  The moduli space of stable relative maps to the projective line combines
features of stable maps and admissible covers. We prove all standard
Gromov-Witten classes on these moduli spaces of stable relative maps have
tautological push-forwards to the moduli space of curves. In particular, the
fundamental classes of all moduli spaces of admissible covers push-forward to
tautological classes. Consequences for the tautological rings of the moduli
spaces of curves include methods for generating new relations, uniform
derivations of the socle and vanishing statements of the Gorenstein conjectures
for the complete, compact type, and rational tail cases, tautological boundary
terms for Ionel's, Looijenga's, and Getzler's vanishings, and applications to
Gromov-Witten theory.
