String topology and the based loop space

TL;DR

This paper establishes a deep connection between string topology of manifolds and the Hochschild cohomology of their based loop space, revealing a BV algebra structure that unifies algebraic and topological perspectives.

Contribution

It constructs an isomorphism between the BV algebra of string topology and Hochschild cohomology, using homotopy-theoretic and duality techniques.

Findings

Hochschild cohomology of C_*ΩM has a BV algebra structure.

The BV algebra matches that of the loop homology of M.

Compatibility with cup product and Gerstenhaber bracket is demonstrated.

Abstract

For M a closed, connected, oriented manifold, we obtain the Batalin-Vilkovisky (BV) algebra of its string topology through homotopy-theoretic constructions on its based loop space. In particular, we show that the Hochschild cohomology of the chain algebra C_*\Omega M carries a BV algebra structure isomorphic to that of the loop homology $\mathbb{H}_*(LM)$. Furthermore, this BV algebra structure is compatible with the usual cup product and Gerstenhaber bracket on Hochschild cohomology. To produce this isomorphism, we use a derived form of Poincar\'e duality with C_*\Omega M-modules as local coefficient systems, and a related version of Atiyah duality for parametrized spectra connects the algebraic constructions to the Chas-Sullivan loop product.