, Volume 56, Issue 2, pp 177-191

Sur l'homotopie rationnelle des espaces fonctionnels

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Abstract

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.