# The number of fiberings of a surface bundle over a surface

## TL;DR

This paper computes the number of distinct surface bundle structures on certain 3-manifolds, revealing cases where this number is finite and greater than one, including explicit examples with two and four fiberings.

## Contribution

It provides the first explicit calculations of Fib$(M)$ for non-product manifolds with finite fiberings, including the Atiyah-Kodaira manifold.

## Key findings

- Fib$(M)=2$ for the Atiyah-Kodaira manifold
- Fib$(M)=2$ for finite covers of trivial surface bundles
- An example with Fib$(M)=4$

## Abstract

For a closed manifold $M$, let Fib$(M)$ be the number of distinct fiberings of $M$ as a fiber bundle with fiber a closed surface. In this paper we give the first computation of Fib$(M)$ where $1<\text{Fib}(M)<\infty$ but $M$ is not a product. In particular, we prove Fib$(M)=2$ for the Atiyah-Kodaira manifold and any finite cover of a trivial surface bundle. We also give an example where Fib$(M)=4$.