Cohomological uniqueness, Massey products and the modular isomorphism problem for 2-groups of maximal nilpotency class

TL;DR

This paper demonstrates that iterated Massey products in the cohomology of classifying spaces uniquely determine the homotopy type of 2-groups with maximal nilpotency class, providing new insights into the modular isomorphism problem.

Contribution

It shows that Massey products characterize the homotopy type of classifying spaces for these 2-groups, offering an alternative proof for the modular isomorphism problem.

Findings

Massey products determine the homotopy type of BG.

An alternative proof of the modular isomorphism problem.

Cohomological invariants characterize 2-groups of maximal nilpotency class.

Abstract

Let $G$ be a finite 2-group of maximal nilpotency class, and let $BG$ be its classifying space. We prove that iterated Massey products in $H^*(BG;\F_2)$ do characterize the homotopy type of $BG$ among 2-complete spaces with the same cohomological structure. As a consequence we get an alternative proof of the modular isomorphism problem for 2-groups of maximal nilpotency class.