The fourth tautological group of $\bar{\mathcal{M}_{g,n}}$ and relations with the cohomology

Abstract

We give a complete description of the tautological subgroup of the fourth cohomology group of the moduli space of pointed stable curves, and prove that for g \geq 8 it coincides with the cohomology group itself. We further give a conjectural upper bound depending on the genus for the degree of new tautological relations.