The Hochschild cohomology of a closed manifold
Publications Mathématiques de l'IHÉS, Volume 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, $H\phantom{\rule{-0.166667em}{0ex}}{H}^{*}\left({𝒞}^{*}\left(M\right);{𝒞}^{*}\left(M\right)\right)$ is a graded commutative and associative algebra. The augmentation map $\epsilon :{𝒞}^{*}\left(M\right)\to 𝑘$ induces a morphism of algebras $I:H\phantom{\rule{-0.166667em}{0ex}}{H}^{*}\left({𝒞}^{*}\left(M\right);{𝒞}^{*}\left(M\right)\right)\to H\phantom{\rule{-0.166667em}{0ex}}{H}^{*}\left({𝒞}^{*}\left(M\right);𝑘\right)$. 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 $H\phantom{\rule{-0.166667em}{0ex}}{H}^{*}\left({𝒞}^{*}\left(M\right);𝑘\right)$, which is in general quite small. The algebra $H\phantom{\rule{-0.166667em}{0ex}}{H}^{*}\left({𝒞}^{*}\left(M\right);{𝒞}^{*}\left(M\right)\right)$ 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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author = {Felix, Yves and Thomas, Jean-Claude and Vigu\'e-Poirrier, Micheline},
title = {The {Hochschild} cohomology of a closed manifold},
journal = {Publications Math\'ematiques de l'IH\'ES},
pages = {235--252},
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year = {2004},
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Felix, Yves; Thomas, Jean-Claude; Vigué-Poirrier, Micheline. The Hochschild cohomology of a closed manifold. Publications Mathématiques de l'IHÉS, Volume 99 (2004), pp. 235-252. doi : 10.1007/s10240-004-0021-y. http://archive.numdam.org/articles/10.1007/s10240-004-0021-y/

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