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It is known by Ehresmann's result, that proper surjective submersions are fiber bundles. The properness of a map somehow relates to the compactness of the fibers or the level sets. So my question is as follows:

Is there an example of a differentiable map that is a submersion between differentiable manifolds $f: M \to N$ such that all level sets are diffeomorphic but $f$ is not a fiber bundle?

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    $\begingroup$ Yes, I have answered this several times on MO. Remove from a projective line bundle two sections that intersect. Over the divisor in the base where they intersect, remove a disjoint section. $\endgroup$
    – Jason Starr
    Commented Nov 15, 2021 at 10:41
  • $\begingroup$ @JasonStarr can you share the link? $\endgroup$
    – piper1967
    Commented Nov 15, 2021 at 10:43
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    $\begingroup$ An interesting special case when a submersion $f$ is a fiber bundle: All fibers are diffeomorphic to $R^k$ for some fixed $k$. See: G. Meigniez, Submersions, fibrations and bundles. Trans. Amer. Math. Soc. 354 (2002), no. 9, 3771–3787. $\endgroup$
    – Moishe Kohan
    Commented Nov 15, 2021 at 13:34
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    $\begingroup$ @JasonStarr: Yes, I noticed this. Maybe these are vector bundles in the holomorphic category. Unfortunately, Fischer and Grauert require properness to get from a smooth bundle to a holomorphic bundle. $\endgroup$
    – Moishe Kohan
    Commented Nov 16, 2021 at 16:11
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    $\begingroup$ In case you want to have more examples: There are surjective entire functions ${\mathbb C}\to {\mathbb C}$ without critical points, see here. It is easy to see that fibers of these maps are countably infinite. But also, clearly, these are not covering maps, hence, not fiber bundles. $\endgroup$
    – Moishe Kohan
    Commented Nov 16, 2021 at 16:17

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