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In this paper, we give some new genus-3 universal equations for Gromov-Witteninvariants of compact symplectic manifolds. These equations were obtained bystudying new relations in the tautological ring of the moduli space of2-pointed genus-3 stable curves. A byproduct of our search for genus-3equations is a new genus-2 universal equation for Gromov-Witten invariants.

Topological Recursion Relations on $\bar{\cal M}_{3,2}$