The cohomology class of the mod 4 braid group

Abstract

The mod 4 braid group, [Formula: see text], is defined to be the quotient of the braid group by the subgroup of the pure braid group generated by squares of all elements, [Formula: see text]. Kordek and Margalit proved [Formula: see text] is an extension of the symmetric group by [Formula: see text]. For [Formula: see text], we construct a 2-cocycle in the group cohomology of the symmetric group with twisted coefficients classifying [Formula: see text]. We show this cocycle is the mod 2 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 that 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 [Formula: see text].