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 ${\mathbb{H}}_{\ast }(LM)$. 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 ${\mathbb{H}}_{\ast }(L{S}^2;{\mathbb{F}}_2)$ 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.

22 pages. Minor corrections. An appendix by Gerald Gaudens and Luc Menichi has been added. Final version. To appear in Comment. Math. Helv

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