$A_3$-formality for Demushkin groups at odd primes

TL;DR

This work determines when the differential graded algebra of continuous cochains on pro-$p$ Demushkin groups at odd primes is $A_3$-formal, linking formality to the canonical Hochschild class $\gamma_G$. It combines Koszul algebra techniques, the Benson–Krause–Schwede canonical class, and Dwyer’s Massey-product framework to compute obstructions explicitly. The main results show $A_3$-formality for Demushkin groups with $q\neq 3$ (including $q=0$ or $q\ge 5$) and non-$A_3$-formality when $q=3$, with a detailed computation of the canonical class across even ranks. The findings illuminate how formal properties of group cohomology interact with Koszul structure, and highlight that $A_3$-formality does not always pass through semidirect products, even when underlying factors are formalisms on their own. This has implications for understanding the extent of Massey-vanishing phenomena and for the structure of absolute Galois groups in number theory.

Abstract

We study a weak form of formality for differential graded algebras, called $A_3$-formality, for the cohomology of pro-p Demushkin groups at odd primes p. We show that the differential graded $\mathbb{F}_p$-algebras of continuous cochains of Demushkin groups with q-invariant not equal 3 are $A_3$-formal, whereas Demushkin groups with q-invariant 3 are not $A_3$-formal. We prove these results by an explicit computation of the Benson-Krause-Schwede canonical class in Hochschild cohomology.

Theorems & Definitions (110)

• Definition 1.1
• Theorem 1.2
• Example 1.3
• Example 1.4
• Example 1.5
• Remark 1.8
• Remark 1.9
• Remark 1.10
• Definition 2.1
• Example 2.2