Surface bundles over surfaces with a fixed signature

TL;DR

This paper constructs new surface bundles over surfaces with signature 4, providing insights into the topology of 4-manifolds and improving bounds on minimal genus surfaces in mapping class groups.

Contribution

It introduces new smooth 4-manifolds with signature 4 as surface bundles over surfaces with small genera, advancing understanding of their topology.

Findings

Constructed smooth 4-manifolds with signature 4

Derived improved upper bounds for minimal genus surfaces

Enhanced understanding of surface bundle signatures

Abstract

The signature of a surface bundle over a surface is known to be divisible by 4. It is also known that the signature vanishes if the fiber genus is less than or equal to 2 or the base genus is less than or equal to 1. In this article, we construct new smooth 4-manifolds with signature 4 which are surface bundles over surfaces with small fiber and base genera. From these we derive improved upper bounds for the minimal genus of surfaces representing the second homology classes of a mapping class group.