Some algebraic constructions in rational homotopy theory

Abstract

In this note we describe constructions in the category of differential graded commutative algebras over the rational numbers Q which are analogs of the space F( X, Y) of continuous maps of X to Y, the component F(X, Y,ƒ) containing ƒ ϵ F(X, Y), fibrations, induced fibrations, the space Γ(π) of sections of a fibration π: E → X, and the component Γ(π,σ) containing σ ϵ Γ (π). As a focus, we address the problem of expressing π ∗(F(X, Y, ƒ)) = Hom(π ∗(F(X,Y, ƒ)),Q) in terms of differential graded algebra models for X and Y.