The universal cover of the Poincaré sphere is the standard 3-sphere. This is exhibited in the following diagram:
where the double cover is the universal cover of , and the double cover of is called the binary icosahedral group. It follows that , and since is a perfect group, its abelianization is trivial, and therefore induces an isomorphism in first homology. It induces an isomorphism in homotopy groups for by an easy computation involving the long exact homotopy sequence applied to the top row.