The Hochschild cohomology of a closed manifold (Q1880961)

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.