# 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 $$N \approx M$$?

(I have asked this question before here, but there were no replies.)

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 $$P\times I$$.
• 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? – kaba Apr 15 at 23:45
• How do I find a compact connected manifold $P$ with a non-trivial Whitehead group $Wh(\pi_1(P))$? – kaba Apr 24 at 20:22