An orbifold partition of $\bar M_g^n$

Abstract

We define a partition of $\bar M_g^n$ and show that the cohomology of $\bar M_g^n$ 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 ${H^k}(\bar M_g^n) \to {H^k}(M_g^n)$ is onto and that the Hodge structure of H ^k(M ⁿ _g ) is pure if g ≥ 2k +1. The main ingredient is the Stability Theorem of Harer and Ivanov.