Skip to main content
SpringerLink
Log in

Sur les espaces fonctionnels dont la source est le classifiant d’un groupe de Lie compact commutatif

  • Published:
Publications Mathématiques de l'Institut des Hautes Études Scientifiques Aims and scope Submit manuscript

Résumé

Nous montrons dans cet article comment les connaissances acquises sur les espaces fonctionnels de source le classifiant du groupeZ/p ([La2], [DS]) et l’utilisation de MU-résolutions instables permettent d’obtenir des résultats sur les espaces fonctionnels de source le classifiant d’unp-groupe abélien fini ou d’un tore si l’on impose au but d’avoir une cohomologie à coefficientsp-adiques sans torsion. Nous montrons notamment que l’ensemble des classes d’homotopie d’applications du classifiant X d’un tore dans un espace Y simplement connexe dont l’homologie entière est nulle en degré impair et un groupe abélien libre de dimension finie en chaque degré pair, et dont la cohomologie rationnelle est polynomiale, s’identifie à l’ensemble des applications de la K-théorie de Y dans la K-théorie de X qui préservent la structure de λ-anneau.

Abstract

We show in this paper how the acquired knowledge on the mapping spaces with source the classifying space ofZ/p ([La2], [DS]) and the use of unstable MU-resolutions give results on the mapping spaces with source the classifying space of a finite abelianp-group or a torus if the target space is required to have a torsion freep-adic cohomology. We prove among other things that the set of homotopy classes of maps from the classifying space X of a torus to some simply connected space Y whose ordinary homology is null in odd degrees and a finite-dimensional free abelian group in each even degree, and whose rational cohomology is polynomial, identifies with the set of maps from the K-theory of Y to the K-theory of X which preserve the λ-ring structure.

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

Références

  1. J. F. Adams,Stable Homotopy and Generalized Homology, University of Chicago Press, 1974.

  2. J. F. Adams, Maps between classifying spaces II,Invent. Math.,49 (1978), 1–65.

    Article  MATH  Google Scholar 

  3. J. F. Adams,Lectures on Lie Groups, Mathematics Lecture Note Series, 1969.

  4. J. F. Adams etZ. Mahmud, Maps between classifying spaces,Invent. Math.,35 (1976), 1–41.

    Article  MATH  Google Scholar 

  5. M. F. Atiyah etG. B. Segal, Equivariant K-theory and completion,J. Diff. Geom.,3 (1969), 1–18.

    MATH  Google Scholar 

  6. E. H. Brown etA. H. Copeland, Homology analogue of Postnikov systems,Mich. Math. Journ. 6 (1959), 313–330.

    Article  MATH  Google Scholar 

  7. M. Bendersky, E. B. Curtis etH. R. Miller, The unstable Adams spectral sequence for generalized homology,Topology,3 (1978), 229–248.

    Article  Google Scholar 

  8. A. K. Bousfield etD. M. Kan,Homotopy limits, Completions, and Localizations, Springer LNM,304, 1972.

  9. N. Bourbaki,Algèbre, chapitres 4 à 7, Masson 1981.

  10. Conner etFloyd,The relation of cobordism to K-theories, Springer LNM,28, 1966.

  11. F.-X. Dehon, Espaces fonctionnels de source le classifiant de S1 et de but un espace sansp-torsion, Thèse, Paris 7 (1999).

  12. W. G. Dwyer, H. R. Miller etJ. A. Neisendorfer, Fibrewise completion and unstable Adams spectral sequence,Israël J. Math.,66 (1989), 160–178.

    MATH  Google Scholar 

  13. E. Dror Farjoun, Pro-nilpotent representation of homology types,Proc. Amer. Math. Soc.,38 (1973), 657–660.

    Article  MATH  Google Scholar 

  14. E. Dror Farjoun etJ. Smith, A Geometric Interpretation of Lannes’ Functor T,Théorie de l’Homotopie, Astérisque,191 (1990), 87–95.

    Google Scholar 

  15. W. G. Dwyer etC. W. Wilkerson, Smith theory and the functor T,Comm. Math. Helv.,66 (1991), 1–17.

    Article  MATH  Google Scholar 

  16. W. G. Dwyer etC. W. Wilkerson, Homotopy fixed-point methods for Lie groups and finite loop spaces,Annals of Math.,139 (1994), 395–442.

    Article  MATH  Google Scholar 

  17. W. G. Dwyer etA. Zabrodsky, Maps between classifying spaces, Algebraic Topology, Barcelona 1986 (proceedings),Springer LNM,1298 (1987), 106–119.

    Google Scholar 

  18. N. J. Kuhn etM. Winstead, On the torsion in the cohomology of certain mapping spaces,Topology,35 (1996), 875–881.

    Article  MATH  Google Scholar 

  19. P. Hilton,Homotopy Theory and Duality, Notes on mathematics and its applications, 1965.

  20. J. Lannes, Sur la cohomologie modulop desp-groupes abéliens élémentaires, Proc. Durham Symposium on Homotopy Theory 1985,London Math. Soc. LNS,117, Camb. Univ. Press, 1987, 97–116.

    Google Scholar 

  21. J. Lannes, Sur les espaces fonctionnels dont la source est le classifiant d’unp-groupe abélien élémentaire,Publ. Math. IHES,75 (1992), 135–244.

    MATH  Google Scholar 

  22. J. Lannes etS. Zarati, Sur lesU-injectifs,Ann. Scient. Ec. Norm. Sup.,19 (1986), 1–31.

    Google Scholar 

  23. J. Lannes etS. Zarati, Théorie de Smith algébrique et classification des H*V —U-injectifs,Bull. Soc. Math. France,123 (1995), 189–223.

    MATH  Google Scholar 

  24. H. R. Miller, The Sullivan conjecture on maps from classifying spaces,Annals of Math.,120 (1984), 39–87.

    Article  Google Scholar 

  25. F. Morel, Quelques remarques sur la cohomologie modulop des pro-p-espaces et les résultats de Jean Lannes concernant les espaces fonctionnelshom(BV, X),Ann. Scient. Ec. Norm. Sup.,26 (1993), 309–360.

    MATH  Google Scholar 

  26. F. Morel, Ensembles profinis simpliciaux et interprétation géométrique du foncteur T,Bull. Soc. Math. France,124 (1996), 347–373.

    MATH  Google Scholar 

  27. D. Notbohm, Maps between classifying spaces,Math. Z.,207 (1991), 153–168.

    Article  MATH  Google Scholar 

  28. S. P. Novikov, The methods of algebraic topology from the viewpoint of cobordism theories,Math. USSR, Izv.,1 (1967), 827–913.

    Article  MATH  Google Scholar 

  29. D. Notbohm etL. Smith, Fake Lie groups and maximal tori I,Math. Ann.,288 (1990), 637–661.

    Article  MATH  Google Scholar 

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

  31. R. M. Switzer,Algebraic Topology — Homotopy and Homology, Grundlehren,212 (1975).

  32. C. W. Wilkerson, Lambda-ring, binomial domains, and vector bundles over CP{su∞},Commun. Algebra,10 (1982), 311–328.

    Article  MATH  Google Scholar 

  33. W. S. Wilson, The ω-spectrum for Brown-Peterson cohomology, Part I,Comm. Math. Helv.,48 (1973), 45–55.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre de Mathématiques, UMR 7640 du CNRS, École Polytechnique, 91128, Palaiseau, cedex, France

    François-Xavier Dehon & Jean Lannes

Authors
  1. François-Xavier Dehon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jean Lannes
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Le premier auteur a bénéficié pendant l’achèvement de ce travail d’une allocation de recherche de l’École Polytechnique.

About this article

Cite this article

Dehon, FX., Lannes, J. Sur les espaces fonctionnels dont la source est le classifiant d’un groupe de Lie compact commutatif. Publications Mathématiques de L’Institut des Hautes Scientifiques 89, 127–177 (1999). https://doi.org/10.1007/BF02698856

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02698856

Mots clés

Keywords

Search

Navigation