# nLab Poincaré sphere

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 $\mathrm{SO}\left(3\right)/I$ 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:

$\begin{array}{ccccc}B& ↪& \mathrm{SU}\left(2\right)\cong {S}^{3}& \stackrel{p}{\to }& S\\ ↓& & ↓\pi & & ↓\cong \\ I& ↪& \mathrm{SO}\left(3\right)& \to & \mathrm{SO}\left(3\right)/I\end{array}$\array{ B & \hookrightarrow & SU(2) \cong S^3 & \stackrel{p}{\to} & S \\ \downarrow & & \downarrow \pi & & \downarrow \cong \\ I & \hookrightarrow & SO(3) & \to & SO(3)/I }

where the double cover $\pi$ is the universal cover of $\mathrm{SO}\left(3\right)$, and the double cover $B$ of $I$ is called the binary icosahedral group. It follows that ${\pi }_{1}\left(S\right)\cong B$, 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 $i\ge 2$ by an easy computation involving the long exact homotopy sequence applied to the top row.

Revised on May 30, 2010 02:47:19 by Todd Trimble (69.118.56.215)