Integral cohomology of the Milnor fibre of the discriminant bundle associated with a finite Coxeter group

Callegaro, Filippo; Salvetti, Mario

Comptes Rendus. Mathématique, Tome 339 (2004) no. 8, pp. 573-578.

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 Δ:Cn/W→C 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.

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 Δ:Cn/W→C, 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.

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