Let $I$ be the group of orientation preserving symetries of the dodecahedron. I've been tasked to proving that:
- $SO(3)/I$ is a manifold
- The first homology group of $SO(3)/I$ is trivial yet its fundamental group isn't
The teacher left as a hint to try finding the universal cover of $SO(3)$ and using its properties to deduce the results.
I'm aware of the result that if a compact topological group $G$ acts in a properly discontinuous manner on a space $X$, then $X$ being Hausdorff implies $X/G$ is also Hausdorff. Since I is finite it is compact and the fact that $SO(3) \to SO(3)/I$ is a covering and $SO(3)$ is connected I believe to already be enough to conclude that $SO(3)$ is a manifold $\textbf{if I manage to prove SO(3) is a manifold}$. But that is the problem, I've not managed to prove $SO(3)$ is a manifold.
If we find a universal cover that is a manifold, since we know $SO(3)$ is Hausdorff that would already be enough to conclude $SO(3)$ is a manifold as well. But right now I'm kinda lost on how to do it.
For context I'm studying via Lee's "Introduction to Topological Manifold". Any help would be appreciated.
