Rational BV-algebra in string topology

Abstract

Let M be a 1-connected closed manifold of dimension m and LM be the space of free loops on M. M. Chas and D. Sullivan defined a structure of BV-algebra on the singular homology of LM, H*(LM;k). When the ring of coefficients is a field of characteristic zero, we prove that there exists a BV-algebra structure on the Hochschild cohomology H H*(C*(M);C*(M)) which extends the canonical structure of Gerstenhaber algebra. We construct then an isomorphism of BV-algebras between H H*(C*(M);C*(M)) and the shifted homology H*+m(LM;k). We also prove that the Chas-Sullivan product and the BV-operator behave well with a Hodge decomposition of H*(LM). ©SMF.