is a finite subgroup of O(4) acting freely by rotations on the 3-sphere
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.
As such, the spherical three-manifolds are precisely the closed 3-manifolds with finite fundamental group.
This divides the set of such manifolds into five classes, described in the following sections.
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
are: 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.
Alternatively, the fundamental group has presentation for coprime integers m, n with m ≥ 1, n ≥ 2.
The simplest example is m = 1, n = 2, when π1(M) is the quaternion group of order 8.
Prism manifolds are uniquely determined by their fundamental groups: if a closed 3-manifold has the same fundamental group as a prism manifold M, it is homeomorphic to M. Prism manifolds can be represented as Seifert fiber spaces in two ways.
Alternatively, the fundamental group has presentation for an odd integer m ≥ 1.
The elements x and y generate a normal subgroup isomorphic to the quaternion group of order 8.
They can all be represented in an essentially unique way as Seifert fiber spaces: the quotient manifold is a sphere and there are 3 exceptional fibers of orders 2, 3, and 3.
They can all be represented in an essentially unique way as Seifert fiber spaces: the quotient manifold is a sphere and there are 3 exceptional fibers of orders 2, 3, and 4.
They can all be represented in an essentially unique way as Seifert fiber spaces: the quotient manifold is a sphere and there are 3 exceptional fibers of orders 2, 3, and 5.