Little discs operads, graph complexes and Grothendieck-Teichmüller groups

This chapter aims to survey applications of the little discs operads which were motivated by the works of Kontsevich and Tamarkin on the deformation-quantization of Poisson manifolds and by the Goodwillie–Weiss embedding calculus in topology. Kontsevich used an explicit definition of such a comparison map in his first proof of the existence of deformation-quantizations. The theory of E₂-operads actually occurs in a second generation of proofs of this theorem. The existence of associators can be used to get insights into the structure of the rational Grothendieck–Teichmuller group. The chapter discusses connection reflects a finer identity between the Grothendieck—Teichmuller group and the group of homotopy automorphisms of E₂-operads. The Grothendieck–Teichmuller group is defined as a group of automorphisms of the parenthesized braid operad. The hairy graph complex HGCmn explicitly consists of formal series of connected graphs with internal vertices.