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Journal of Homotopy and Related Structures

, Volume 11, Issue 3, pp 493–501 | Cite as

On Hilali’s conjecture related to Halperin’s

  • Badr Ben El Krafi
  • Mohamed Rachid Hilali
  • My Ismail Mamouni
Article
  • 72 Downloads

Abstract

In this paper, we focus on Hilali’s conjecture that, for any simply-connected elliptic CW-complex X, the total sum of the rational Betti numbers is at least as large as the total rank of its rational homotopy. We investigate this conjecture for coformal spaces and suggest some research directions to resolve it completely. Finally, we put up a bridge between the Hilali conjecture and that of Halperin: the toral rank conjecture and use it to establish the latter holds for all manifolds of dimension less than 16 and whose toral rank is equal to 4.

Keywords

Rational homotopy theory Minimal models Coformal spaces  Toral rank conjecture 

Mathematics Subject Classification

Primary 55P62  Secondary 55T05 

Notes

Acknowledgments

The authors would like to thank all members of the research group Moroccan Area in Algebraic Topology (MAAT) for the beautiful ambiance and atmosphere of work that prevailed within the group. This work is based on several discussions exchanged during their monthly seminar. We are also indebted to thank Thomas for his continuing support. The coformal case was deeply discussed with him during a short Angers-stay of the first author.Finally, we are grateful to our anonymous referee(s) for the suitable advice and assistance during the refereeing process.

References

  1. 1.
    Amann, M.: A note on the Hilali conjecture. arXiv:1501.02975 [math.AT]
  2. 2.
    Allday, C., Halperin, S.: Lie group actions on spaces of finite rank. Quart. J. Math. Oxford 28, 69–76 (1978)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Allday, C., Puppe, V.: Bounds on the torus rank, transformation groups Pozñan 1985. Lect. Notes Math. 1217, 1–10 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    De Bobadilla, J.F., Fresan, J., Munõz, V., Murillo, A.: The Hilali conjecture for hyperelliptic spaces. In: Pardalos, P., Rassias, T.M. (eds.) Mathematics Without Boundaries: Surveys in Pure Mathematics, pp. 21–36. Springer, Berlin (2014)Google Scholar
  5. 5.
    Deninger, C., Singhof, W.: On the cohomology of nilpotent Lie algebra. Bull. S.M.F 116(1), 3–14 (1988)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Friedlander, J., Halperin, S.: An arithmetic characterization of the rational homotopy groups of certain spaces. Invent. Math. 53, 117–133 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Félix, Y., Halperin, S., Jacobsson, C., Löfwall, C.: The radical of the homotopy lie algebra. Am. J. Math. 110(2), 301–322 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Félix, Y., Halperin, S., Thomas, J.-C.: Rational Homotopy Theory, Graduate Texts in Mathematics, vol. 205. Springer, New York (2001)CrossRefGoogle Scholar
  9. 9.
    Félix, Y., Halperin, S., Thomas, J.C.: Hopf algebras and a counterexample to a conjecture of anick. J. Algebra 169(1), 176–193 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Halperin, S.: Finitness in the minimal models of Sullivan. Trans. Am. Math. Soc. 230, 173–199 (1983)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Halperin, S.: Rational homotopy and torus actions, aspects of topology. In: James, I.M., Kronheimer, E.H. (eds.) London Math. Soc. Lecture Notes Ser., vol. 93 (1985)Google Scholar
  12. 12.
    Halperin, S.: Le complexe de Koszul en algébre et topologie. Ann. l’Inst. Fourier 37, 77–97 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hilali, M.R.: Action du tore \(\mathbb{T}^n\) sur les espaces simplement connexes. Thesis. University of catholique de Louvain, Belgique (1990)Google Scholar
  14. 14.
    Hilali, M.R., Mamouni, M.I.: A conjectured lower bound for the cohomological dimension of elliptic spaces. J. Homotopy Relat. Struct. 3(1), 379–384 (2008)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Jessup, B., Lupton, G.: Free torus actions and two-stage spaces. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 137, pp. 191–207, Cambridge University Press, Cambridge (2004)Google Scholar
  16. 16.
    Nakamura, O., Yamaguchi, T.: Lower bounds of Betti numbers of elliptic spaces with certain formal dimensions. Kochi J. Math. 6, 9–28 (2011)MathSciNetzbMATHGoogle Scholar

Copyright information

© Tbilisi Centre for Mathematical Sciences 2015

Authors and Affiliations

  • Badr Ben El Krafi
    • 1
  • Mohamed Rachid Hilali
    • 1
  • My Ismail Mamouni
    • 2
  1. 1.Département de Mathématiques et d’InformatiqueFaculté des Sciences Ain ChockCasablancaMorocco
  2. 2.Centre de Préparation à l’Agrégation, CRMEF RabatRabatMorocco

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