Topological recursive relations in $H^{2g}(M_{g,n})$

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.