Generalized Bockstein maps and Massey products

Abstract

Given a profinite group G of finite p-cohomological dimension and a pro-p quotient H of G by a closed normal subgroup N, we study the filtration on the Iwasawa cohomology of N by powers of the augmentation ideal in the group algebra of H. We show that the graded pieces are related to the cohomology of G via analogues of Bockstein maps for the powers of the augmentation ideal. For certain groups H, we relate the values of these generalized Bockstein maps to Massey products relative to a restricted class of defining systems depending on H. We apply our study to prove lower bounds on the p-ranks of class groups of certain nonabelian extensions of $\mathbb {Q}$ and to give a new proof of the vanishing of Massey triple products in Galois cohomology.

On pro-p groups with quadratic cohomology

“…Moreover, there are number-theoretic analogues of the Borromean rings, giving essential Massey products in Galois cohomology (see, e.g., [12,29,40]). Further interesting results on Massey products in Galois cohomology have been obtained by various authors (see, e.g., [13,14,18,22,23,25,42]).…”

Demushkin groups of uncountable rank

“…By [13], it is enough to prove that the sequence is exact at 𝐻 2 (𝐺, 𝐹) for every 𝛼 ∈ 𝐻 1 (𝐺). Again, as every subgroup of a Demushkin group is either free or Demushkin, it is enough to prove the property holds for every Demushkin group.…”

“…Proof A pro‐$$p$$ group $$G$$ is said to be of $$p$$‐absolute Galois type if for every $$\alpha \in H^1(G)$$, the following sequence is exact: By [13], it is enough to prove that the sequence is exact at $$H^2(G,F)$$ for every $$\alpha \in H^1(G)$$. Again, as every subgroup of a Demushkin group is either free or Demushkin, it is enough to prove the property holds for every Demushkin group.…”