The computation of Stiefel-Whitney classes

Guillot, Pierre

Annales de l'Institut Fourier, Tome 60 (2010) no. 2, pp. 565-606.

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.

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.

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