  Let f: X -> Y be a based map of simply connected spaces. The corresponding
evaluation map w: map(X,Y;f) -> Y induces a homomorphism of homotopy groups
whose image in pi_n(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 = 1_X. 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.
