$A_3$-formality for Demushkin groups at odd primes

TL;DR

This paper investigates $A_3$-formality in the cohomology of pro-p Demushkin groups at odd primes, revealing conditions under which these groups exhibit this property through explicit Hochschild cohomology computations.

Contribution

It establishes $A_3$-formality for certain Demushkin groups and identifies the role of the q-invariant in this property, using explicit Hochschild cohomology calculations.

Findings

Demushkin groups with q-invariant not equal to 3 are $A_3$-formal.

Demushkin groups with q-invariant 3 are not $A_3$-formal.

Explicit computation of the Benson-Krause-Schwede canonical class was used.

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.