The Hochschild cohomology of a closed manifold

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

doi:10.1007/s10240-004-0021-y

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

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

Résumé

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

MR Zbl | 5 citations dans Numdam

DOI : 10.1007/s10240-004-0021-y

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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},
     publisher = {Springer},
     volume = {99},
     year = {2004},
     doi = {10.1007/s10240-004-0021-y},
     mrnumber = {2075886},
     zbl = {1060.57019},
     language = {en},
     url = {https://www.numdam.org/articles/10.1007/s10240-004-0021-y/}
}

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Felix, Yves; Thomas, Jean-Claude; Vigué-Poirrier, Micheline. The Hochschild cohomology of a closed manifold. Publications Mathématiques de l'IHÉS, Tome 99 (2004), pp. 235-252. doi : 10.1007/s10240-004-0021-y. https://www.numdam.org/articles/10.1007/s10240-004-0021-y/

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