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Thom's first isotopy lemma says that given $f:M\to P$ a smooth map between smooth manifolds and a closed Whitney stratified subset $S$ of $M$, such that $f|_S:S\to P$ is proper and $f|_X:X\to P$ is a submersion for any stratum $X$ of $S$, then $f|_S:S\to f(S)$ is a locally trivial fibration. Does this imply that $f|_X:X\to f(X)$ is a locally trivial fibration, for any stratum $X$ of $S$?

Edit (Jan 03, 2024): Reviving this post. Kindly refer me to some works where the above lemma is used to prove fibrations in some concrete examples, which are not proven using some other method.

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  • $\begingroup$ I must admit, I'm struggling a bit with your notation. ... $\endgroup$
    – user102126
    Commented Jul 5, 2018 at 9:28
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    $\begingroup$ The notation is clear. $\endgroup$
    – Ben McKay
    Commented Jul 5, 2018 at 9:44
  • $\begingroup$ Instead of editing in a new question to an old post, you should post the new question separately. $\endgroup$
    – Sam Hopkins
    Commented Jan 3 at 17:33

2 Answers 2

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Yes. Notice that since $f|_S: S \to S$ is proper, so is $f|_X$ (because $X$ is closed). Furthermore, $f|_X$ is a submersion and is surjective onto its image. So by a direct application of Ehresmann's fibration lemma you get your result.

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  • $\begingroup$ I can use Ehresmann's fibration theorem if X is compact... $\endgroup$
    – RKS
    Commented Sep 4, 2018 at 2:40
  • $\begingroup$ You don't need X to be compact to use Ehresmann's fibration theorem. You need $f|_X$ to be a submersion (which is by hypothesis) and you need $f|_X$ to be proper (have a look at the link in the answer). Equivalently, you need that $f|_{S}^{-1}(p) \cap X= f|_X^{-1}(p)$ is a compact for all $p$ in $f(X)$. But this is true because $f|_S$ is proper, so $f|_S^{-1}(p)$ is a compact in $S$ and the intersection of a compact set with a closed set is a compact set. $\endgroup$
    – Paul
    Commented Sep 4, 2018 at 4:25
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Also note that because any fibration is also a compactification of a subset of all strata, we can always divide any subsets of these strata into an infinite number of compactified stratified and homotopically self similar spaces. Also note that the isotopy from strata 1 to strata 2 is a homotopy as well.

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