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A Serre fibration is clearly defined with motivation from homotopy theory, but we can consider smooth versions $f\colon M\to N$ in the category of (finite-dimensional, paracompact etc) smooth manifolds, namely any smooth homotopy $H\colon I^n\times I \to N$ and smooth map $I^n\times\lbrace0\rbrace \to M$ lifting $H(-,0)$, there is a smooth lift of $H$ to $M$. For $N$ connected and $M$ non-empty, it is necessarily surjective.

Clearly smooth fibre bundles are Serre fibrations, but arbitrary surjective submersions are not (for example: $[0,\frac{2}{3})\coprod (\frac{1}{3},1] \to [0,1]$). However, I have reasons to consider fibrations with connected fibre.

I have a rough sketch of a proof in my head that smooth Serre fibrations are not far off being submersions, but that may just be the result of an overactive imagination.

I'd like to know what's the ugliest that a smooth Serre fibration with connected fibres can be. Single examples are welcome, as are characterisation theorem.

Secondly, what sort of fibres do we get? (Clearly they just might not be manifolds)

Conversely, what nice properties are smooth Serre fibrations with connected fibres guaranteed to have?

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# Given a Serre fibration between manifolds, how ugly can it be?

A Serre fibration is clearly defined with motivation from homotopy theory, but we can consider smooth versions $f\colon M\to N$ in the category of (finite-dimensional, paracompact etc) smooth manifolds, namely any smooth homotopy $H\colon I^n\times I \to N$ and smooth map $I^n\times\lbrace0\rbrace \to M$ lifting $H(-,0)$, there is a smooth lift of $H$ to $M$. For $N$ connected and $M$ non-empty, it is necessarily surjective.

Clearly smooth fibre bundles are Serre fibrations, but arbitrary surjective submersions are not (for example: $[0,\frac{2}{3})\coprod (\frac{1}{3},1] \to [0,1]$).

I have a rough sketch of a proof in my head that smooth Serre fibrations are not far off being submersions, but that may just be the result of an overactive imagination.

I'd like to know what's the ugliest that a smooth Serre fibration can be. Single examples are welcome, as are characterisation theorem.

Secondly, what sort of fibres do we get? (Clearly they just might not be manifolds)

Conversely, what nice properties are smooth Serre fibrations guaranteed to have?