I would like to know if there are compact (n-1)-manifolds $N$ that are not spheres but such that there is a manifold with boundary $M$ which satisfies the following two properties:


* $\partial M\cong N$
* $M-\partial M\cong \mathbb{R}^n$


I am primarily interested in this question in the category of smooth manifolds but I would be interested to know the answer in the topological case as well.



What manifolds are boundaries of euclidian spaces ?
$N$ has to be a homotopy-sphere. So as long as it's dimension isn't $4$, there's a proof that it has to be the standard $S^{n-1}$. 


These arguments appear in the Kosinski book on smooth manifolds. The basic idea goes like this. 


1) $N$ is simply connected. There's the inclusion map $N \to M$, but if $p \in int(M)$ then there's also a retraction map $M \setminus \{p\} \to N$. Provided $n \geq 3$ removal of a point does not affect the fundamental group. 


2) $N$ is a homology sphere by Alexander/Poincare duality of the pair $(M,N)$. Part (1) technically gives us this as well but this argument works even if $M$ is not contractible. 


So by the Whitehead theorem, $N$ is a homotopy sphere. Moreover, $M$ is a contractible manifold whose boundary is a homotopy sphere. So $M$ is a disc by the h-cobordism theorem, and the disc has a unique smooth structure (not known in dimension $5$ still). 


So that resolves all cases except $dim(N)=4$. 


