The Hochschild cohomology of a closed manifold

Let M be a closed orientable manifold of dimension d and l*(M) be the usual cochain algebra on M with coefficients in a field k. The Hochschild cohomology of M, HH* (l* (M); l* (M)) is a graded commutative and associative algebra. The augmentation map E : l* (M) -> k induces a morphism of algebras I : HH* (l* (M); l* (M)) -> HH* (l* (M); k). In this paper we produce a chain model for the morphism I. We show that the kernel of I is a nilpotent ideal and that the image of I is contained in the center of HH*(l*(M); k), which is in general quite small. The algebra HH*(l*(M); l*(M)) is expected to be isomorphic to the loop homology constructed by Chas and Sullivan. Thus our results would be translated in terms of string homology.