Publications Mathématiques de l'Institut des Hautes Études Scientifiques

, Volume 99, Issue 1, pp 235–252

The Hochschild cohomology of a closed manifold

Authors

    • Département de mathématiqueUniversité Catholique de Louvain
    • Département de mathématiqueUniversité d’Angers
    • Institut GaliléeUniversité de Paris-Nord
Article

DOI: 10.1007/s10240-004-0021-y

Cite this article as:
Felix, Y., Thomas, J. & Vigué-Poirrier, M. Publ. Math., Inst. Hautes Étud. Sci. (2004) 99: 235. doi:10.1007/s10240-004-0021-y

Abstract

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.

Copyright information

© Institut des Hautes Études Scientifiques and Springer-Verlag 2004