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 $|3\sigma (E)|\le |\chi (E)|$.

26 pages