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Article Dans Une Revue Manuscripta mathematica Année : 2011

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

Résumé

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 $W$.
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Dates et versions

hal-01198821 , version 1 (16-10-2015)

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Vincent Beck. Abelianization of Subgroups of Reflection Group and their Braid Group; an Application to Cohomology. Manuscripta mathematica, 2011, 136 (3), pp.273-293. ⟨10.1007/s00229-011-0438-9⟩. ⟨hal-01198821⟩
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