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Suppose $M, N$ are compact manifolds with boundary, with $M \subset \text{int}(N)$, and assume that $M$ is diffeomorphic to $N$.

Is $N\setminus M$ automatically diffeomorphic to $\partial N \times [0,1]$? If not, what is a counterexample?

I suspect that the answer to the first question is negative, but I haven't managed to come up with a counterexample despite drawing a lot of pictures, albeit two-dimensional ones. I also attempted to prove that the question has a positive answer, by following the gradient flow of an appropriate smooth function on $N$, but I don't know how to guarantee that such a function has no critical points, and this seems to spoil my proof idea.

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One counterexample is an annulus $N=S^1\times [0,1]$ (think of a flat disk with a smaller open disk removed), and $M$ lying inside a small disk in $N$, therefore not boundary-parallel.

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  • $\begingroup$ That's an easy one :). Thanks a lot...I was making things too complicated. $\endgroup$ – Matthew Kvalheim Jun 12 '18 at 23:19
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    $\begingroup$ You're welcome. I'm still wondering about the case when $\partial N$ has to be connected though. $\endgroup$ – Arnaud Mortier Jun 12 '18 at 23:22
  • $\begingroup$ Good point. I'm also curious about that case. $\endgroup$ – Matthew Kvalheim Jun 12 '18 at 23:27

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