High Quality Content by WIKIPEDIA articles! In mathematics, a spherical 3-manifold M is a 3-manifold of the form M = S3 / ? where ? is a finite subgroup of SO(4) acting freely by rotations on the 3-sphere S3. All such manifolds are prime, orientable, and closed. Spherical 3-manifolds are sometimes called elliptic 3-manifolds or Clifford-Klein manifolds. A spherical 3-manifold has a finite fundamental group isomorphic to ? itself. The elliptization conjecture, proved by Grigori Perelman, states that conversely all 3-manifolds with finite fundamental group are spherical manifolds. The fundamental group is either cyclic, or is a central extension of a dihedral, tetrahedral, octahedral, or icosahedral group by a cyclic group of even order. This divides the set of such manifolds into 5 classes, described in the following sections. The spherical manifolds are exactly the manifolds with spherical geometry, one of the 8 geometries of Thurston's geometrization conjecture.

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