Cap products in string topology

Abstract

Chas and Sullivan showed that the homology of the free loop space $LM$ of an oriented closed smooth manifold $M$ admits the structure of a Batalin–Vilkovisky (BV) algebra equipped with an associative product (loop product) and a Lie bracket (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 the loop product as well as on the loop bracket. We show that Poisson identities and Jacobi identities hold for the cap product action, turning $H^{\ast }\left(M\right)\oplus {ℍ}_{\ast }\left(LM\right)$ into a BV algebra. Finally, we describe cap products in terms of the BV algebra structure in the loop homology.