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

where ${\displaystyle \Gamma }$ is a finite subgroup of O(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.

A special case of the Bonnet–Myers theorem says that every smooth manifold which has a smooth Riemannian metric which is both geodesically complete and of constant positive curvature must be closed and have finite fundamental group. William Thurston's elliptization conjecture, proven by Grigori Perelman using Richard Hamilton's Ricci flow, states a converse: every closed three-dimensional manifold with finite fundamental group has a smooth Riemannian metric of constant positive curvature. (This converse is special to three dimensions.) As such, the spherical three-manifolds are precisely the closed 3-manifolds with finite fundamental group.

According to Synge's theorem, every spherical 3-manifold is orientable, and in particular ${\displaystyle \Gamma }$ must be included in SO(4). 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 five classes, described in the following sections.

The spherical manifolds are exactly the manifolds with spherical geometry, one of the eight geometries of Thurston's geometrization conjecture.

The manifolds ${\displaystyle S^{3}/\Gamma }$ 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 ${\displaystyle S^{3}\subset \mathbb {C} ^{2}}$ by the action of the group that is generated by elements of the form

where ${\displaystyle \omega =e^{2\pi i/p}}$. Such a lens space ${\displaystyle L(p;q)}$ has fundamental group ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$ for all ${\displaystyle q}$, so spaces with different ${\displaystyle p}$ are not homotopy equivalent. Moreover, classifications up to homeomorphism and homotopy equivalence are known, as follows. The three-dimensional spaces ${\displaystyle L(p;q_{1})}$ and ${\displaystyle L(p;q_{2})}$ are: