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

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

B SU(2)S 3 p S π I SO(3) SO(3)/I\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 π\pi is the universal cover of SO(3)SO(3), and the double cover BB of II is called the binary icosahedral group. It follows that π 1(S)B\pi_1(S) \cong B, and since BB is a perfect group, its abelianization is trivial, and therefore pp induces an isomorphism in first homology. It induces an isomorphism in homotopy groups π i\pi_i for i2i \geq 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)