Real projective groups are formal

TL;DR

The paper proves that for real projective profinite groups $G$, the mod $2$ cohomology algebra $H^*(G,\mathbb{F}_2)$ is intrinsically formal, which implies the formality of the differential graded algebra $C^*(G,\mathbb{F}_2)$. The authors achieve this by exploiting Scheiderer’s real-projective structure to express $H^*(G,\mathbb{F}_2)$ as a connected sum $B_*\sqcap V_*$ of a Boolean and a dual graded algebra, then establishing vanishing results for graded Hochschild cohomology through finite and infinite case analyses using Kadeishvili’s criterion. A crucial tool is a detailed study of Hochschild cohomology for connected sums, together with Mittag-Leffler arguments and transfinite recursion to extend finite-case results to arbitrary subrings. Consequently, they deduce strong Massey vanishing in arbitrary degrees for cohomology and derive formality and Koszulity results for fields with virtual cohomological dimension at most $1$, providing new positive cases for formality questions in Galois cohomology. The work thus links group-theoretic structure, Hochschild cohomology, and arithmetic applications, offering a robust framework for future investigations into formality phenomena in Galois theory and related cohomology theories.

Abstract

We prove that the mod 2-cohomology algebras of real projective groups are formal. As a consequence we derive the Hopkins-Wickelgren formality and the strong Massey vanishing conjecture for a large class of fields.

Theorems & Definitions (141)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Definition 1.7
• Theorem 1.8
• Remark 1.9
• Theorem 1.10