Cap Products in String Topology

TL;DR

This paper explores how the cap product interacts with the BV algebra structure in the homology of free loop spaces, extending the algebraic framework of string topology.

Contribution

It demonstrates the compatibility of the cap product with loop products and brackets, extending the BV algebra structure to include cohomology of the base manifold.

Findings

Cap product acts as a derivation on loop product and bracket.

Poisson and Jacobi identities hold for cap product action.

Cap product can be described within the BV algebra framework.

Abstract

Chas and Sullivan showed that the homology of the free loop space LM of an oriented closed smooth finite dimensional manifold M admits the structure of a Batalin-Vilkovisky (BV) algebra equipped with an associative product called the loop product and a Lie bracket called the loop bracket. We show that the cap product is compatible with the above two products in the loop homology. Namely, the cap product with cohomology classes coming from M via the circle action acts as derivations on loop products as well as on loop brackets. We show that Poisson identities and Jacobi identities hold for the cap product action, extending the BV structure in the loop homology to the one including the cohomology of M. Finally, we describe the cap product in terms of the BV algebra structure in the loop homology.