Spherical 3-manifold

In mathematics, a spherical 3-manifold M is a 3-manifold of the form

where ${\displaystyle \Gamma }$ is a finite subgroup of SO(4) acting freely by rotations on the 3-sphere ${\displaystyle S^{3}}$. 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 Γ itself. The elliptization conjecture, proved by Grigori Perelman, states that conversely all 3-manifolds with finite fundamental group are spherical manifolds.

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