Oriented pro-$\ell$ groups with the Bogomolov-Positselski property (2103.12438v3)
Abstract: For a prime number $\ell$ we say that an oriented pro-$\ell$ group $(G,\theta)$ has the Bogomolov-Positselski property if the kernel of the canonical projection on its maximal $\theta$-abelian quotient $\pi{ab}_{G,\theta}\colon G\to G(\theta)$ is a free pro-$\ell$ group contained in the Frattini subgroup of $G$. We show that oriented pro-$\ell$ groups of elementary type have the Bogomolov-Positselski property. This shows that Efrat's Elementary Type Conjecture implies a positive answer to Positselski's version of Bogomolov's Conjecture on maximal pro-$\ell$ Galois groups of a field $K$ in case that $K\times/(K\times)\ell$ is finite. Secondly, it is shown that for an $H\bullet$-quadratic oriented pro-$\ell$ group $(G,\theta)$ the Bogomolov-Positselski property can be expressed by the injectivity of the transgression map $d_2{2,1}$ in the Hochschild-Serre spectral sequence.