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Let $M$ be a smooth manifold and let $f_0, f_1 \colon [0, 1] \times \partial M \to M$ be two smooth embeddings that are the identity map on $\partial M \times\{0\} = \partial M$ . If $\partial M$ is non-compact does there exist a diffeotopy $H \colon M × [0, 1] \to M$ relative to $\partial M$ from $H_0=\text{id}_M$ to a diffeomorphism $H_1\colon M\to M$ with $H_1\circ f_0=f_1$?

The case for compact boundary can be found in most textbooks. Can someone suggest to me some references for the non-compact boundary case?

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    $\begingroup$ As stated the answer is no. You also need to assume that the image of the collaring maps is closed. Otherwise it is fairly straightforward to find counterexamples. I also spent some time trying to find a reference for this statement, but I was not successful either. $\endgroup$
    – Stefan Friedl
    Commented Apr 25, 2021 at 14:40
  • $\begingroup$ Maybe the boundary needs to be slightly stronger than paracompact? $\endgroup$
    – Shijie Gu
    Commented Apr 25, 2021 at 16:44

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