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 S^d , d ≥1 . In particular, we show that ℍ_∗(LS^2;\mathbb{F}_2) and the Hochschild cohomology HH^∗(H^∗(S^2);H^∗(S^2)) 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_∗(Ω^2S^3;\mathbb{F}_2) that we compute in the Appendix.