Whitehead products in function spaces: Quillen model formulae

Abstract

We study Whitehead products in the rational homotopy groups of a general component of a function space. For the component of any based map $f:X\to Y$, in either the based or free function space, our main results express the Whitehead product directly in terms of the Quillen minimal model of $f$. These results follow from a purely algebraic development in the setting of chain complexes of derivations of differential graded Lie algebras, which is of interest in its own right. We apply the results to study the Whitehead length of function space components.