Signature of surface bundles and bounded cohomology

Abstract

Extending a result of Morita, we show that all tautological classes of the moduli space of genus g curves are bounded. As an application, we obtain that for a surface bundle $E\to B$ over a closed surface, the Eulder characteristic $\chi(E)$ and the signature $\sigma(E)$ are related by $\vert 3 \sigma(E)\vert \leq \vert \chi(E)\vert$.