The computation of Stiefel-Whitney classes

[Le calcul des classes de Stiefel-Whitney]

Guillot, Pierre

Annales de l'Institut Fourier, Tome 60 (2010), p. 565-606 / Harvested from Numdam

Résumé

L’anneau de cohomologie d’un groupe fini, modulo un nombre premier, peut être calculé à l’aide d’un ordinateur, comme l’a montré Carlson. Ici «  calculer  » signifie trouver une présentation en termes de générateurs et relations, et seul l’anneau (gradué) sous-jacent est en jeu. Nous proposons une méthode pour déterminer certains éléments de structure supplémentaires : classes de Stiefel-Whitney et opérations de Steenrod. Les calculs sont concrètement menés pour une centaine de groupes (les résultats sont consultables en détails sur Internet).

Nous donnons ensuite une application : à l’aide des nouvelles informations obtenues, nous pouvons dans de nombreux cas déterminer quelles sont les classes de cohomologie qui sont supportées par des cycles algébriques.

Abstract

The cohomology ring of a finite group, with coefficients in a finite field, can be computed by a machine, as Carlson has showed. Here “compute” means to find a presentation in terms of generators and relations, and involves only the underlying (graded) ring. We propose a method to determine some of the extra structure: namely, Stiefel-Whitney classes and Steenrod operations. The calculations are explicitly carried out for about one hundred groups (the results can be consulted on the Internet).

Next, we give an application: thanks to the new information gathered, we can in many cases determine which cohomology classes are supported by algebraic varieties.

Détail

Publié le : 2010-01-01

MR 2667787 | Zbl pre05726205

DOI : https://doi.org/10.5802/aif.2533
Classification:  20J06,  57R20,  65K05,  14C15
Mots clés: cohomologie des groupes, classes caractéristiques, algorithmes, ordinateurs, anneaux de Chow

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