Kontsevich formality and cohomologies for graphs (Q703997)

The paper under review provides a nice discussion of the ideas of graph cohomology involved in the construction of formalities on \({\mathbb R}^d\). We recall that a formality on a smooth manifold \(M\) is a formal nonlinear mapping from the polyvector fields on \(M\) into the space of polydifferential operators on \(M\), satisfying a so-called formality equation. Objects of this kind occur in connection with the deformation quantization of Poisson manifolds; see the paper by \textit{M.~Kontsevich} [Lett. Math. Phys. 66, No. 3, 157--216 (2003; Zbl 1058.53065)]. The results in the present paper are presented without detailed proofs. Some of them were already published in the paper by \textit{D.~Arnal} and \textit{M.~Masmoudi} [Bull. Soc. Math. Fr. 130, No. 1, 49--69 (2002; Zbl 1028.46099)].