Skip to main content
SpringerLink
Log in

The cellular structure of the classifying spaces of finite groups

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper we obtain a description of the Bℤ/p-cellularization (in the sense of Dror-Farjoun) of the classifying spaces of all finite groups, for all primes p.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Adem and R. J. Milgram, Cohomology of Finite Groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 309, Springer-Verlag, Berlin, 1994.

    Google Scholar 

  2. A. K. Bousfield and D. M. Kan, Homotopy Limits, Completions and Localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin, 1972.

    Google Scholar 

  3. C. Broto and N. Kitchloo, Classifying spaces of Kac-Moody groups, Mathematische Zeitschrift 240 (2002), 621–649.

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Benson, H. Krause and S. Iyengar, Local cohomology and support for triangulated categories, Annales Scientifiques de l’École Normale Supérieure 41 (2008), 573–619.

    MathSciNet  Google Scholar 

  5. C. Broto, R. Levi, and B. Oliver, The homotopy theory of fusion systems, Journal of the American Mathematical Society 16 (2003), 779–856 (electronic).

    Article  MathSciNet  MATH  Google Scholar 

  6. C. Broto, R. Levi and B. Oliver, Discrete models for the p-local homotopy theory of compact Lie groups and p-compact groups, Geometry and Topology 11 (2007), 315–427.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. K. Bousfield, The localization of spaces with respect to homology, Topology 14 (1975), 133–150.

    Article  MathSciNet  MATH  Google Scholar 

  8. A. K. Bousfield, Localization and periodicity in unstable homotopy theory, Journal of the American Mathematical Society 7 (1994), 831–873.

    Article  MathSciNet  MATH  Google Scholar 

  9. D. J. Benson and C.W. Wilkerson, Finite simple groups and Dickson invariants, in Homotopy Theory and its Applications (Cocoyoc, 1993), Contemporary Mathematics, Vol. 188, American Mathematical Society, Providence, RI, 1995, pp. 39–50.

    Google Scholar 

  10. N. Castellana, J. A. Crespo, and J. Scherer, Postnikov pieces and Bℤ/p-homotopy theory, Transactions of the American Mathematical Society 359 (2007), 1791–1816.

    Article  MathSciNet  Google Scholar 

  11. W. Chachólski, W. G. Dwyer and M. Intermont, The A-complexity of a space, Journal of the London Mathematical Society. Second Series 65 (2006), 204–222.

    Article  Google Scholar 

  12. W. Chachólski, On the functors CWA and PA, Duke Mathematical Journal 84 (1996), 599–631.

    Article  MathSciNet  MATH  Google Scholar 

  13. W. G. Dwyer, J. P. C. Greenlees and S. Iyengar, Duality in algebra and topology, Advances in Mathematics 200 (2006), 357–402.

    Article  MathSciNet  MATH  Google Scholar 

  14. E. Dror-Farjoun, R. Göbel and Y. Segev, Cellular covers of groups, Journal of Pure and Applied Algebra 208 (2007), 61–76.

    Article  MathSciNet  Google Scholar 

  15. E. Dror-Farjoun, R. Göbel, Y. Segev and S. Shelah, On kernels of cellular covers, Groups, Geometry, and Dynamics 4 (2007), 409–419.

    Article  Google Scholar 

  16. W.G. Dwyer, The centralizer decomposition of BG, in Algebraic Topology: New Trends in Localization and Periodicity (Sant Feliu de Guíxols, 1994), Progress in Mathematics, Vol. 136, Birkhäuser, Basel, 1996, pp. 167–184.

    Google Scholar 

  17. E. Dror-Farjoun, Cellular Spaces, Null Spaces and Homotopy Localization, Lecture Notes in Mathematics, Vol. 1622, Springer-Verlag, Berlin, 1996.

    Google Scholar 

  18. E. Dror-Farjoun, R. Göbel and Y. Segev, Cellular covers of spaces, Journal of Pure and Applied Algebra, to appear.

  19. R.J. Flores, Nullification and cellularization of classifying spaces of finite, groups, Transactions of the American Mathematical Society 359 (2007), 1791–1816.

    Article  MathSciNet  MATH  Google Scholar 

  20. R.J. Flores and R. Foote, Strongly closed subgroups of finite groups, Advances in Mathematics 222 (2009), 453–484.

    Article  MathSciNet  MATH  Google Scholar 

  21. R. Foote, A characterization of finite groups containing a strongly closed 2-subgroup, Communications in Algebra 25 (1997), 593–606.

    Article  MathSciNet  MATH  Google Scholar 

  22. R.J. Flores and J. Scherer, Cellularization of classifying spaces and fusion properties of finite groups, Journal of the London Mathematical Society. Second Series 76 (2007), 41–56.

    Article  MathSciNet  MATH  Google Scholar 

  23. D. Goldschmidt, 2-Fusion in finite groups, Annals of Mathematics 99 (1974), 70–117.

    Article  MathSciNet  Google Scholar 

  24. D. Goldschmidt, Strongly closed 2-subgroups of finite groups, Annals of Mathematics 102 (1975), 475–489.

    Article  MathSciNet  Google Scholar 

  25. J. Kiessling, Classification of certain cellular classes of chain complexes, Israel Journal of Mathematics 174 (2009), 179–188.

    Article  MathSciNet  MATH  Google Scholar 

  26. J. Martino and S. Priddy, On the cohomology and homotopy of Swan groups, Mathematische Zeitschrift 225 (1997), 277–288.

    Article  MathSciNet  MATH  Google Scholar 

  27. H. Miller, The Sullivan conjecture on maps from classifying spaces, Annals of Mathematics. Second Series 120 (1984), 39–87.

    Article  MathSciNet  Google Scholar 

  28. G. Mislin and C. Thomas, On the homotopy set [Bπ,BG] with π finite and G a compact connected Lie group, The Quarterly Journal of Mathematics. Oxford. Second Series 40 (1989), 65–78.

    Article  MathSciNet  MATH  Google Scholar 

  29. J. L. Rodríguez and J. Scherer, Cellular approximations using Moore spaces, in Cohomological Methods in Homotopy Theory (Bellaterra, 1998), Progress in Mathematics, Vol. 196, Birkhäuser, Basel, 2001, pp. 357–374.

    Chapter  Google Scholar 

  30. J. L. Rodríguez and J. Scherer, A connection between cellularization for groups and spaces via two-complexes, Journal of Pure and Applied Algebra 212 (2008), 1664–1673.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Estadística, Universidad Carlos III de Madrid, Avda. de la Universidad Carlos III, 22, E-28903, Colmenarejo (Madrid), Spain

    Ramón J. Flores

  2. Department of Mathematics and Statistics, University of Vermont, 16 Colchester Avenue, Burlington, Vermont, 05405, USA

    Richard M. Foote

Authors
  1. Ramón J. Flores
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Richard M. Foote
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ramón J. Flores.

Additional information

The first author was supported by MEC grant MTM2004-06686.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Flores, R.J., Foote, R.M. The cellular structure of the classifying spaces of finite groups. Isr. J. Math. 184, 129–156 (2011). https://doi.org/10.1007/s11856-011-0062-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-011-0062-0

Keywords

Search

Navigation