Profinite completion of operads and the Grothendieck-Teichmüller group (Q1673990)

Let \(E_2\) denote the operad of little \(2\)-disks. The \(n\)-th space of \(E_2\) has the homotopy type of the space of configurations of \(n\) points in \({\mathbb{R}}^2\). This configuration space is known to be equivalent to the classifying space of the pure braid group on \(n\) strands. It follows that there are groupoid models for \(E_2\), one such model being the operad \(PaB\) of parenthesized braids. In the paper under review, the author studies the automorphisms of the profinite completion of \(E_2\). Profinite completion of spaces is a homotopical analogue of profinite completion of groups. A model for profinite completion of spaces, suitable for the purposes of this work, was constructed by \textit{G. Quick} [Doc. Math 13, 585--612 (2008; Zbl 1173.55008)]. The author shows that the group of homotopy automorphisms of the profinite completion of \(E_2\) is isomorphic to the profinite Grothendieck-Teichmüller group \(\widehat{\text{GT}}\). He uses Drinfeld's construction for \(\widehat{\text{GT}}\). That construction is of operadic nature and relies on the operad \(PaB\): Let \(\widehat{PaB}\) be the operad in profinite groupoids obtained by applying profinite completion on each arity of \( PaB\). The group \(\widehat{\text{GT}}\) is the group of automorphisms of \(\widehat{PaB}\) that induce the identity on objects. The action of \(\widehat{\text{GT}}\) on \(\widehat{PaB}\) then induces an isomorphism from \(\widehat{\text{GT}}\) to the group of homotopy automorphisms of \(\widehat{PaB}\). The author has to deal with various technical issues. For example, the profinite completion functor from spaces to profinite spaces does not preserve products. To overcome this problem he works with weak operads, and defines the profinite completion functor as a functor from weak operads in spaces to weak operads in profinite spaces. The results in this paper are analogous to results in the rational case, due to \textit{B. Fresse} [Homotopy of operads and Grothendieck-Teichmüller groups. Part 1: The algebraic theory and its topological background. Providence, RI: American Mathematical Society (AMS) (2017; Zbl 1373.55014), Homotopy of operads and Grothendieck-Teichmüller groups. Part 2: The applications of (rational) homotopy theory methods. Providence, RI: American Mathematical Society (AMS) (2017; Zbl 1375.55007)], and \textit{T. Willwacher} [Invent. Math. 200, No. 3, 671--760 (2015; Zbl 1394.17044)].