# Commentarii Mathematici Helvetici

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**Volume 84, Issue 1, 2009, pp. 135–157**

**DOI: 10.4171/CMH/155**

Published online: 2009-03-31

String topology for spheres

Luc Menichi^{[1]}(1) Université d'Angers, France

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

^{2}

`S`

^{3};\mathbb{F}

_{2}) that we compute in the Appendix.

*Keywords: *String topology, Batalin–Vilkovisky algebra, Gerstenhaber algebra, Hochschild cohomology, free loop space

Menichi Luc: String topology for spheres. *Comment. Math. Helv.* 84 (2009), 135-157. doi: 10.4171/CMH/155