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 $H_{\ast }(LM)$. Extending work of Cohen, Jones and Yan, we compute this Batalin–Vilkovisky algebra structure when $M$ is a sphere $S^d$, $d\ge 1$. In particular, we show that $H_{\ast }(L{S}^2;{\mathbb{F}}_2)$ and the Hochschild cohomology $H{H}^{\ast }(H^{\ast }(S^2);H^{\ast }(S^2))$ 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 $H_{\ast }({\Omega }^2{S}^3;{\mathbb{F}}_2)$ that we compute in the Appendix.