Abelianization of subgroups of reflection groups and their braid groups: an application to Cohomology

Abstract

The final result of this article gives the order of the extension

$$ 1 \longrightarrow P/[P,P]{\mathop{\longrightarrow} \limits^j} B/[P,P] {\mathop{\longrightarrow} \limits^p} W \longrightarrow 1$$

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 first refine Stanley-Springer’s theorem on the abelianization of a reflection group to describe the abelianization of the stabilizer N _H of a hyperplane H. The second step is to describe the abelianization 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 (where p: B → 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.