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— Let X be an n-connected elliptic space and Y a non rationally contractible, finite-type, g-dimensional CW complex, where q < n. We show that the function space X of free maps from Y into X is formal if and only if the rational cohomology algebra H*[X',Q) is free, that is, X has the rational homotopy type of a product of odd dimensional spheres. RESUME. — FORMALITE DES ESPACES DE FONCTIONS LIBRES DANS UN ESPACE ELLIPTIQUE. — Soient X un espace elliptique n-connexe et Y un CW complexe non rationnellement contractile, de type fini et de dimension q < n. Nous montrons que Pespace X des fonctions libres de Y dans X est formel si et seulement si Palgebre H*(X,Q) est libre, i.e. X a Ie type d'homotopie rationnelle d'un produit de spheres de dimensions impaires.

Formality of the function space of free maps into an elliptic space