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

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

2004-05-15

arxiv.org/abs/math/0405307v1

J. Pure Appl. Algebra 197 (2005), no. 1-3, 323--332

Mathematics

Algebraic Topology

9 pages

Scientific paper

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.

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