# Is a manifold-with-boundary with given interior and non-empty boundary essentially unique?

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 $$N \approx M$$?

EDIT: Since there were no replies here, I asked the same question on MathOverflow. The surprising answer is no.