Cap products in string topology

Tamanoi, Hirotaka¹

¹ Department of Mathematics, University of California Santa Cruz, Santa Cruz, CA 95064

Algebraic and Geometric Topology, Tome 9 (2009) no. 2, pp. 1201-1224

Résumé

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∗(M) ⊕ ℍ∗(LM) into a BV algebra. Finally, we describe cap products in terms of the BV algebra structure in the loop homology.

Détail

DOI : 10.2140/agt.2009.9.1201

Keywords: Batalin–Vilkovisky algebra, cap product, intersection product, loop bracket, loop product, string topology, Batalin–Vilkovisky algebra, cap product, intersection product, loop bracket, loop product, string topology