#### Volume 18, issue 4 (2018)

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The number of fiberings of a surface bundle over a surface

### Lei Chen

Algebraic & Geometric Topology 18 (2018) 2245–2263
##### 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 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 $SFib\left(M\right)=4$.

##### Keywords
surface bundle, branched cover
##### Mathematical Subject Classification 2010
Primary: 57R22, 57M50
Secondary: 57M10, 55N25