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The Hochschild cohomology of a closed manifold

Yves Felix;  Jean-Claude Thomas;  Micheline Vigué-Poirrier — 2004

Publications Mathématiques de l'IHÉS

Let M be a closed orientable manifold of dimension and 𝒞 * ( M ) be the usual cochain algebra on M with coefficients in a field. The Hochschild cohomology of M, H H * ( 𝒞 * ( M ) ; 𝒞 * ( M ) ) is a graded commutative and associative algebra. The augmentation map ε : 𝒞 * ( M ) → 𝑘 induces a morphism of algebras I : H H * ( 𝒞 * ( M ) ; 𝒞 * ( M ) ) → 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...

Rational string topology

Yves Félix;  Jean-Claude Thomas;  Micheline Vigué-Poirrier — 2007

Journal of the European Mathematical Society

We use the computational power of rational homotopy theory to provide an explicit cochain model for the loop product and the string bracket of a simply connected closed manifold M . We prove that the loop homology of M is isomorphic to the Hochschild cohomology of the cochain algebra C * ( M ) with coefficients in C * ( M ) . Some explicit computations of the loop product and the string bracket are given.

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