The computation of Stiefel-Whitney classes

Pierre Guillot[1]

  • [1] Université de Strasbourg-CNRS IRMA 7 rue René Descartes 67084 Strasbourg (France)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 2, page 565-606
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Guillot, Pierre. "The computation of Stiefel-Whitney classes." Annales de l’institut Fourier 60.2 (2010): 565-606. <http://eudml.org/doc/116282>.

@article{Guillot2010,
abstract = {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.},
affiliation = {Université de Strasbourg-CNRS IRMA 7 rue René Descartes 67084 Strasbourg (France)},
author = {Guillot, Pierre},
journal = {Annales de l’institut Fourier},
keywords = {Cohomology of groups; characteristic classes; algorithms; computers; chow rings; Steenrod algebras; cohomology of groups; computation; Chow rings; cohomology rings; Stiefel-Whitney classes; Steenrod operations},
language = {eng},
number = {2},
pages = {565-606},
publisher = {Association des Annales de l’institut Fourier},
title = {The computation of Stiefel-Whitney classes},
url = {http://eudml.org/doc/116282},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Guillot, Pierre
TI - The computation of Stiefel-Whitney classes
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 2
SP - 565
EP - 606
AB - 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.
LA - eng
KW - Cohomology of groups; characteristic classes; algorithms; computers; chow rings; Steenrod algebras; cohomology of groups; computation; Chow rings; cohomology rings; Stiefel-Whitney classes; Steenrod operations
UR - http://eudml.org/doc/116282
ER -

References

top
  1. William W. Adams, Philippe Loustaunau, An introduction to Gröbner bases, 3 (1994), American Mathematical Society, Providence, RI Zbl0803.13015MR1287608
  2. M. F. Atiyah, Characters and cohomology of finite groups, Inst. Hautes Études Sci. Publ. Math. (1961), 23-64 Zbl0107.02303MR148722
  3. Patrick Brosnan, Steenrod operations in Chow theory, Trans. Amer. Math. Soc. 355 (2003), 1869-1903 (electronic) Zbl1045.55005MR1953530
  4. Jon F. Carlson, personal webpage 
  5. Jon F. Carlson, Calculating group cohomology: tests for completion, J. Symbolic Comput. 31 (2001), 229-242 Zbl0979.20047MR1806218
  6. Jon F. Carlson, Lisa Townsley, Luis Valeri-Elizondo, Mucheng Zhang, Cohomology rings of finite groups, 3 (2003), Kluwer Academic Publishers, Dordrecht Zbl1056.20039MR2028960
  7. Leonard Evens, On the Chern classes of representations of finite groups, Trans. Amer. Math. Soc. 115 (1965), 180-193 Zbl0133.28403MR212099
  8. Leonard Evens, Daniel S. Kahn, Chern classes of certain representations of symmetric groups, Trans. Amer. Math. Soc. 245 (1978), 309-330 Zbl0402.20009MR511412
  9. Zbigniew Fiedorowicz, Stewart Priddy, Homology of classical groups over finite fields and their associated infinite loop spaces, 674 (1978), Springer, Berlin Zbl0403.55010MR513424
  10. William Fulton, Robert MacPherson, Characteristic classes of direct image bundles for covering maps, Ann. of Math. (2) 125 (1987), 1-92 Zbl0628.55010MR873377
  11. David J. Green, personal webpage 
  12. Pierre Guillot, personal webpage 
  13. Pierre Guillot, The Chow rings of G 2 and Spin(7), J. Reine Angew. Math. 604 (2007), 137-158 Zbl1122.14005MR2320315
  14. Pierre Guillot, Addendum to the paper: “The Chow rings of G 2 and Spin ( 7 ) ” [J. Reine Angew. Math. 604 (2007), 137–158;], J. Reine Angew. Math. 619 (2008), 233-235 Zbl1142.14303MR2414952
  15. Bruno Kahn, Classes de Stiefel-Whitney de formes quadratiques et de représentations galoisiennes réelles, Invent. Math. 78 (1984), 223-256 Zbl0557.12014MR767193
  16. Andrzej Kozlowski, The Evens-Kahn formula for the total Stiefel-Whitney class, Proc. Amer. Math. Soc. 91 (1984), 309-313 Zbl0514.57005MR740192
  17. Andrzej Kozlowski, Transfers in the group of multiplicative units of the classical cohomology ring and Stiefel-Whitney classes, Publ. Res. Inst. Math. Sci. 25 (1989), 59-74 Zbl0687.55005MR999350
  18. Jean Lannes, Sur les espaces fonctionnels dont la source est le classifiant d’un p -groupe abélien élémentaire, Inst. Hautes Études Sci. Publ. Math. (1992), 135-244 Zbl0857.55011MR1179079
  19. John Milnor, The Steenrod algebra and its dual, Ann. of Math. (2) 67 (1958), 150-171 Zbl0080.38003MR99653
  20. John W. Milnor, James D. Stasheff, Characteristic classes, (1974), Princeton University Press, Princeton, N. J. Zbl0298.57008MR440554
  21. Daniel Quillen, The Adams conjecture, Topology 10 (1971), 67-80 Zbl0219.55013MR279804
  22. Daniel Quillen, The mod 2 cohomology rings of extra-special 2 -groups and the spinor groups, Math. Ann. 194 (1971), 197-212 Zbl0225.55015MR290401
  23. Lionel Schwartz, Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture, (1994), University of Chicago Press, Chicago, IL Zbl0871.55001MR1282727
  24. Jean-Pierre Serre, Représentations linéaires des groupes finis, (1978), Hermann, Paris Zbl0407.20003MR543841
  25. C. B. Thomas, Characteristic classes and the cohomology of finite groups, 9 (1986), Cambridge University Press, Cambridge Zbl0618.20036MR878978
  26. Burt Totaro, The Chow ring of a classifying space, Algebraic -theory (Seattle, WA, 1997) 67 (1999), 249-281, Amer. Math. Soc., Providence, RI Zbl0967.14005MR1743244

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.