The Hochschild cohomology of a closed manifold



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.


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© Institut des Hautes Études Scientifiques and Springer-Verlag 2004

Authors and Affiliations

  1. 1.Département de mathématiqueUniversité Catholique de LouvainLouvain la NeuveBelgium
  2. 2.Département de mathématiqueUniversité d’AngersAngersFrance
  3. 3.Institut GaliléeUniversité de Paris-NordVilletaneuseFrance

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