The Hochschild cohomology of a closed manifold
Publications Mathématiques de l'IHÉS, Volume 99 (2004), p. 235-252
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, HH * (𝒞 * (M);𝒞 * (M)) is a graded commutative and associative algebra. The augmentation map ε:𝒞 * (M)𝑘 induces a morphism of algebras I:HH * (𝒞 * (M);𝒞 * (M))HH * (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 HH * (𝒞 * (M);𝑘), which is in general quite small. The algebra HH * (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.
@article{PMIHES_2004__99__235_0,
     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},
     publisher = {Springer},
     volume = {99},
     year = {2004},
     pages = {235-252},
     doi = {10.1007/s10240-004-0021-y},
     zbl = {1060.57019},
     mrnumber = {2075886},
     language = {en},
     url = {http://www.numdam.org/item/PMIHES_2004__99__235_0}
}
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://www.numdam.org/item/PMIHES_2004__99__235_0/

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