On the formality of the little disks operad in positive characteristic
Pedro Boavida de Brito, Geoffroy Horel
TL;DR
The paper addresses the formality of the little disks operad $E_n$ in positive characteristic by constructing a pro-$p$ (and rational) action of the Grothendieck–Teichmüller group on the completed operad $E_n^{ abla}$ and exploiting purity phenomena to derive formality ranges. It develops a pro-space, pro-operad framework with localization, completion, and a Boardman–Vogt–style tensor product that behaves well under completion, enabling a transfer of automorphisms from $E_2$ to higher $E_n$ via additivity. The main results are: (i) $(n-1)(p-2)$-formality of $C_*(E_n,Fp)$ and absolute formality over $Q$; (ii) relative formality for maps $E_n o E_{n+d}$ with explicit ranges depending on $ ext{gcd}(n-1,d)$; (iii) Hopf-formality in the rational setting, and (iv) consequences for the formality of chains on configuration spaces and connections to embedding calculus. These results extend Kontsevich–Tamarkin–Lambrechts–Volić formality to positive characteristic and illuminate the arithmetic structure controlling operadic formality, with potential applications to topological field theories and embedding spaces.
Abstract
Using a variant of the Boardman-Vogt tensor product, we construct an action of the Grothendieck-Teichmüller group on the completion of the little n-disks operad $E_n$. This action is used to establish a partial formality theorem for $E_n$ with mod $p$ coefficients and to give a new proof of the formality theorem in characteristic zero.