Afrika Matematika

, Volume 27, Issue 5–6, pp 851–864 | Cite as

The rational homotopy type of elliptic spaces up to cohomological dimension 8

  • Mohamed Rachid Hilali
  • My Ismail Mamouni
  • Hicham Yamoul


Our goal in this paper is to give a full classification of the rational homotopy type of any elliptic and simply connected space when the sum of its Betti numbers is less or equal than 8.


Rational homotopy theory Sullivan models Elliptic spaces Rational homotopy type 

Mathematics Subject Classification

Primary 55P62 55P15 Secondary 55P10 55Q05 55Q52 


  1. 1.
    Bazzoni, G., Muñoz, V.: Classification of minimal algebras over any field up to dimension 6. Trans. Am. Math. Soc. 364(2), 1007–1028 (2012)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Friedlander, J., Halperin, S.: An arithmetic characterization of the rational homotopy groups of certain spaces. Invent. Math. 53, 117–133 (1979)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Félix, Y., Halperin, S., Thomas, J.-C.: Rational Homotopy Theory, Graduate Texts in Mathematics, vol. 205. Springer, New York (2001)CrossRefGoogle Scholar
  4. 4.
    Halperin, S.: Finitness in the minimal models of Sullivan. Trans. Am. Math. Soc. 230, 173–199 (1983)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hilali, M.R., Lamane, H., Mamouni, M.I.: Classification of rational homotopy type for 8-cohomological dimension elliptic spaces. Adv. Pure Math. 2, 15–21 (2012)CrossRefMATHGoogle Scholar
  6. 6.
    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)MathSciNetMATHGoogle Scholar
  7. 7.
    James, I.M.: Reduced product spaces. Ann. Math. 62(1), 170–197 (1955)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Kadeishvili, T.: Cohomology \(C_\infty \)-algebra and rational homotopy type (2008). arXiv:0811.1655v1 [math.AT]
  9. 9.
    Kono, S., Tamamura, A.: Graded commutative \({\mathbb{Q}} \)-algebras \({\mathbb{Q}} [x_1, \ldots, x_m]/(f_1, \ldots, f_n)\) of dimension 10 over \({\mathbb{Q}}\). Bull. Okayama Univ. Sci. 36(A), 11–18 (2000)Google Scholar
  10. 10.
    Kono, S., Tamamura, A.: Elliptic graded commutative \({\mathbb{Q}}\)-algebras of dimension 11 and 13 over \({\mathbb{Q}}\). Bull. Okayama Univ. Sci. 37(A), 29–34 (2001)Google Scholar
  11. 11.
    Mimuraz, M., Shiga, H.: On the classification of rational homotopy types of elliptic spaces with homotopy Euler characteristic zero for \(\dim < 8\). Bull. Belg. Math. Soc. Simon Stevin 18(5), 925–939 (2011)MathSciNetMATHGoogle Scholar
  12. 12.
    Powell, G.M.L.: Elliptic spaces with the rational homotopy type of spheres. Bull. Belg. Math. Soc. Simon Stevin 4(2), 251–263 (1997)MathSciNetMATHGoogle Scholar
  13. 13.
    Schlessinger, M., Stasheff, J.: Deformation theory and rational homotopy type (2012). arXiv:1211.1647v1 [math.QA]
  14. 14.
    Shiga, H., Yamaguchi, T.: The set of rational homotopy types with given cohomology algebra. Homology Homotopy Appl. 5(1), 423–436 (2003)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Mohamed Rachid Hilali
    • 1
  • My Ismail Mamouni
    • 2
  • Hicham Yamoul
    • 1
  1. 1.Département de Mathématiques et d’InformatiqueFaculté des Sciences Ain ChockMaarif, CasablancaMorocco
  2. 2.Département de Didactique des Mathématiques, Centre de Préparation à l’AgrégationCRMEF RabatRabatMorocco

Personalised recommendations