The computation of Stiefel-Whitney classes
Annales de l'Institut Fourier, Volume 60 (2010) no. 2, p. 565-606
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.
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.
DOI : https://doi.org/10.5802/aif.2533
Classification:  20J06,  57R20,  65K05,  14C15
Keywords: Cohomology of groups, characteristic classes, algorithms, computers, chow rings
@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},
     mrnumber = {2667787},
     zbl = {pre05726205},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2010__60_2_565_0}
}
Guillot, Pierre. The computation of Stiefel-Whitney classes. Annales de l'Institut Fourier, Volume 60 (2010) no. 2, pp. 565-606. doi : 10.5802/aif.2533. http://www.numdam.org/item/AIF_2010__60_2_565_0/

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