The number of fiberings of a surface bundle over a surface

Abstract

For a closed manifold $M$, let $SFib\left(M\right)$ be the number of ways that $M$ can be realized as a surface bundle, up to ${\pi }_1$–fiberwise diffeomorphism. We consider the case when $dim\left(M\right)=4$. We give the first computation of $SFib\left(M\right)$ where $1<SFib\left(M\right)<\infty$ but $M$ is not a product. In particular, we prove $SFib\left(M\right)=2$ for the Atiyah–Kodaira manifold and any finite cover of a trivial surface bundle. We also give an example where $SFib\left(M\right)=4$.