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From https://pdfs.semanticscholar.org/e737/a4f8b93242910c050c2faf761236dcf60f64.pdf (Theorem 2.1) it follows:

(*) If $\pi:P\rightarrow M$ is a topological fibre bundle over a paracompat hausdorff topological space $M$ then it is a Hurewicz fibration and hence a Serre fibration. (In the category of topological spaces)

I found the analogous notion of homotopy lifting property and Serre/Hurewiz fibration in the category of smooth manifolds here When is a fibration a fiber bundle?.

My question is the following:

Does (*) holds in the category of smooth manifolds also?

I got an affirmative answer at least for Serre fibration in https://mathoverflow.net/questions/116231/given-a-serre-fibration-between-manifolds-how-ugly-can-it-be (where in the question it is mentioned "Clearly smooth fibre bundles are Serre fibrations" by @David Roberts.) But (no proof or any reference where such proof is discussed) is mentioned in the question.

So can anyone please give a proof of (*) or suggest any reference where such proof is discussed (at least for Serre fibration)?

Also it would be very helpful if someone can suggest some references or literature resources where the notion of Serre/Hurewicz fibration in the category of smooth manifolds is discussed.

Thanks in Advance.

Adittya Chaudhuri
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  • A Hurewicz fibration is always a Serre fibration by definition, regardless of what category of spaces you work in. This is tautological. – Tyrone Sep 23 '19 at 12:25
  • @Tyrone Yh that is true.. – Adittya Chaudhuri Sep 23 '19 at 12:26
  • Perhaps I am confused, since my comment above seems to answer your question completely. – Tyrone Sep 23 '19 at 12:29
  • @Tyrone I asked for reference for smooth fibration theory and I asked whether in that theory are locally trivial smooth fibre bundles Serre\Hurewicz smooth fibrations ? I already know that this statement is true when we consider locally trivial topological fibre bundle...so your comment does not answer my question . – Adittya Chaudhuri Sep 23 '19 at 13:01
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    Any locally trivial fibration is a Hurewicz fibration. You can basically just construct lifts locally and then patch them together. If your local data is smooth, and your change of charts are smooth, then your output lift is smooth. Now you have a (smooth) Hurewicz fibration, so you also have a (smooth) Serre fibration. – Tyrone Sep 23 '19 at 13:17
  • @Tyrone Thanks. Do you know any reference where fibration in the category of smooth manifolds is mentioned? – Adittya Chaudhuri Sep 23 '19 at 13:19
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    A paper you might enjoy is this one by G. Meigniez. Note that what the author calls fibrations are Hurewicz fibrations. Bonus points: the paper has some really cool counterexamples in it. – Tyrone Sep 23 '19 at 13:22
  • @Tyrone Thanks for sharing. I will surely look into it. – Adittya Chaudhuri Sep 23 '19 at 13:24

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