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.
Keywords: Cohomology of groups, characteristic classes, algorithms, computers, chow rings
Mot 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}, pages = {565--606}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {60}, number = {2}, year = {2010}, doi = {10.5802/aif.2533}, mrnumber = {2667787}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2533/} }
TY - JOUR AU - Guillot, Pierre TI - The computation of Stiefel-Whitney classes JO - Annales de l'Institut Fourier PY - 2010 SP - 565 EP - 606 VL - 60 IS - 2 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2533/ DO - 10.5802/aif.2533 LA - en ID - AIF_2010__60_2_565_0 ER -
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. http://www.numdam.org/articles/10.5802/aif.2533/
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