Sur l'homotopie rationnelle des espaces fonctionnels

Abstract

Let X be a nilpotent space such that it exists k⩾1 with H^p (X,ℚ) = 0 p > k and H^k (X,ℚ) ≠ 0, let Y be a (m−1)-connected space with m⩾k+2, then the rational homotopy Lie algebra of Y^X (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 Y^X 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.