The Hochschild cohomology of a closed manifold

Felix, Yves; Thomas, Jean-Claude; Vigué-Poirrier, Micheline

Publications Mathématiques de l'IHÉS, Tome 99 (2004), pp. 235-252.

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

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