The Poincaré sphere is a “fake” 3-dimensional sphere: it is a smooth 3-dimensional manifold which has the same homology as the usual 3-sphere, but which is not homeomorphic to it.
Specifically, it is obtained as where is the icosahedral group of rotations (the subgroup of rotations which stabilize an icosahedron).
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.