String topology for spheres (Q1001225)

Let \(M\) be a compact oriented \(d\)-dimensional smooth manifold. Extending the work of Cohen, Jones and Yan, the author computes the Batalin-Vilkovisky (BV for short) algebra structure defined by Chas and Sullivan on the homology of the free loop space on \(M\), \(LM\). He shows that the BV structures on \(H_*(LM;\mathbb F_2)\) and on the Hochschild cohomology \(H\!H^*(H^*(M;\mathbb F_2);H^*(M;\mathbb F_2))\) are not isomorphic for \(M=S^2\), but, as expected, that the underlying Gerstenhaber algebras are isomorphic. The main idea consists in using the double loop space \(\Omega^2S^3\). The action of \(S^1\) by rotation on the sphere extends to an action of \(S^1\) on \(\Omega^2X\) for any space \(X\). Getzler has proved that \(H_*(\Omega^2X)\) equipped with the induced operator \(\Delta\) and the Pontryagin product is a BV algebra. In [Topology Appl. 156, No.~2, 365--374 (2008; Zbl 1191.55004)], \textit{G. Gaudens} and \textit{L. Menichi} give a complete computation of the BV structure of \(H_*(\Omega^2S^3)\). Now since the natural surjection \(r : S^2 \to (S^1\times S^1)/(S^1\times *) \) is compatible with the \(S^1\)-action we have a map of \(S^1\)-spaces \(L\Omega X \to \Omega^2X\). Combined with the adjunction \(X \to \Omega \Sigma X\), this gives an \(S^1\)-equivariant map \(LX \to \Omega^2\Sigma X\). For \(X = S^2\), this map is an important tool in the above computation.