Generalized Bockstein maps and Massey products

TL;DR

This paper explores the relationship between Iwasawa cohomology filtrations, Bockstein maps, and Massey products in the context of profinite groups, providing new insights into Galois cohomology and class group structures.

Contribution

It introduces generalized Bockstein maps for powers of the augmentation ideal and relates them to Massey products, advancing understanding of cohomological structures in Galois theory.

Findings

Lower bounds on p-ranks of class groups established

New proof of vanishing triple Massey products in Galois cohomology

Connections between cohomology filtrations and Massey products demonstrated

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 the rational numbers and to give a new proof of the vanishing of triple Massey products in Galois cohomology.