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Abelianization of subgroups of reflection groups and their braid groups: an application to cohomology. (English) Zbl 1241.20038
The final result of this article gives the order of the extension $1\to P/[P,P]@>j>>B/[P,P]@>p>>W\to 1$ as an element of the cohomology group $$H^2(W,P/[P,P])$$ (where $$B$$ and $$P$$ stand for the braid group and the pure braid group associated to the complex reflection group $$W$$). To obtain this result, the author first refines 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, he just needs a group homomorphism from the inverse image of $$N_H$$ by $$p$$ (where $$p\colon B\to W$$ 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. He also suggests a lifting construction for every element of the centralizer of a reflection in $$W$$.

##### MSC:
 20F36 Braid groups; Artin groups 20F55 Reflection and Coxeter groups (group-theoretic aspects) 20J06 Cohomology of groups
GAP; CHEVIE
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##### References:
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