Date of Graduation

8-2022

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics (PhD)

Degree Level

Graduate

Department

Mathematical Sciences

Advisor

Matthew B. Day

Committee Member

Matthew Clay

Second Committee Member

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.

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