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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 \(W\).

20F36 Braid groups; Artin groups
20F55 Reflection and Coxeter groups (group-theoretic aspects)
20J06 Cohomology of groups
Full Text: DOI
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