An Algebraic Chain Model of String Topology

TL;DR

This paper develops an algebraic chain complex model for the free loop space of manifolds, capturing string topology structures such as Gerstenhaber and Batalin-Vilkovisky algebras, including non-simply-connected cases.

Contribution

It introduces a comprehensive algebraic chain model for string topology that extends to non-simply-connected manifolds, linking homology structures with algebraic models.

Findings

Defines Gerstenhaber and Batalin-Vilkovisky algebra structures on homology

Models the gravity algebra on equivariant homology

Includes non simply-connected case in the algebraic framework

Abstract

A chain complex model for the free loop space of a connected, closed and oriented manifold is presented, and on its homology, the Gerstenhaber and Batalin-Vilkovisky algebra structures are defined and identified with the string topology structures. The gravity algebra on the equivariant homology of the free loop space is also modeled. The construction includes non simply-connected case, and therefore gives an algebraic and chain level model of Chas-Sullivan's String Topology.