The top-cohomology of Artin groups with coefficients in rank-1 local systems over [formula omitted

Abstract

Let W be a Coxeter group and let G w be the associated Artin group. We consider the local system over k( G w , 1) with coefficients in R = Z[q, q −1] which associates to the standard generators of G w the multiplication by q. For the all list of finite irreducible Coxeter groups we calculate the top-cohomology of this local system. It turns out that the ideal which we compute is a sort of Alexander ideal for a hypersurface. In case of the classical braid group Br n this ideal is the principal ideal generated by the nth cyclotomic polynomial. We use these results to calculate the topological category of k( G w , 1): we prove that it equals the obvious bound given by obstruction theory (so, in case of braid group Br n , it is exactly n).