Oriented 3-manifold fibering over two different surfaces?

Question

In a previous post I asked if a closed connected 3-manifold could have fibrations over surfaces $\Sigma_1$ and $\Sigma_2$ with $\Sigma_1 \ncong \Sigma_2$. I got the great example of $K \times S^1$ where $K$ is the Klein bottle to show that this can indeed occur, however this manifold is not orientable. Are there any orientable examples of this phenomenon?

Answer 1

Yes.

The only example of a closed connected orientable 3 manifold that fibers over two different surfaces is the lens space $ L_{4,1} $ which is a circle bundle over both $ S^2 $ and $ \mathbb{R}P^2 $.

$ L_{4,1} $ is a circle bundle over the sphere $ S^2 $, the $ n=4 $ case of of the family of bundles $$ S^1 \to L_{n,1} \to S^2 $$ But $ L_{4,1} $ is also a circle bundle over $ \mathbb{R}P^2 $ $$ S^1 \to L_{4,1} \to \mathbb{R}P^2 $$ Indeed $ L_{4,1} $ is isometric to the unit tangent bundle of the projective plane $ UT(\mathbb{R}P^2 ) $. See

https://www.maths.ed.ac.uk/~v1ranick/papers/konno.pdf

This is related to the fact that $ S^2 $ isometrically double covers $ \mathbb{R}P^2 $ while $ UT(S^2) \cong \mathbb{R}P^3 \cong L_{2,1} $ double covers $ L_{4,1} $.