Callegaro, F. On the cohomology of Artin groups in local systems and the associated Milnor fiber. (English) Zbl 1109.20027 J. Pure Appl. Algebra 197, No. 1-3, 323-332 (2005). 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. Cited in 9 Documents MSC: 20F36 Braid groups; Artin groups 20J06 Cohomology of groups 32S55 Milnor fibration; relations with knot theory Keywords:finite irreducible Coxeter groups; classifying spaces; Artin groups; local systems; cohomology; Milnor fibers; discriminant bundles; reflection groups PDFBibTeX XMLCite \textit{F. Callegaro}, J. Pure Appl. Algebra 197, No. 1--3, 323--332 (2005; Zbl 1109.20027) Full Text: DOI arXiv References: [1] Bourbaki, N., Groupes et Algebres de Lie (1981), Masson: Masson Paris, (Chapter 4-6) · Zbl 0483.22001 [2] E. Brieskorn, Sur les groupes des tresses, Seminoure Bourbaki, Lecture Notes in Math. 317 (1973) 21-44.; E. Brieskorn, Sur les groupes des tresses, Seminoure Bourbaki, Lecture Notes in Math. 317 (1973) 21-44. · Zbl 0277.55003 [3] Brieskorn, E.; Saito, K., Artin-Gruppen und Coxeter-Gruppen, Invent. Math., 17, 245-271 (1972) · Zbl 0243.20037 [4] Brown, K. S., Cohomology of Groups (1982), Springer: Springer Berlin · Zbl 0367.18012 [5] F. Callegaro, Proprietá intere della coomologia dei gruppi di Artin e della fibra di Milnor associata, Master Thesis, Dipartimento di Matematica, Univ. di Pisa, June 2003.; F. Callegaro, Proprietá intere della coomologia dei gruppi di Artin e della fibra di Milnor associata, Master Thesis, Dipartimento di Matematica, Univ. di Pisa, June 2003. [6] Cohen, D. C.; Suciu, A. I., Homology of iterated semidirect products of free groups, J. Pure Appl. Algebra, 126, 87-120 (1998) · Zbl 0908.20033 [7] C. De Concini, Private communication.; C. De Concini, Private communication. [8] De Concini, C.; Salvetti, M., Cohomology of Artin groups, Math. Res. Lett., 3, 293-297 (1996) · Zbl 0870.57004 [9] De Concini, C.; Procesi, C.; Salvetti, M., Arithmetic properties of the cohomology of braid groups, Topology, 40, 739-751 (2001) · Zbl 0999.20046 [10] De Concini, C.; Procesi, C.; Salvetti, M.; Stumbo, F., Arithmetic properties of the cohomology of Artin groups, Ann. Scuola Norm. Sup. Pisa Cl. Sci., XXVIII, 4, 695-717 (1999) · Zbl 0973.20025 [11] Deligne, P., Les immeubles des groupes de tresses generalises, Invent. Math., 17, 273-302 (1972) · Zbl 0238.20034 [12] Frenkel, E. V., Cohomology of the commutator subgroup of the braids group, Functional Anal. Appl., 22, 3, 248-250 (1988) · Zbl 0675.20042 [13] Salvetti, M., Topology of the complement of the real hyperplanes in \(C^n\), Invent. Math., 88, 167-189 (1987) · Zbl 0594.57009 [14] Salvetti, M., The homotopy type of Artin groups, Math. Res. Lett., 1, 565-577 (1994) · Zbl 0847.55011 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.