  We show that any degree at least $g$ polynomial in descendant or tautological
classes vanishes on $M_{g,n}$ when $g\ge 2$. This generalizes a result of
Looijenga and proves a version of Getzler's conjecture. The method we use is
the study of the relative Gromov-Witten invariants of $P^1$ relative 2 points
combined with the degeneration formulas of [IP1]. At the end of the paper, we
also included a quick proof of a very recent conjecture made by Vakil.
