Surface bundles over surfaces: new inequalities between signature, simplicial volume and Euler characteristic

TL;DR

This paper establishes three new inequalities relating the signature, simplicial volume, and Euler characteristic of surface bundles over surfaces, with two valid generally and one specific to ramified coverings, key examples with non-zero signature.

Contribution

It introduces three new inequalities connecting key topological invariants of surface bundles, including the first known bounds for ramified covering bundles.

Findings

Two inequalities hold for all surface bundles.

One inequality applies specifically to ramified covering bundles.

Examples include the main known cases with non-zero signature.

Abstract

We present three new inequalities tying the signature, the simplicial volume and the Euler characteristic of surface bundles over surfaces. Two of them are true for any surface bundle, while the third holds on a specific family of surface bundles, namely the ones that arise through a ramified covering. These are the main known examples of bundles with non-zero signature.