Let W be a finite Coxeter group generated by real reflections in a complex vector space. We compute the integral cohomology of the Milnor fibre of the discriminant bundle , together with the action of the monodromy, for the whole list of exceptional groups. Here Δ is the map induced by the square of the polynomial defining the arrangement of reflection hyperplanes of W. The computation is equivalent to that of the cohomology, with suitable local coefficients, of the corresponding Artin group. These computations complete, for the exceptional cases, those performed by De Concini et al. for rational coefficients.
Soit W un groupe de Coxeter fini engendré par des réflexions réelles dans un espace vectoriel complexe. On calcule la cohomologie entière de la fibre de Milnor du fibré discriminant et l'action de la monodromie, pour tous les groupes exceptionnels. Ici Δ est l'application induite par le carré du polynôme qui définit l'arrangement des hyperplans de réflexion de W. Le calcul équivaut à celui de la cohomologie, à coefficients locaux bien choisis, du groupe d'Artin correspondant. Ces calculs complètent, pour les cas exceptionnels, ceux de De Concini et al. à coefficients rationnels.
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Filippo Callegaro 1; Mario Salvetti 2
@article{CRMATH_2004__339_8_573_0, author = {Filippo Callegaro and Mario Salvetti}, title = {Integral cohomology of the {Milnor} fibre of the discriminant bundle associated with a finite {Coxeter} group}, journal = {Comptes Rendus. Math\'ematique}, pages = {573--578}, publisher = {Elsevier}, volume = {339}, number = {8}, year = {2004}, doi = {10.1016/j.crma.2004.09.008}, language = {en}, }
TY - JOUR AU - Filippo Callegaro AU - Mario Salvetti TI - Integral cohomology of the Milnor fibre of the discriminant bundle associated with a finite Coxeter group JO - Comptes Rendus. Mathématique PY - 2004 SP - 573 EP - 578 VL - 339 IS - 8 PB - Elsevier DO - 10.1016/j.crma.2004.09.008 LA - en ID - CRMATH_2004__339_8_573_0 ER -
%0 Journal Article %A Filippo Callegaro %A Mario Salvetti %T Integral cohomology of the Milnor fibre of the discriminant bundle associated with a finite Coxeter group %J Comptes Rendus. Mathématique %D 2004 %P 573-578 %V 339 %N 8 %I Elsevier %R 10.1016/j.crma.2004.09.008 %G en %F CRMATH_2004__339_8_573_0
Filippo Callegaro; Mario Salvetti. Integral cohomology of the Milnor fibre of the discriminant bundle associated with a finite Coxeter group. Comptes Rendus. Mathématique, Volume 339 (2004) no. 8, pp. 573-578. doi : 10.1016/j.crma.2004.09.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.09.008/
[1] Singularities of Differentiable Maps, vol. II, Birkhäuser, Boston, 1988
[2] Groupes et algèbres de Lie, Masson, Paris, 1981 (Ch. 4–6)
[3] Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe, Invent. Math., Volume 12 (1971), pp. 57-61
[4] Artin-Gruppen und Coxeter-Gruppen, Invent. Math., Volume 17 (1972), pp. 245-271
[5] Cohomology of Groups, Graduate Texts in Math., vol. 87, Springer-Verlag, 1982
[6] 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
[7] Homology of iterated semidirect products of free groups, J. Pure Appl. Alg., Volume 126 (1998), pp. 87-120
[8] Cohomology of Artin groups, Math. Res. Lett., Volume 3 (1996), pp. 293-297
[9] Arithmetic properties of the cohomology of braid groups, Topology, Volume 40 (2001), pp. 739-751
[10] Arithmetic properties of the cohomology of Artin groups, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Volume XXVIII (1999) no. 4, pp. 695-717
[11] Les immeubles des groupes des tresses généralisés, Invent. Math., Volume 17 (1972), pp. 273-302
[12] Regular elements and monodromy of discriminants of the finite reflection groups, Indag. Math. (N.S.), Volume 6 (1995) no. 2, pp. 129-143
[13] Cohomology of the commutator subgroup of the braids group, Functional Anal. Appl., Volume 22 (1988) no. 3, pp. 248-250
[14] Reflection Groups and Coxeter Groups, Cambridge University Press, 1990
[15] Singular Points of Complex Hypersurfaces, Ann. of Math. Stud., vol. 61, Princeton University Press, Princeton, 1968
[16] Topology of the complement of real hyperplanes in , Invent. Math., Volume 88 (1987), pp. 167-189
[17] The homotopy type of Artin groups, Math. Res. Lett., Volume 1 (1994), pp. 565-577
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