On the Bogomolov-Positselski Conjecture
Julian Feuerpfeil
TL;DR
The paper develops new group-theoretic criteria for identifying the Bogomolov-Positselski property within $p$-oriented pro-$p$ groups, bridging Positselski’s Koszulity framework with the Hochschild–Serre approach of Quadrelli–Weigel. It introduces a central exact-sequence criterion tying $ ext{ker}\,d_2^{2,1}$ to the vanishing of a $ ext{Tor}$-group and the isomorphism of a key map, and leverages conilpotent coalgebra techniques to derive a weaker, yet computably verifiable Bogomolov–Positselski criterion under Koszul-like hypotheses. Additionally, the work provides a route to relax Positselski’s conditions and makes the criteria computationally accessible in finite-dimension settings by reducing checks to a handful of Ext/Tor computations. These results connect Galois-cohomology, Milnor $K$-theory, and quadratic Koszulity concepts, offering practical tools for verifying the Bogomolov–Positselski property and advancing understanding of maximal pro-$p$ Galois groups.
Abstract
Let $p$ be a prime, we say that a Kummerian oriented pro-$p$ group $(G,θ)$ has the Bogomolov-Positselski property if $I_θ(G)$ is a free pro-$p$ group. We give a new criterion for an oriented pro-$p$ group to have the Bogomolov-Positselski property based on previous work by Positselski (arXiv:1405.0965) and Quadrelli and Weigel (arXiv:2103.12438) linking their seemingly unrelated approaches and thereby answering a question posed by Quadrelli and Weigel. Under further assumptions, we derive two additional criteria. The first of which strongly resembles an analogue of the Merkujev-Suslin theorem. The second allows to relax the conditions given by Positselski in Theorem 2 of arXiv:1405.0965. In addition, we show how to make those weaker assumptions computationally effective in some special cases.