The Hochschild cohomology of a closed manifold. (English) Zbl 1060.57019

Felix, Yves; Thomas, Jean-Claude; Vigué-Poirrier, Micheline

Publ. Math., Inst. Hautes Étud. Sci. 99, 235-252 (2004).

Let \(M\) be a simply connected closed oriented \(d\)-dimensional smooth manifold and let \(\mathcal C^*(M)\) be the cochain algebra of \(M\) with coefficients in the field \(\mathbf k\). The augmentation \(\mathcal C^*(M)\to\mathbf k\) induces in Hochschild cohomology a morphism of graded algebras \[ I: HH^*(\mathcal C^*(M);\mathcal C^*(M))\to HH^*(\mathcal C^*(M);\mathbf k). \] The authors give a chain model for \(I\) from which they prove rather formally that the kernel of \(I\) is a nilpotent ideal of nilpotency index less than or equal to \(d/2\), and that the image of \(I\) is central. When \(k\) is of characteristic zero, they refine this result, and prove that \(I\) is surjective if and only if \(M\) has the rational homotopy type of a product of odd dimensional spheres.
The interest in Hochschild cohomology of the cochain algebra comes mainly from the correspondence with the cohomology of the free loop space.
The paper ends with a discussion on Hochschild cohomology of Poincaré duality spaces.

Reviewer: Bjørn Dundas (Bergen)

MSC:

57R19	Algebraic topology on manifolds and differential topology
16E40	(Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
55N35	Other homology theories in algebraic topology
19D55	\(K\)-theory and homology; cyclic homology and cohomology

Keywords:

Hochschild cohomology; free loop space

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