The Hochschild cohomology of a closed manifold

Abstract

Let M be a closed orientable manifold of dimension d and $\mathcal{C}^*(M)$ be the usual cochain algebra on M with coefficients in a field k. The Hochschild cohomology of M, $H\!H^*(\mathcal{C}^*(M);\mathcal{C}^*(M))$ is a graded commutative and associative algebra. The augmentation map $\varepsilon: \mathcal{C}^*(M) \to{\textbf{\textit{k}}}$ induces a morphism of algebras $I : H\!H^*(\mathcal{C}^*(M);\mathcal{C}^*(M)) \to{H\!H^*(\mathcal{C}^*(M);{\textbf{\textit{k}}})}$. 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 $H\!H^*(\mathcal{C}^*(M);{\textbf{\textit{k}}})$, which is in general quite small. The algebra $H\!H^*(\mathcal{C}^*(M);\mathcal{C}^*(M))$ 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.