From Deformation Theory of Wheeled Props to Classification of Kontsevich Formality Maps

Abstract

Abstract We study the homotopy theory of the wheeled prop controlling Poisson structures on formal graded finite-dimensional manifolds and prove, in particular, that the Grothendieck–Teichmüller group acts on that wheeled prop faithfully and homotopy nontrivially. Next, we apply this homotopy theory to the study of the deformation complex of an arbitrary Kontsevich formality map and compute the full cohomology group of that deformation complex in terms of the cohomology of a certain graph complex introduced earlier by Kontsevich [ 3] and studied by Willwacher [ 18].