On the rational homotopy type of function spaces

Abstract:

The main result of this paper is the construction of a minimal model for the function space $\mathcal {F}(X,Y)$ of continuous functions from a finite type, finite dimensional space $X$ to a finite type, nilpotent space $Y$ in terms of minimal models for $X$ and $Y$. For the component containing the constant map, $\pi _{*}(\mathcal {F}(X,Y))\otimes Q =\pi _{*}(Y)\otimes H^{-*}(X;Q)$ in positive dimensions. When $X$ is formal, there is a simple formula for the differential of the minimal model in terms of the differential of the minimal model for $Y$ and the coproduct of $H_{*}(X;Q)$. We also give a version of the main result for the space of cross sections of a fibration.