Let $M$ be a compact connected manifold-with-boundary such that $\circ M \neq \emptyset$, where $\circ M$ is the boundary of $M$. Let $N$ be a compact connected manifold-with-boundary such that $\circ N \neq \emptyset$ and $\bullet M \approx \bullet N$, where $\bullet M$ denotes the interior of $M$ and $\approx$ denotes homeomorphic. Does it necessarily hold that $N \approx M$?


(I have asked this question before [here](https://math.stackexchange.com/q/3169301/51161), but there were no replies.)



Is a manifold-with-boundary with given interior and non-empty boundary essentially unique?
No, there are examples detected by Whitehead torsion. If $P$ is a compact connected $(n-1)$-manifold with empty boundary, then (assuming $n\ge 6$) for every element $\tau$ of the Whitehead group of $\pi\_1(P)$ there is an $h$-cobordism $M$ on $P$ such that $\tau$ is the Whitehead torsion of the pair $(M,P)$. The interior of $M$ will be isomorphic to $P\times\mathbb R$, but if $\tau$ is nontrivial then $M$ will not be isomorphic to $P\times I$.


