Let $M^n$ be a smooth compact manifold such that the boundary $\partial M$ is diffeomorphic to a spherical space-form $S^{n-1}/\Gamma$, where $\Gamma \subset O(n)$ is a finite subgroup acting freely on $S^{n-1}$.

If the interior part $M-\partial M$ is diffeomorphic to $\mathbb R^n$, can we prove that $\Gamma$ must be trivial?

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    $\begingroup$ If $n=2$ of course not. If $n\ge 3$, yes. This is because $\mathbf{R}^n$ is simply connected at infinity, so whenever it's homeomorphic to the interior of a topological manifold with boundary, this boundary has a single component and is simply connected, excluding $\Gamma\neq 1$. $\endgroup$ – YCor Apr 30 at 22:59

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