Symplectic $A_\infty$-algebras and string topology operations

TL;DR

This paper develops a homotopy-invariant framework for string topology operations on free loop space homology using obstruction theory and rational homotopy theory, extending previous algebraic structures.

Contribution

It introduces a new approach to constructing string topology operations via symplectic $A_inf$-algebras, ensuring homotopy invariance.

Findings

Existence of string topology structures on homology of free loop spaces.

Homotopy invariance of the constructed operations.

Application of obstruction theory and rational homotopy theory to these structures.

Abstract

In this paper we establish the existence of certain structures on the ordinary and equivariant homology of the free loop space on a manifold or, more generally, a formal Poincar\'e duality space. These structures; namely the loop product, the loop bracket and the string bracket, were introduced and studied by Chas and Sullivan under the general heading `string topology'. Our method is based on obstruction theory for $C_\infty$-algebras and rational homotopy theory. The resulting string topology operations are manifestly homotopy invariant.