The computation of Stiefel-Whitney classes  [ Le calcul des classes de Stiefel-Whitney ]
Annales de l'Institut Fourier, Tome 60 (2010) no. 2, p. 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.
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
@article{AIF_2010__60_2_565_0,
     author = {Guillot, Pierre},
     title = {The computation of Stiefel-Whitney classes},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {2},
     year = {2010},
     pages = {565-606},
     doi = {10.5802/aif.2533},
     zbl = {pre05726205},
     mrnumber = {2667787},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2010__60_2_565_0}
}
Guillot, Pierre. The computation of Stiefel-Whitney classes. Annales de l'Institut Fourier, Tome 60 (2010) no. 2, pp. 565-606. doi : 10.5802/aif.2533. https://aif.centre-mersenne.org/item/AIF_2010__60_2_565_0/

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