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?

4$\begingroup$ Did you want to say that $M$ is compact? $\endgroup$ – Anton Petrunin Jun 19 '16 at 20:55

1$\begingroup$ Consider closed interval $[0,1]$. It's interior is topologically equivalent to the real line. $\endgroup$ – Vít Tuček Jun 19 '16 at 20:59

2$\begingroup$ 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

6$\begingroup$ This is basically the same as the following previous question: mathoverflow.net/q/81714/317 $\endgroup$ – Andy Putman Jun 19 '16 at 21:06