Cap Products in String Topology

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.

19 pages. Revised version 2 with added references, improved exposition, and simplified signs

References in corpus (5)

• The differential topology of loop spaces
• A Homotopy Theoretic Proof of the BV Identity in Loop Homology
• Loop coproducts in string topology and triviality of higher genus TQFT operations
• Batalin-Vilkovisky Lie Algebra Structure on the Loop Homology of Complex Stiefel Manifolds
• The loop homology algebra of spheres and projective spaces

Cited by in corpus (9)

• String topology for complex projective spaces
• TQFT string operations in open-closed string topology
• Loop coproducts in string topology and triviality of higher genus TQFT operations
• Derived string topology and the Eilenberg-Moore spectral sequence
• The bv algebra in string topology of classifying spaces
• String Topology, Euler Class and TNCZ free loop fibrations
• A reduction of the string bracket to the loop product
• Loop homology of some global quotient orbifolds
• String topology on Gorenstein spaces