Batalin-Vilkovisky coalgebra of string topology

TL;DR

This paper establishes that the reduced Hochschild and cyclic homologies of certain algebraic structures naturally form Batalin-Vilkovisky and gravity coalgebras, providing algebraic models for string topology operations.

Contribution

It introduces a novel algebraic framework linking Hochschild and cyclic homologies to Batalin-Vilkovisky and gravity coalgebras in string topology.

Findings

Reduced Hochschild homology has a Batalin-Vilkovisky coalgebra structure.

Reduced cyclic homology has a gravity coalgebra structure.

Provides algebraic models for string topology operations.

Abstract

We show that the reduced Hochschild homology of a DG open Frobenius algebra has the natural structure of a Batalin-Vilkovisky coalgebra, and the reduced cyclic homology has the natural structure of a gravity coalgebra. This gives an algebraic model for a Batalin-Vilkovisky coalgebra structure on the reduced homology of the free loop space of a simply connected closed oriented manifold, and a gravity coalgebra structure on the reduced equivariant homology.