Tautological and non-tautological cohomology of the moduli space of curves

Abstract

After a short exposition of the basic properties of the tautological ring of Mg,n, we explain three methods of detecting non-tautological classes in cohomology. The first is via curve counting over finite fields. The second is by obtaining length bounds on the action of the symmetric group n on tautological classes. The third is via classical boundary geometry. Several new non-tautological classes are found.