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.
Keywords: Cohomology of groups, characteristic classes, algorithms, computers, chow rings
Mot clés : cohomologie des groupes, classes caractéristiques, algorithmes, ordinateurs, anneaux de Chow
Guillot, Pierre 1
@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 = {https://aif.centre-mersenne.org/articles/10.5802/aif.2533/}
}
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%0 Journal Article %A Guillot, Pierre %T The computation of Stiefel-Whitney classes %J Annales de l'Institut Fourier %D 2010 %P 565-606 %V 60 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2533/ %R 10.5802/aif.2533 %G en %F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.2533/
[1] An introduction to Gröbner bases, Graduate Studies in Mathematics, 3, American Mathematical Society, Providence, RI, 1994 | MR | Zbl
[2] Characters and cohomology of finite groups, Inst. Hautes Études Sci. Publ. Math. (1961) no. 9, pp. 23-64 | DOI | Numdam | MR | Zbl
[3] Steenrod operations in Chow theory, Trans. Amer. Math. Soc., Volume 355 (2003) no. 5, p. 1869-1903 (electronic) | DOI | MR | Zbl
[4] personal webpage (http://www.math.uga.edu/~lvalero/cohointro.html)
[5] Calculating group cohomology: tests for completion, J. Symbolic Comput., Volume 31 (2001) no. 1-2, pp. 229-242 Computational algebra and number theory (Milwaukee, WI, 1996) | DOI | MR | Zbl
[6] Cohomology rings of finite groups, Algebras and Applications, 3, Kluwer Academic Publishers, Dordrecht, 2003 (With an appendix: Calculations of cohomology rings of groups of order dividing 64 by Carlson, Valeri-Elizondo and Zhang) | MR | Zbl
[7] On the Chern classes of representations of finite groups, Trans. Amer. Math. Soc., Volume 115 (1965), pp. 180-193 | DOI | MR | Zbl
[8] Chern classes of certain representations of symmetric groups, Trans. Amer. Math. Soc., Volume 245 (1978), pp. 309-330 | DOI | MR | Zbl
[9] Homology of classical groups over finite fields and their associated infinite loop spaces, Lecture Notes in Mathematics, 674, Springer, Berlin, 1978 | MR | Zbl
[10] Characteristic classes of direct image bundles for covering maps, Ann. of Math. (2), Volume 125 (1987) no. 1, pp. 1-92 | DOI | MR | Zbl
[11] personal webpage (http://www.math.uni-wuppertal.de/~green/Coho_v2/)
[12] personal webpage (http://www-irma.u-strasbg.fr/~guillot/research/cohomology_of_groups/index.html)
[13] The Chow rings of and Spin(7), J. Reine Angew. Math., Volume 604 (2007), pp. 137-158 | DOI | MR | Zbl
[14] Addendum to the paper: “The Chow rings of and ” [J. Reine Angew. Math. 604 (2007), 137–158;], J. Reine Angew. Math., Volume 619 (2008), pp. 233-235 | DOI | MR | Zbl
[15] Classes de Stiefel-Whitney de formes quadratiques et de représentations galoisiennes réelles, Invent. Math., Volume 78 (1984) no. 2, pp. 223-256 | DOI | MR | Zbl
[16] The Evens-Kahn formula for the total Stiefel-Whitney class, Proc. Amer. Math. Soc., Volume 91 (1984) no. 2, pp. 309-313 | DOI | MR | Zbl
[17] Transfers in the group of multiplicative units of the classical cohomology ring and Stiefel-Whitney classes, Publ. Res. Inst. Math. Sci., Volume 25 (1989) no. 1, pp. 59-74 | DOI | MR | Zbl
[18] Sur les espaces fonctionnels dont la source est le classifiant d’un -groupe abélien élémentaire, Inst. Hautes Études Sci. Publ. Math. (1992) no. 75, pp. 135-244 (With an appendix by Michel Zisman) | DOI | Numdam | MR | Zbl
[19] The Steenrod algebra and its dual, Ann. of Math. (2), Volume 67 (1958), pp. 150-171 | DOI | MR | Zbl
[20] Characteristic classes, Princeton University Press, Princeton, N. J., 1974 (Annals of Mathematics Studies, No. 76) | MR | Zbl
[21] The Adams conjecture, Topology, Volume 10 (1971), pp. 67-80 | DOI | MR | Zbl
[22] The cohomology rings of extra-special -groups and the spinor groups, Math. Ann., Volume 194 (1971), pp. 197-212 | DOI | MR | Zbl
[23] Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1994 | MR | Zbl
[24] Représentations linéaires des groupes finis, Hermann, Paris, 1978 | MR | Zbl
[25] Characteristic classes and the cohomology of finite groups, Cambridge Studies in Advanced Mathematics, 9, Cambridge University Press, Cambridge, 1986 | MR | Zbl
[26] The Chow ring of a classifying space, Algebraic -theory (Seattle, WA, 1997) (Proc. Sympos. Pure Math.), Volume 67, Amer. Math. Soc., Providence, RI, 1999, pp. 249-281 | MR | Zbl
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