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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna


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Volume 29, Issue 2

Issues

A note on the Hilali conjecture

Manuel Amann
  • Corresponding author
  • Fakultät für Mathematik, Institut für Algebra und Geometrie, Karlsruher Institut für Technologie, Englerstraße 2, 76131 Karlsruhe, Germany
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Published Online: 2016-05-21 | DOI: https://doi.org/10.1515/forum-2015-0078

Abstract

In this short note we observe that the Hilali conjecture holds for 2-stage spaces, i.e. we argue that the dimension of the rational cohomology is at least as large as the dimension of the rational homotopy groups for these spaces. We also prove the Hilali conjecture for a class of spaces which puts it into the context of fibrations.

Keywords: Hilali conjecture; rational cohomology groups; rational homotopy groups

MSC 2010: 55Q52; 55P62

References

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    Félix Y., Halperin S. and Thomas J.-C., Rational Homotopy Theory, Grad. Texts in Math. 205, Springer, New York, 2001. Google Scholar

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    Félix Y., Oprea J. and Tanré D., Algebraic Models in Geometry, Oxf. Grad. Texts Math. 17, Oxford University Press, Oxford, 2008. Google Scholar

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    Fernández de Bobilla J., Fresán J., Munõz V. and Murillo A., The Hilali conjecture for hyperelliptic spaces, Mathematics Without Boundaries, Springer, New York (2014), 21–36. Google Scholar

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    Hilali M. R., Action du tore Tn sur les espaces simplement connexes, PhD thesis, Université Catholique de Louvain, Louvain-la-Neuve, 1980. Google Scholar

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    Hilali M. R. and Mamouni M. I., A conjectured lower bound for the cohomological dimension of elliptic spaces, J. Homotopy Relat. Struct. 3 (2008), no. 1, 379–384. Google Scholar

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    Hilali M. R. and Mamouni M. I., A lower bound of cohomologic dimension for an elliptic space, Topology Appl. 156 (2008), no. 2, 274–283. Google Scholar

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    Hilali M. R. and Mamouni M. I., A conjectured lower bound for the cohomological dimension of elliptic spaces – Recent results in some simple cases, preprint 2013, http://arxiv.org/abs/0803.3821.

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    Hilali M. R. and Mamouni M. I., La conjecture H: Une minoration de la dimension cohomologique pour un espace elliptique, preprint 2013, https://arxiv.org/abs/0712.3784.

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    Jessup B. and Lupton G., Free torus actions and two-stage spaces, Math. Proc. Cambridge Philos. Soc. 137 (2004), no. 1, 191–207. Google Scholar

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    Nakamura O. and Yamaguchi T., Lower bounds of Betti numbers of elliptic spaces with certain formal dimensions, Kochi J. Math. 6 (2001), 9–28. Google Scholar

About the article


Received: 2015-04-27

Revised: 2016-02-02

Published Online: 2016-05-21

Published in Print: 2017-03-01


Citation Information: Forum Mathematicum, Volume 29, Issue 2, Pages 251–257, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2015-0078.

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