Automorphisms of the little disks operad with torsion coefficients

TL;DR

This paper characterizes the automorphisms of the p-completed little disks operad, revealing their connection to the pro-p Grothendieck-Teichmüller group and demonstrating its faithful action on the p-complete stable operad.

Contribution

It establishes a precise link between automorphisms of the p-completed little disks operad and the pro-p Grothendieck-Teichmüller group, a novel insight in operad symmetry.

Findings

Automorphisms are given by the pro-p Grothendieck-Teichmüller group.

The Grothendieck-Teichmüller group acts faithfully on the p-complete stable little disks operad.

Provides a new understanding of symmetries in p-completed operads.

Abstract

We compute the automorphisms of the Bousfield-Kan completion at a prime p of the little two-disks operads and show that they are given by the pro-p Grothendieck-Teichm\"uller group. We also show that the Grothendieck-Teichm\"uller group acts faithfully on the p-complete stable little disks operad.