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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 $N \approx M$?

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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 $P\times I$.

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  • $\begingroup$ Thank you. I do not yet understand the answer; I will have to read about Whitehead torsion tomorrow. Would the same answer apply if $M$ and $N$ were orientable? $\endgroup$
    – kaba
    Commented Apr 15, 2019 at 23:45
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    $\begingroup$ Yes, it has nothing to do with orientability. $\endgroup$
    – Tom Goodwillie
    Commented Apr 16, 2019 at 0:30
  • $\begingroup$ How do I find a compact connected manifold $P$ with a non-trivial Whitehead group $Wh(\pi_1(P))$? $\endgroup$
    – kaba
    Commented Apr 24, 2019 at 20:22
  • $\begingroup$ You can use a lens space. Many finite cyclic groups have nontrivial Whitehead groups. $\endgroup$
    – Tom Goodwillie
    Commented Apr 24, 2019 at 22:35

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