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Binary icosahedral group
Applications
The coset space Spin(3) / 2I = S3 / 2I is a spherical 3-manifold called the Poincaré homology sphere.
It is an example of a homology sphere, i.e. a 3-manifold whose homology groups are identical to those of a 3-sphere.
The fundamental group of the Poincaré sphere is isomorphic to the binary icosahedral group, as the Poincaré sphere is the quotient of a 3-sphere by the binary icosahedral group.