On the cohomology of Artin groups in local systems and the associated Milnor fiber

Abstract

Let W be a finite irreducible Coxeter group and let X W be the classifying space for G W , the associated Artin group. If A is a commutative unitary ring, we consider the two local systems L q and L q ′ over X W , respectively over the modules A [ q , q - 1 ] and A [ [ q , q - 1 ] ] , given by sending each standard generator of G W into the automorphism given by the multiplication by q. We show that H * ( X W , L q ′ ) = H * + 1 ( X W , L q ) and we generalize this relation to a particular class of algebraic complexes. We remark that H * ( X W , L q ′ ) is equal to the cohomology with trivial coefficients A of the Milnor fiber of the discriminant bundle of the associated reflection group.