Whitehead products in function spaces: Quillen model formulae

TL;DR

This paper provides algebraic formulas for Whitehead products in the rational homotopy groups of function space components, linking them to Quillen models and derivation complexes.

Contribution

It introduces explicit Quillen model formulae for Whitehead products in function spaces, advancing the algebraic understanding of their rational homotopy groups.

Findings

Formulas for Whitehead products in terms of Quillen models

Algebraic development in chain complexes of derivations

Application to Whitehead length of function space components

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.