Journal of the
Korean Mathematical Society JKMS

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 $\leq 2$ or the base genus $\leq 1$. 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.

Keywords: surface bundle, mapping class group, signature, Lefschetz fibration

MSC numbers: Primary 57R22, 57R55, 20F12, 57M07