Rationalized Evaluation Subgroups of a Map and the Rationalized G-sequence

Let f : X → Y be a based map of simply connected spaces. The corresponding evaluation map ω : map(X, Y ; f)→ Y induces a homomorphism of homotopy groups whose image in πn(Y ) is called the nth evaluation subgroup of f . The nth Gottlieb group of X occurs as the special case in which Y = X and f = 1X . We identify the homomorphism induced on rational homotopy groups by this evaluation map, in terms of a map of complexes of derivations constructed using Sullivan minimal models. Our identification allows for the characterization of the rationalization of the nth evaluation subgroup of f . It also allows for the identification of several long exact sequences of rational homotopy groups, including the long exact sequence induced on rational homotopy groups by the evaluation fibration. As a consequence, we obtain an identification of the rationalization of the so-called G-sequence of the map f . This is a sequence—in general not exact—of groups and homomorphisms that includes the Gottlieb groups of X and the evaluation subgroups of f . We use these results to study the G-sequence in the context of rational homotopy theory. We give new examples of non-exact G-sequences and uncover a relationship between the homology of the rational G-sequence and negative derivations of rational cohomology. We also relate the splitting of the rational G-sequence of a fibre inclusion to a well-known conjecture in rational homotopy theory.