# String topology for spheres

### Luc Menichi

Université d'Angers, France

## 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 $H_{∗}(LS_{2};F_{2})$ 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 $H_{∗}(Ω_{2}S_{3};F_{2})$ that we compute in the Appendix.

## Cite this article

Luc Menichi, String topology for spheres. Comment. Math. Helv. 84 (2009), no. 1, pp. 135–157

DOI 10.4171/CMH/155