Four-fold Massey products in Galois cohomology

TL;DR

This paper introduces a new criterion for the vanishing of 4-Massey products in mod-2 Galois cohomology, leading to a splitting variety and proving their universal vanishing over number fields, thus constraining Galois group structures.

Contribution

It develops a necessary and sufficient condition for 4-Massey product vanishing and constructs a splitting variety satisfying a local-to-global principle, with implications for Galois groups.

Findings

All defined 4-Massey products vanish over number fields.

Introduces a splitting variety satisfying local-to-global principles.

Provides explicit restrictions on Galois group structures.

Abstract

In this paper, we develop a new necessary and sufficient condition for the vanishing of 4-Massey products of elements in the mod-2 Galois cohomology of a field. This new description allows us to define a splitting variety for 4-Massey products, which is shown in the Appendix to satisfy a local-to-global principle over number fields. As a consequence, we prove that, for a number field, all such 4-Massey products vanish whenever they are defined. This provides new explicit restrictions on the structure of absolute Galois groups of number fields.