Skip to main content

Serre’s theorem and the Nil l filtration of Lionel Schwartz

  • Conference paper
Cohomological Methods in Homotopy Theory

Part of the book series: Progress in Mathematics ((PM,volume 196))

Abstract

We give three different cohomological characterizations of classifying spaces of p-compact toral groups amongst finite Postnikov systems satisfying mild conditions. This leads to a unifying generalization of previous versions of Serre’s theorem on the homotopy groups of a finite complex.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. K. Bousfield. The localization of spaces with respect to homology. Topology, 14:133–150, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. K. Bousfield and D. M. Kan. Homotopy Limits, Completions and Localizations. Lecture Notes in Math. vol. 304. Springer, 1972.

    Google Scholar 

  3. W. G. Dwyer. Strong convergence of the Eilenberg—Moore spectral sequence. Topology, 13:255–265, 1974.

    Article  MathSciNet  MATH  Google Scholar 

  4. W. G. Dwyer and C. W. Wilkerson. Spaces of null homotopic maps. Astérisque, 191:97–108, 1990. International Conference on Homotopy Theory (Marseille-Luminy, 1988).

    Google Scholar 

  5. W. G. Dwyer and C. W. Wilkerson. Homotopy fixed point methods for Lie groups and finite loop spaces.Ann.of Math., 139:395–442, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Grodal. The transcendence degree of the mod p cohomology of finite Postnikov systems. In Stable and unstable homotopy (Toronto,ON, 1996), pages 111–130. Amer. Math. Soc., Providence, RI, 1998.

    Google Scholar 

  7. D. K. Harrison. Infinite abelian groups and homological methods. Ann. of Math.,69:366–391, 1959.

    Article  MathSciNet  MATH  Google Scholar 

  8. H.-W. Henn, J. Lannes, and L. Schwartz. The categories of unstable modules and unstable algebras over the Steenrod algebra modulo nilpotent objects. Amer. J. Math., 115:1053–1106, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Lannes. Sur les espaces fonctionnels dont la source est le classifiant d’un p-groupe abélien élémentaire. Inst. Hautes Études Sci. Publ. Math., 75:135–244, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Lannes and L. Schwartz. A propos de conjectures de Serre et Sullivan. Invent. Math., 83:593–603, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Lannes and L. Schwartz. Sur les groupes d’homotopie des espaces dont la cohomologie modulo 2 est nilpotente. Israel J. Math., 66:260–273, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  12. C. A. McGibbon and J. Neisendorfer. On the homotopy groups of a finite dimensional space. Comment. Math. Helv., 59:253–257, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  13. F. Morel. Quelques remarques sur la cohomologie modulop continue des pro-pespaces et les résultats de J. Lannes concernant les espaces fonctionnels hom(BV,X).. Ann Sci. École Norm. Sup. (4), 26(3):309–360, 1993.

    MathSciNet  MATH  Google Scholar 

  14. D. Rector. Steenrod operations in the Eilenberg-Moore spectral sequence. Comment. Math. Helv., 45:540–552, 1970.

    Article  MathSciNet  MATH  Google Scholar 

  15. L. Schwartz. La filtration nilpotente de la categorie U et la cohomologie des espaces de lacets. In Algebraic topology —rational homotopy (Louvain-la-Neuve, 1986), Lecture Notes in Math. vol. 1318, pages 208–218. Springer, 1988.

    Google Scholar 

  16. L. Schwartz. Unstable Modules over the Steenrod Algebra and Sullivan’s Fixed Point Set Conjecture. The University of Chicago Press, 1994.

    Google Scholar 

  17. J.-P. Serre. Cohomologie modulo 2 des complexes d’Eilenberg-Mac Lane.Comment.Math.Helv.,27:198–232, 1953.

    Article  MathSciNet  MATH  Google Scholar 

  18. B. Shipley. On Serre’s conjecture. 1992 MIT Thesis proposal.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer Basel AG

About this paper

Cite this paper

Grodal, J. (2001). Serre’s theorem and the Nil l filtration of Lionel Schwartz. In: Aguadé, J., Broto, C., Casacuberta, C. (eds) Cohomological Methods in Homotopy Theory. Progress in Mathematics, vol 196. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8312-2_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8312-2_13

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9513-2

  • Online ISBN: 978-3-0348-8312-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics