An orbifold partition of ${\overline{M}_g^n}$

We define a partition of ${\overline{M}_g^n}$ and show that the cohomology of ${\overline{M}_g^n}$ in a given degree admits a filtration whose respective quotients are isomorphic to the shifted cohomology groups of the parts if $g$ is sufficiently large. This implies that the map $H^k({\overline{M}_g^n}) \ra H^k(M_g^n)$ is onto and that the Hodge structure of $H^k(M_g^n)$ is pure of weight $k$ if $g \geq 2k+1$. Our main ingredient is the stability theorem of Harer and Ivanov.