From deformation theory of wheeled props to classification of Kontsevich formality maps

TL;DR

This paper explores the homotopy theory of wheeled props related to Poisson structures, demonstrating the Grothendieck-Teichmueller group's action and analyzing deformation complexes of Kontsevich formality maps with cohomology computations.

Contribution

It introduces a homotopy-theoretic framework for wheeled props controlling Poisson structures and links the deformation complex cohomology to graph complexes, revealing new structural insights.

Findings

Grothendieck-Teichmueller group acts faithfully on the wheeled prop

Computed the full cohomology of the deformation complex of Kontsevich formality maps

Connected deformation complex cohomology to graph complex cohomology

Abstract

We study homotopy theory of the wheeled prop controlling Poisson structures on arbitrary formal graded finite-dimensional manifolds and prove, in particular, that Grothendieck-Teichmueller group acts on that wheeled prop faithfully and homotopy non-trivially. Next we apply this homotopy theory to the study of the deformation complex of an arbitrary Maxim 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 Maxim Kontsevich in [K1] and studied by Thomas Willwacher in [W1].