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.