Gaps in the Milnor–Moore spectral sequence and the Hilali conjecture

Abstract

Let X be a simply-connected space, \((\Lambda V, d)\) its minimal Sullivan model and \(d_k\) (\(k\ge 2\)) the first non-zero homogeneous part of the differential d. In this paper, assuming that \((\Lambda V, d_k)\) is elliptic, we show that \(H(\Lambda V,d)\) has no \(e_0\)-gap and consequently we confirm the Hilali conjecture when \(V = V^{odd}\) or else when \(k\ge 3\).

Résumé

Soit X un espace topologique simplement connexe, \((\Lambda V, d)\) son modéle minimal de Sullivan et \(d_k\) (\(k\ge 2\)) la premiére partie homogéne non nulle de la différentielle d. Dans cet article, en supposant que \((\Lambda V, d_k)\) est elliptique, nous montrons que \(H(\Lambda V,d)\) n’a aucun \(e_0\)-écart et par conséquent, nous confirmons la conjecture de Hilali lorsque \(V = V^{impair}\) ou lorsque \(k\ge 3\)

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Notes

  1. 1.

    The case where \(V=V^{odd}\) was treated under some restrictive hypothesis by Amann in [2], who I thank for informing and inspiring me to add such a generalization here.

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Acknowledgements

I would like to thank the referee for his valuable remarks and suggestions. I am also grateful to all members of the Moroccan Area of Algebraic Topology group for several discussions that contributed to the achievement of my results.

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Correspondence to Youssef Rami.

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Rami, Y. Gaps in the Milnor–Moore spectral sequence and the Hilali conjecture. Ann. Math. Québec 43, 435–442 (2019). https://doi.org/10.1007/s40316-018-0107-4

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Keywords

  • Milnor-Moore spectral sequence
  • Hilali conjecture
  • Sullivan models
  • Elliptic spaces

Mathematics Subject Classification

  • 55P62
  • 55T99
  • 55Q15