In mathematics, a spherical 3-manifold M is a 3-manifold of the form where is a finite subgroup of SO(4) acting freely by rotations on the 3-sphere . All such manifolds are prime, orientable, and closed. Spherical 3-manifolds are sometimes called elliptic 3-manifolds or Clifford-Klein manifolds. Properties A spherical 3-manifold has a finite fundamental group isomorphic to Γ itself. The elliptization conjecture, proved by Grigori Perelman, states that conversely all compact 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. Cyclic case (lens spaces) The manifolds with Γ cyclic are precisely the 3-dimensional lens spaces. A lens space is not determined by its fundamental group (there are non-homeomorphic lens spaces with isomorphic fundamental groups); but any other spherical manifold is. Three-dimensional lens spaces arise as quotients of by the action of the group that is generated by elements of the form where . Such a lens space has fundamental group for all , so spaces with different are not homotopy equivalent. Moreover, classifications up to homeomorphism and homotopy equivalence are known, as follows. The three-dimensional spaces and are: homotopy equivalent if and only if for some homeomorphic if and only if In particular, the lens spaces L(7,1) and L(7,2) give examples of two 3-manifolds that are homotopy equivalent but not homeomorphic. The lens space L(1,0) is the 3-sphere, and the lens space L(2,1) is 3 dimensional real projective space. Lens spaces can be represented as Seifert fiber spaces in many ways, usually as fiber spaces over the 2-sphere with at most two exceptional fibers, though the lens space with fundamental group of order 4 also has a representation as a Seifert fiber space over the projective plane with no exceptional fibers. Dihedral case (prism manifolds) A prism manifold is a closed 3-dimensional manifold M whose fundamental group is a central extension of a dihedral group. The fundamental group π1(M) of M is a product of a cyclic group of order m with a group having presentation for integers k, m, n with k ≥ 1, m ≥ 1, n ≥ 2 and