Publications Mathématiques de l'Institut des Hautes Études Scientifiques

, Volume 99, Issue 1, pp 235-252

First online:

The Hochschild cohomology of a closed manifold

  • Yves FelixAffiliated withDépartement de mathématique, Université Catholique de Louvain Email author 
  • , Jean-Claude ThomasAffiliated withDépartement de mathématique, Université d’Angers Email author 
  • , Micheline Vigué-PoirrierAffiliated withInstitut Galilée, Université de Paris-Nord Email author 

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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.