures.

Contribution

It introduces a Morse theoretic approach to string topology, extending previous algebraic descriptions to a geometric, flow-based perspective.

Findings

Morse theoretic description of string topology operations

Connections between gradient flows and topological structures

Descriptions of Thom and Euler classes via Morse theory

Abstract

Let M be a closed, oriented, n-dimensional manifold. In this paper we give a Morse theoretic description of the string topology operations introduced by Chas and Sullivan, and extended by the first author, Jones, Godin, and others. We do this by studying maps from surfaces with cylindrical ends to M, such that on the cylinders, they satisfy the gradient flow equation of a Morse function on the loop space, LM. We then give Morse theoretic descriptions of related constructions, such as the Thom and Euler classes of a vector bundle, as well as the shriek, or unkehr homomorphism.