# 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 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∗(Ω2S3;\mathbb{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