Sur l'homotopie rationnelle des espaces fonctionnels
The author studies the rational homotopy type of the (pointed) function spaces \((Y,y_ 0)^{(X,x_ 0)}\) and \(Y^ X\) where: X is a nilpotent (pointed) space such that there exists \(k\geq 1\) with \(H^ p(X; {\mathbb{Q}})=0\), \(p>k\) and \(H^ k(X; {\mathbb{Q}})\neq 0\), Y is an (m-1)-connected (pointed) space with \(m\geq k+2\). She shows that the rational homotopy Lie algebra of \((Y,y_ 0)^{(X,x_ 0)}\) is isomorphic (as Lie algebra) to \(H^+(X; {\mathbb{Q}})\otimes (\pi_*(\Omega Y)\otimes {\mathbb{Q}})\). She also finds sufficient conditions for the exponential growth of the sequence of Betti numbers. The context is the theory of minimal model [\textit{D. Sullivan}, Publ. Math., Inst. Hautes Étud. Sci. 47, 269-331 (1977; Zbl 0374.57002)]. More precisely, the main tool is the Haefliger's model for \(Y^ X\) [\textit{A. Haefliger}, Lect. Notes Math. 484, 121-152 (1975; Zbl 0316.57009; Trans. Am. Math. Soc. 273, 609-620 (1982; Zbl 0508.55019)] soon used by \textit{K. Shibata} for the study of Gel'fand-Fuchs cohomology [Jap. J. Math., New Ser. 7, 379-415 (1981; Zbl 0525.57025)].
scientific article; zbMATH DE number 3962749
scientific article