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 F ( X , Y ) \mathcal {F}(X,Y) of continuous functions from a finite type, finite dimensional space X X to a finite type, nilpotent space Y Y in terms of minimal models for X X and Y Y . For the component containing the constant map, π ∗ ( F ( X , Y ) ) ⊗ Q = π ∗ ( Y ) ⊗ H − ∗ ( X ; Q ) \pi _{*}(\mathcal {F}(X,Y))\otimes Q =\pi _{*}(Y)\otimes H^{-*}(X;Q) in positive dimensions. When X 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 Y and the coproduct of H ∗ ( X ; Q ) H_{*}(X;Q) . We also give a version of the main result for the space of cross sections of a fibration.