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On the cohomology of Artin groups in local systems and the associated Milnor fiber. (English) Zbl 1109.20027

Summary: Let \(W\) be a finite irreducible Coxeter group and let \(\mathbf X_W\) be the classifying space for \(G_W\), the associated Artin group. If \(A\) is a commutative unitary ring, we consider the two local systems \(\mathcal L_q\) and \(\mathcal L'_q\) over \(\mathbf X_W\), 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 \(H^*(\mathbf X_W,\mathcal L'_q)=H^{*+1}(\mathbf X_W,\mathcal L_q)\) and we generalize this relation to a particular class of algebraic complexes. We remark that \(H^*(\mathbf X_W,\mathcal L'_q)\) is equal to the cohomology with trivial coefficients \(A\) of the Milnor fiber of the discriminant bundle of the associated reflection group.

MSC:

20F36 Braid groups; Artin groups
20J06 Cohomology of groups
32S55 Milnor fibration; relations with knot theory
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