Profinite completion of operads and the Grothendieck-Teichm\"uller group

TL;DR

This paper establishes a deep connection between the homotopy automorphisms of the profinite completion of the little 2-disks operad and the profinite Grothendieck-Teichmüller group, linking algebraic and topological structures.

Contribution

It proves an isomorphism between the homotopy automorphisms of the profinite little 2-disks operad and the profinite Grothendieck-Teichmüller group, revealing new insights into Galois actions.

Findings

Homotopy automorphisms of the profinite little 2-disks operad are isomorphic to the profinite Grothendieck-Teichmüller group.

The absolute Galois group of $ ext{Q}$ acts faithfully on the profinite completion of $E_2$.

Establishes a link between algebraic Galois groups and topological operad automorphisms.

Abstract

In this paper, we prove that the group of homotopy automorphisms of the profinite completion of the operad of little $2$-disks is isomorphic to the profinite Grothendieck-Teichm\"uller group. In particular, the absolute Galois group of $\mathbb{Q}$ acts faithfully on the profinite completion of $E_2$ in the homotopy category of profinite weak operads.