String topology for spheres

2006/09/11 by Luc Menichi, Menichi, Luc, Gerald Gaudens +1
Mathematics · #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #math.AT #math.GT

paper · pdf · doi:10.48550/arxiv.math/0609304

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

arxiv created 2007/11/13 · arxiv updated 2009/12/01

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_*(LS²;\mathbbF₂) 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_*(Ω² S³;\mathbbF₂) that we compute in the Appendix.

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