The rational homotopy Lie algebra of function spaces
Denote by \({\mathcal F}(X,Y)\) (resp. \({\mathcal F}_{*}(X,Y)\)) the space of free (resp. based) maps between nilpotent complexes \(X\) and \(Y\), with \(X\) a finite complex and \(Y\) a complex of finite type over the rationals \({\mathbb Q}\). The assumptions on \(X\) and \(Y\) assure that all the components of both \({\mathcal F}(X,Y)\) and \({\mathcal F}_{*}(X,Y)\) are nilpotent CW-complexes of finite type over \({\mathbb Q}\) and hence can be rationalized in the classical sense. The authors give a complete description of the rational homotopy Lie algebra of any component of the function spaces \({\mathcal F}(X,Y)\) and \({\mathcal F}_{*}(X,Y)\). They also give a full and explicit description of the Lie bracket on the rational homotopy Lie algebras of these spaces as well as of higher order Whitehead products in them. As an application they deduce the existence of \(H\)-space structures on a given component of a pointed mapping space \({\mathcal F}_{*}(X,Y;f)\) between rational spaces, assuming that the cone length of \(X\) is smaller than the order of any non trivial generalized Whitehead product in \({\pi}_{*}(Y)\). The paper is well written with due acknowledgement to previous work in this direction.
scientific article; zbMATH DE number 5361850