Abelianization of Subgroups of Reflection Group and their Braid Group; an Application to Cohomology

TL;DR

This paper investigates the abelianization of subgroups related to complex reflection groups and their braid groups, providing new descriptions and applications to cohomology extensions.

Contribution

It refines Stanley-Springer's theorem for complex reflection groups and describes the abelianization of key subgroups of braid groups, with applications to cohomology.

Findings

Order of the extension in cohomology is explicitly determined.

Refined abelianization descriptions for stabilizers and subgroups of braid groups.

A lifting construction for elements of the centralizer of a reflection.

Abstract

The final result of this article gives the order of the extension $$\xymatrix{1\ar[r] & P/[P,P] \ar^{j}[r] & B/[P,P] \ar^-{p}[r] & W \ar[r] & 1}$$ as an element of the cohomology group $H^2(W,P/[P,P])$ (where $B$ and $P$ stands for the braid group and the pure braid group associated to the complex reflection group $W$). To obtain this result, we describe the abelianization of the stabilizer $N_H$ of a hyperplane $H$. Contrary to the case of Coxeter groups, $N_H$ is not in general a reflection subgroup of the complex reflection group $W$. So the first step is to refine Stanley-Springer's theorem on the abelianization of a reflection group. The second step is to describe the abelianization of various types of big subgroups of the braid group $B$ of $W$. More precisely, we just need a group homomorphism from the inverse image of $N_H$ by $p$ with values in $\QQ$ (where $p : B \ra W$ is the canonical morphism) but a slight enhancement gives a complete description of the abelianization of $p^{-1}(W')$ where $W'$ is a reflection subgroup of $W$ or the stabilizer of a hyperplane. We also suggest a lifting construction for every element of the centralizer of a reflection in $W$.