Can a 3-manifold fiber over two different surfaces?

Question

Let $M$ be a closed connected 3-manifold and let $\Sigma$ and $\Sigma'$ be two closed connected (not necessarily orientable) surfaces. Could there be fiber bundles $\phi : M \to \Sigma$ and $\phi': M \to \Sigma'$ with $\Sigma \ncong \Sigma'$?

Answer 1

Yes, it can. Consider $M=S^1 \times K^2$ where $K^2$ is the Klein bottle. This manifold fibers over both $T^2$ (since $K^2$ fibers over $S^1$) and over $K^2$. Of course, if the base of the fibration is required to have negative Euler characteristic, then $M$ cannot fiber over nonhomeomorphic surfaces.

Answer 2

No. You can check look at my old paper https://arxiv.org/abs/1106.4595, and look at the argument following Theorem 3.9. In the case you are interested in, this reduces to the statement that the fundamental group of a compact surface has no finite or infinite cyclic normal subgroup.