Oriented pro-ℓgroups with the Bogomolov–Positselski property

For a prime number ℓwe say that an oriented pro-ℓgroup (G,θ)has the Bogomolov–Positselski property if the kernel of the canonical projection on its maximal θ-abelian quotient πG,θab:G→G(θ)is a free pro-ℓgroup contained in the Frattini subgroup of G. We show that oriented pro-ℓgroups of elementary type have the Bogomolov–Positselski property (cf. Theorem 1.2). This shows that Efrat’s Elementary Type Conjecture implies a positive answer to Positselski’s version of Bogomolov’s Conjecture on maximal pro-ℓGalois groups of a field Kin case that K×/(K×)ℓis finite. Secondly, it is shown that for an H∙-quadratic oriented pro-ℓgroup (G,θ)the Bogomolov–Positselski property can be expressed by the injectivity of the transgression map d22,1in the Hochschild–Serre spectral sequence (cf. Theorem 1.4).