The Cohomology of the Mod 4 Braid Group

TL;DR

This paper investigates the cohomology of the mod 4 braid group, constructing specific 2-cocycles that classify its extensions, and provides presentations for these extensions and related subgroups.

Contribution

It constructs explicit 2-cocycles in the symmetric group cohomology that classify the mod 4 braid group extensions and offers presentations for these groups and subgroups.

Findings

Constructed a 2-cocycle classifying the mod 4 braid group extension.

Identified the cocycle as the mod 2 reduction of a related cocycle.

Provided presentations for the extensions and a generating set for the level 4 congruence subgroup.

Abstract

The mod 4 braid group, $\mathcal{Z}_{n}$, is defined to be the quotient of the braid group by the subgroup of the pure braid group generated by squares of all elements. Kordek and Margalit proved $\mathcal{Z}_{n}$ is an extension of the symmetric group by $\mathbb{Z}_{2}^{\binom{n}{2}}$. For $n\geq 1$, we construct a 2-cocycle in the group cohomology of the symmetric group with twisted coefficients classifying $\mathcal{Z}_{n}$. We show this cocycle is the$\mod2$ reduction of the 2-cocycle corresponding to the extension of the symmetric group by the abelianization of the pure braid group. We also construct the 2-cocycle corresponding to this second extension and show it represents an order two element in the cohomology of the symmetric group. Furthermore, we give presentations for both extensions and a normal generating set for the level 4 congruence subgroup of the braid group.