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 $SO(3)/I$ where $I$ is the icosahedral group of rotations (the subgroup of rotations which stabilize an icosahedron).

The universal cover of the Poincaré sphere $S$ is the standard 3-sphere. This is exhibited in the following diagram:

where the double cover $\pi$ is the universal cover of $SO(3)$, and the double cover $B$ of $I$ is called the binary icosahedral group. It follows that $\pi_1(S) \cong B$, and since $B$ is a perfect group, its abelianization is trivial, and therefore $p$ induces an isomorphism in first homology. It induces an isomorphism in homotopy groups $\pi_i$ for $i \geq 2$ by an easy computation involving the long exact homotopy sequence applied to the top row.