# Question about nested manifolds with boundary which are diffeomorphic

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.

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.
• You're welcome. I'm still wondering about the case when $\partial N$ has to be connected though. – Arnaud Mortier Jun 12 '18 at 23:22