Cohomology of quasi-abelianized braid groups

TL;DR

This paper studies the rational cohomology of certain quotient braid groups, providing a combinatorial description, establishing dualities, and proving stabilization properties for reflection groups.

Contribution

It offers a new combinatorial framework for understanding the cohomology of quasi-abelianized braid groups and proves duality and stabilization results.

Findings

Combinatorial description of cohomology via graph classes

Established Poincaré dualities for these groups

Proved stabilization properties for infinite reflection groups

Abstract

We investigate the rational cohomology of the quotient of (generalized) braid groups by the commutator subgroup of the pure braid groups. We provide a combinatorial description of it using isomorphism classes of certain families of graphs. We establish Poincar\'e dualities for them and prove a stabilization property for the infinite series of reflection groups.