The rational homotopy Lie algebra of function spaces

Abstract

In this paper we fully describe the rational homotopy Lie algebra of any component of a given (free or pointed) function space. Also, we characterize higher order Whitehead products on these spaces. From this, we deduce the existence of $H$-structures on a given component of a pointed mapping space ${\mathcal{F}}_{\ast }(X,Y;f)$ between rational spaces, assuming the cone length of $X$ is smaller than the order of any non trivial generalized Whitehead product in ${\pi }_{\ast }(Y)$.