# Spherical space-form as the boundary of an Euclidean ball

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?

• 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 $\Gamma\neq 1$. – YCor Apr 30 at 22:59