Signature of surface bundles and bounded cohomology

TL;DR

This paper proves that tautological classes in the moduli space of genus g curves are bounded and establishes a new inequality relating the Euler characteristic and signature of surface bundles over surfaces.

Contribution

It extends Morita's result to show boundedness of tautological classes and derives a novel inequality linking Euler characteristic and signature of surface bundles.

Findings

Tautological classes are bounded in the moduli space of genus g curves.

The inequality |3σ(E)| ≤ |χ(E)| holds for surface bundles over surfaces.

Provides a new connection between topological invariants of surface bundles.

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 $\vert 3 \sigma(E)\vert \leq \vert \chi(E)\vert$.