  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.
