RATIONAL BV-ALGEBRA IN STRING TOPOLOGY

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 oil 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).