# 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 $W$.
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Cited literature [14 references]

https://hal.archives-ouvertes.fr/hal-01198821
Contributor : Vincent Beck <>
Submitted on : Friday, October 16, 2015 - 5:57:24 PM
Last modification on : Thursday, May 3, 2018 - 3:32:07 PM
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braid-cohom-v2.pdf
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### Identifiers

• HAL Id : hal-01198821, version 1
• ARXIV : 1003.0719

### Citation

Vincent Beck. Abelianization of Subgroups of Reflection Group and their Braid Group; an Application to Cohomology. manuscripta mathematica, Springer Verlag, 2011, 136 (3), pp.273-293. ⟨hal-01198821⟩

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