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This question already has an answer here:

Suppose I keep in my pocket a manifold with boundary $M$ , and I provide you access to $int M := M \setminus \partial M$ up to homeomorphism/diffeomorphism. What can you deduce about $\partial M$? can you reconstruct it up to homotopy/homeomorphism/diffeomorphism? can you deduce homotopical / topologial / smooth invariants of it?

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marked as duplicate by Anton Petrunin, Douglas Zare, Alex Degtyarev, Fernando Muro, Sebastian Goette Jun 20 '16 at 6:56

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ Did you want to say that $M$ is compact? $\endgroup$ – Anton Petrunin Jun 19 '16 at 20:55
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    $\begingroup$ Consider closed interval $[0,1]$. It's interior is topologically equivalent to the real line. $\endgroup$ – Vít Tuček Jun 19 '16 at 20:59
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    $\begingroup$ Adding boundaries to open manifolds has been discussed at least in these previous questions: 22441, 34602, 81714, 83356, 158391. So in a sense this is a duplicate. $\endgroup$ – Igor Khavkine Jun 19 '16 at 21:02
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    $\begingroup$ This is basically the same as the following previous question: mathoverflow.net/q/81714/317 $\endgroup$ – Andy Putman Jun 19 '16 at 21:06