  Simple boundary expressions for the k-th power of the cotangent line class on
the moduli space of stable 1-pointed genus g curves are found for k >= 2g. The
method is by virtual localization on the moduli space of maps to the projective
line. As a consequence, nontrivial tautological classes in the kernel of the
push-forward map associated to the irreducible boundary divisor of the moduli
space of stable g+1 curves are constructed. The geometry of genus g+1 curves
then provides universal equations in genus g Gromov-Witten theory. As an
application, we prove all the Gromov-Witten identities conjectured recently by
K. Liu and H. Xu.
