2010/03/03 by Vincent Beck, Beck, Vincent
Mathematics · #20E22 #20F36 #20F55 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #math.GR #math.GT #msc:20E22 #msc:20F36 #msc:20F55
paper · pdf · doi:10.48550/arxiv.1003.0719
16 pages, new results on the stabilizer of a hyperplane added in section 1 and 2, new organisation of the paper, tables and GAP instructions added
arxiv created 2010/08/31 · arxiv updated 2010/09/02
The final result of this article gives the order of the extension \xymatrix1\ar[r] P/[P,P] \arj[r] B/[P,P] \ar-p[r] W \ar[r] 1 as an element of the cohomology group H2(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 NH of a hyperplane H. Contrary to the case of Coxeter groups, NH 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 NH 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.