manuscripta mathematica

, Volume 56, Issue 2, pp 177–191

Sur l'homotopie rationnelle des espaces fonctionnels

  • Micheline Vigué-Poirrier

DOI: 10.1007/BF01172155

Cite this article as:
Vigué-Poirrier, M. Manuscripta Math (1986) 56: 177. doi:10.1007/BF01172155


Let X be a nilpotent space such that it exists k⩾1 with Hp (X,ℚ) = 0 p > k and Hk (X,ℚ) ≠ 0, let Y be a (m−1)-connected space with m⩾k+2, then the rational homotopy Lie algebra of YX (resp.\(\left( {Y, y_0 } \right)^{\left( {X, x_0 } \right)} \) is isomorphic as Lie algebra, to H* (X,ℚ) ⊗ (Π* (ΩY) ⊗ ℚ) (resp.+ (X,ℚ) ⊗ (Π* (ΩY) ⊗ ℚ)). If X is formal and Y Π-formal, then the spaces YX and\(\left( {Y, y_0 } \right)^{\left( {X, x_0 } \right)} \) are Π-formal. Furthermore, if dim Π* (ΩY) ⊗ ℚ is infinite and dim H* (Y,Q) is finite, then the sequence of Betti numbers of\(\left( {Y, y_0 } \right)^{\left( {X, x_0 } \right)} \) grows exponentially.

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Micheline Vigué-Poirrier
    • 1
  1. 1.PalaiseauFrance

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