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

Signature of surface bundles and bounded cohomology