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 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 H2S3;\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