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

Filippo Callegaro
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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.

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Filippo Callegaro. 2004-05-15. On the cohomology of Artin groups in local systems and the associated Milnor fiber. https://arxiv.org/abs/math/0405307

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