String topology for spheres

Abstract

Let M be a compact oriented d-dimensional smooth manifold. Chas and Sullivan have defined a structure of Batalin�Vilkovisky algebra on H*(LM). Extending work of Cohen, Jones and Yan, we compute this Batalin�Vilkovisky algebra structure when M is a sphere Sd, d =1. In particular, we show that H*(LS2;\mathbb{F}2) and the Hochschild cohomology HH*(H*(S2);H*(S2)) are surprisingly not isomorphic as Batalin�Vilkovisky algebras, although we prove that, as expected, the underlying Gerstenhaber algebras are isomorphic. The proof requires the knowledge of the Batalin�Vilkovisky algebra H*(O2S3;\mathbb{F}2) that we compute in the Appendix.