Reference request: Quotients of $S^3=SU(2)$ by finite subgroups

Question

The finite subgroups of $S^3=SU(2)$ are the cyclic groups and the binary dihedral, tetrahedral, octahedral, icosahedral groups (the inverse images of the finite subgroups of $SO(3)$ under the double cover $S^3\to SO(3)$). Is there a reference about topologies of the quotients $S^3/G$ where $G\subset S^3$ is a finite subgroup? I know that if $G$ is cyclic then $S^3$ is a lens space , but I have no information about $S^3/G$ for other finite subgroups $G$. Do these spaces have names? or are there spaces familiar spaces?

Answer 1

You may think about $S^3$ as the unit quaternions. Then, it is clear since quaternions multiplication respects the Euclidean norm on $\mathbb{H}=\mathbb{R}^4$, that a subgroup will act on $S^3$ preserving the usual round metric of $S^3$.

Next, if the action is free, then the quotient space will be a $3$-manifold (note that the finiteness of the group gives automatically that the action is properly discontinuous). Because the group action preserves the metric the quotient manifold may be given a metric which is locally isometric to $S^3$, or equivelantly with sectional curvature $1$ everywhere.

There is a name for such a manifold, a spherical $3$-manifold. They are classified completely, the wikipedia page gives a very nice exposition of this https://en.wikipedia.org/wiki/Spherical_3-manifold. There are four families in addition to Lens spaces, corresponding to the finite groups you mention dihedral, tetrahedral, octahedral, icosahedral.

In the last 3 cases, these $3$-manifolds are uniquely determined by there fundamental groups, which perhaps makes them less well-known than Lens spaces. The fact that two Lens spaces can be homotopy equivelant but not homeomorphic makes them a perfect testing ground for new $3$-manifold invariants, hence their relative fame.

The case of icosahedral spherical $3$-manifold however has a special place in the history of the study of $3$-manifolds. One of them is the ingenious counterexample of Poincaré to his first formulation of the Poincare conjecture. It has the same integral homology as $S^3$, in particular the abelianization of the icosahedral group is the trivial group.