The manifolds with Γ cyclic are precisely the 3-dimensional lens spaces. A lens space is not determined by its fundamental group ; 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:
In particular, the lens spaces L and L give examples of two 3-manifolds that are homotopy equivalent but not homeomorphic. The lens space L is the 3-sphere, and the lens space L is 3 dimensionalreal 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 manifoldM whose fundamental group is a central extension of a dihedral group. The fundamental group π1 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 m coprime to 2n. Alternatively, the fundamental group has presentation for coprime integersm, n with m ≥ 1, n ≥ 2. We continue with the latter presentation. This group is a metacyclic group of order 4mn with abelianization of order 4m. The element y generates a cyclic normal subgroup of order 2n, and the element x has order 4m. The center is cyclic of order 2m and is generated by x2, and the quotient by the center is the dihedral group of order 2n. When m = 1 this group is a binary dihedral or dicyclic group. The simplest example is m = 1, n = 2, when π1 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.
Tetrahedral case
The fundamental group is a product of a cyclic group of order m with a group having presentation for integers k, m with k ≥ 1, m ≥ 1 and m coprime to 6. Alternatively, the fundamental group has presentation for an odd integerm ≥ 1. We continue with the latter presentation. This group has order 24m. The elements x and y generate a normal subgroup isomorphic to the quaternion group of order 8. The center is cyclic of order 2m. It is generated by the elementsz3 and x2 = y2, and the quotient by the center is the tetrahedral group, equivalently, the alternating groupA4. When m = 1 this group is the binary tetrahedral group. These manifolds are uniquely determined by their fundamental groups. 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.
Octahedral case
The fundamental group is a product of a cyclic group of order m coprime to 6 with the binary octahedral group which has the presentation These manifolds are uniquely determined by their fundamental groups. 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.
Icosahedral case
The fundamental group is a product of a cyclic group of order m coprime to 30 with the binary icosahedral group which has the presentation When m is 1, the manifold is the Poincaré homology sphere. These manifolds are uniquely determined by their fundamental groups. 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.
