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

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 $W$.