Extensions of groups by braided 2-groups

TL;DR

This paper classifies extensions of groups by braided 2-groups using homotopy classes of maps and introduces a generalized Pontryagin square in the context of local coefficients, providing explicit cochain descriptions.

Contribution

It provides a classification framework for group extensions by braided 2-groups and generalizes the Pontryagin square to local coefficient settings with explicit cochain formulas.

Findings

Classification of extensions via homotopy classes of maps

Generalization of the Pontryagin square for local coefficients

Explicit cochain-level description of the generalized Pontryagin square

Abstract

We classify extensions of a group $G$ by a braided 2-group $\mathcal{B}$ as defined by Drinfeld, Gelaki, Nikshych, and Ostrik. We describe such extensions as homotopy classes of maps from the classifying space of $G$ to the classifying space of the 3-group of braided $\mathcal{B}$-bitorsors. The Postnikov system of the latter space contains a generalization of the classical Pontryagin square to the setting of local coefficients, which has been previously discussed by Baues and more recently, in a setting close to ours, by Etingof, Nikshych, and Ostrik. We give an explicit cochain-level description of this Pontryagin square for group cohomology.