String topology for spheres

Details String topology for spheres String topology for spheres

2006-09-11

arxiv.org/abs/math/0609304v2

Mathematics

Algebraic Topology

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

Scientific paper

Let $M$ be a compact oriented $d$-dimensional smooth manifold. Chas and Sullivan have defined a structure of Batalin-Vilkovisky algebra on $\mathbb{H}_*(LM)$. Extending work of Cohen, Jones and Yan, we compute this Batalin-Vilkovisky algebra structure when $M$ is a sphere $S^d$, $d\geq 1$. In particular, we show that $\mathbb{H}_*(LS^2;\mathbb{F}_2)$ 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 $H_*(\Omega^2 S^3;\mathbb{F}_2)$ that we compute in the Appendix.

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