8-2022

Dissertation

#### Degree Name

Doctor of Philosophy in Mathematics (PhD)

#### Department

Mathematical Sciences

Matthew B. Day

Matthew Clay

Lance E. Miller

#### Keywords

Braid Groups, Group Cohomology

#### Abstract

In 2014 Brendle and Margalit proved the level $4$ congruence subgroup of the braid group, $B_{n}[4]$, is the subgroup of the pure braid group generated by squares of all elements, $PB_{n}^{2}$. We define the mod $4$ braid group, $\Z_{n}$, to be the quotient of the braid group by the level 4 congruence subgroup, $B_{n}/B_{n}[4]$. In this dissertation we construct a group presentation for $\Z_{n}$ and determine a normal generating set for $B_{n}[4]$ as a subgroup of the braid group. Further work by Kordek and Margalit in 2019 proved $\Z_{n}$ is an extension of the symmetric group, $S_{n}$, by $\mathbb{Z}_{2}^{\binom{n}{2}}$. A classical result of Eilenberg and MacLane classifies group extensions by classes in the second group cohomology with twisted coefficients. We first construct a representative for the cohomology class, $[\phi]$, of $H^{2}(S_{n};\mathbb{Z}^{\binom{n}{2}})$ classifying the extension, $G_{n}$, of the symmetric group by the abelianization of the pure braid group. We then show a representative for the cohomology class in $H^{2}(S_{n};\mathbb{Z}_{2}^{\binom{n}{2}})$ classifying $\Z_{n}$ is the composition of $[\phi]$ with the mod 2 reduction of integers.

COinS