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


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.


Rational homotopy theory Minimal models Coformal spaces  Toral rank conjecture 

Mathematics Subject Classification

Primary 55P62  Secondary 55T05 



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.


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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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