Rational homotopy of the space of self-maps of complexes with finitely many homotopy groups

Abstract

For simply connected CW complexes X with finitely many, finitely generated homotopy groups, 1 ^{1} the path components of the function space M ( X , X ) M(X,X) of free self-maps of X are all of the same rational homotopy type if and only if all the k-invariants of X are of finite order. In case X is rationally a two-stage Postnikov system the space M 0 ( X , X ) {M_0}(X,X) of inessential self-maps of X has the structure of rational H-space if and only if the k-invariants of X are of finite order.