Verifying the Hilali conjecture up to formal dimension twenty

Abstract

We prove that in formal dimension \(\le 20\) the Hilali conjecture holds, i.e. that the total dimension of the rational homology bounds from above the total dimension of the rational homotopy for a simply connected rationally elliptic space.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    de Bobadilla, J.F., Fresán, J., Muñoz, V., Murillo, A.: The Hilali conjecture for hyperelliptic spaces. In: Mathematics Without Boundaries (pp. 21–36). Springer, New York (2014)

  2. 2.

    Félix, Y., Halperin, S., Thomas, J.C.: Rational Homotopy Theory, vol. 205. Springer Science and Business Media, New York (2012)

    Google Scholar 

  3. 3.

    Friedlander, J.B., Halperin, S.: An arithmetic characterization of the rational homotopy groups of certain spaces. Invent. Math. 53(2), 117–133 (1979)

    MathSciNet  Article  Google Scholar 

  4. 4.

    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)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Hilali, M.R., Mamouni, M.I.: A lower bound of cohomologic dimension for an elliptic space. Topol. Appl. 156(2), 274–283 (2008)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Miller, T.J.: On the formality of \( k-1\) connected compact manifolds of dimension less than or equal to \(4 k-2\). Ill. J. Math. 23(2), 253–258 (1979)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Nakamura, O., Yamaguchi, T.: Lower bounds of Betti numbers of elliptic spaces with certain formal dimensions. Kochi J. Math. 6, 9–28 (2011)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    SageMath, the Sage Mathematics Software System (Version 8.8), The Sage Developers. https://www.sagemath.org (2019)

Download references

Acknowledgements

The authors would like to acknowledge the support of the Directed Reading Program at Stony Brook University, under which this project was initiated. Several computations were carried out with the Commutative Differential Graded Algebras module for SageMath [8] written by Miguel Marco and John Palmieri, for which the authors are duly grateful. We thank the referee for their help in streamlining the exposition.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Spencer Cattalani.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Scott Wilson.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cattalani, S., Milivojević, A. Verifying the Hilali conjecture up to formal dimension twenty. J. Homotopy Relat. Struct. 15, 323–331 (2020). https://doi.org/10.1007/s40062-020-00256-2

Download citation

Keywords

  • Rational homotopy theory
  • Rationally elliptic spaces
  • Hilali conjecture