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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.

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  • $\begingroup$ A Hurewicz fibration is always a Serre fibration by definition, regardless of what category of spaces you work in. This is tautological. $\endgroup$
    – Tyrone
    Commented Sep 23, 2019 at 12:25
  • $\begingroup$ @Tyrone Yh that is true.. $\endgroup$ Commented Sep 23, 2019 at 12:26
  • $\begingroup$ Perhaps I am confused, since my comment above seems to answer your question completely. $\endgroup$
    – Tyrone
    Commented Sep 23, 2019 at 12:29
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    $\begingroup$ 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. $\endgroup$
    – Tyrone
    Commented Sep 23, 2019 at 13:17
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    $\begingroup$ 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. $\endgroup$
    – Tyrone
    Commented Sep 23, 2019 at 13:22

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