let $M$ be a smooth $n$-manifold with boundary $\partial M$; I denote by $M^o$ the internal part of $M$, that is $M \smallsetminus \partial M$.
The question is the same as in the title: let $M$ and $N$ be two compact orientable topological manifolds such that $M^o$ is homeomorphic to $N^o$. Does this imply that $M$ is homeomorphic to $N$? Can we say something about the connected components of the boundary?


I think that this question should have a really easy answer, but I cannot find it. One thing that I could prove is that the Euler characteristic of $\partial M$ must be the same as the one of $\partial N$, this being an easy consequence of the fact that $M^o$ is homotopically equivalent to $M$. The next step i tried is to prove that the sequence of Euler characteristics of the boundary components must agree:
$$ ((\partial M)\_1, \ldots , (\partial M)\_r) = ((\partial N)\_1, \ldots , (\partial N)\_r) ;$$
but I do not see a way to prove this. I am specially interested in the case of 3-manifolds, and from this one could conclude that $\partial M$ is homeomorphic to $\partial N$, because Euler characteristic identifies closed surfaces. Do you have any tip or any simple proof?



Does $M^o=N^o$ imply that $\partial M = \partial N$?
In general, this is wrong. 


Take $M=L(7,1)\times S^{2n}\times[0,1]$ and $N=L(7,2)\times S^{2n}\times[0,1]$. 


Their boundaries $L(7,1)\times S^{2n}$ and $L(7,2)\times S^{2n}$ are not homeomorphic but $h$-cobordant, as proven by Milnor in



> 
> J. Milnor, 
>  Two complexes which are homeomorphic but combinatorially distinct.
>  Ann. of Math. (2) **74** (1961) 575–590. 
> 
> 
> 


It follows that $\mathring{M}=L(7,1)\times S^{2n}\times \mathbb R=L(7,2)\times S^{2n}\times \mathbb R=\mathring{N}$.


However, in dimension $3$ this should be true, following from work of Edwards:



> 
> C. Edwards,
>  Concentricity in 3-manifolds.
>  Trans. Amer. Math. Soc. **113** (1964) 406–423.
> 
> 
> 


He showed that two compact $3$-manifolds are homeomorphic if and only if their interiors are homeomorphic. A main ingerdient in his proof is (as you said) that oriented $2$-manifolds are determined by their Euler characteristic.


