The Cohomology of the Mod 4 Braid Group

2021/07/26 by Trevor Nakamura, Nakamura, Trevor · 2 citations
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Advanced Algebra and Geometry #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2107.12263

Abstract

The mod 4 braid group, 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 Z_n is an extension of the symmetric group by ℤ₂^\binomn2. For n≥ 1, we construct a 2-cocycle in the group cohomology of the symmetric group with twisted coefficients classifying 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.

Cited by

• Powers of half-twists and congruence subgroups of braid groups
• Congruence Subgroups of the Virtual Braid Group

Related

• Quotients of the braid group that are extensions of the symmetric group
• Representation stability in the level 4 braid group
• Cohomology of braids, principal congruence subgroups and geometric\n representations
• A rational cohomology which including that of a spin hyperelliptic mapping class group
• The cohomology of the braid group B₃ and of SL₂(Z) with coefficients in a geometric representation