Practical number

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

$M=S^3/\mathrm{\Gamma }$

where $\mathrm{\Gamma }$ is a finite subgroup of SO(4) acting freely by rotations on the 3-sphere $S^3$. 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 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 $S^3/\mathrm{\Gamma }$ 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 $S^3\subset {\mathbb{ℂ}}^2$ by the action of the group that is generated by elements of the form

$\left(\begin{array}{cc}\omega & 0\\ 0& {\omega }^q\end{array}\right).$

where $\omega =e^{2\pi i/p}$. Such a lens space $L(p;q)$ has fundamental group $\mathbb{\mathbb{Z}}/p\mathbb{\mathbb{Z}}$ for all $q$, so spaces with different $p$ are not homotopy equivalent. Moreover, classifications up to homeomorphism and homotopy equivalence are known, as follows. The three-dimensional spaces $L(p;q_1)$ and $L(p;q_2)$ are:

1. homotopy equivalent if and only if $q_1{q}_2\equiv \pm n^2(\mathrm{mod}p)$ for some $n\in \mathbb{\mathbb{N}};$
2. homeomorphic if and only if $q_1\equiv \pm q_2^{\pm 1}(\mathrm{mod}p).$

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

$⟨x,y\mid xy{x}^{-1}=y^{-1},x^{2^k}=y^n⟩$

for integers k, m, n with k ≥ 1, m ≥ 1, n ≥ 2 and m coprime to 2n.

Alternatively, the fundamental group has presentation

$⟨x,y\mid xy{x}^{-1}=y^{-1},x^{2m}=y^n⟩$

for coprime integers m, n with m ≥ 1, n ≥ 2. (The n here equals the previous n, and the m here is 2^k-1 times the previous m.)

We continue with the latter presentation. This group is a metacyclic group of order 4mn with abelianization of order 4m (so m and n are both determined by this group). 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(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.

Tetrahedral case

The fundamental group is a product of a cyclic group of order m with a group having presentation

$⟨x,y,z\mid (xy{)}^2=x^2=y^2,zx{z}^{-1}=y,zy{z}^{-1}=xy,z^{3^k}=1⟩$

for integers k, m with k ≥ 1, m ≥ 1 and m coprime to 6.

Alternatively, the fundamental group has presentation

$⟨x,y,z\mid (xy{)}^2=x^2=y^2,zx{z}^{-1}=y,zy{z}^{-1}=xy,z^{3m}=1⟩$

for an odd integer m ≥ 1. (The m here is 3^k-1 times the previous m.)

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 elements z³ and x² = y², and the quotient by the center is the tetrahedral group, equivalently, the alternating group A₄.

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 (of order 48) which has the presentation

$⟨x,y\mid (xy{)}^2=x^3=y^4⟩.$

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 (order 120) which has the presentation

$⟨x,y\mid (xy{)}^2=x^3=y^5⟩.$

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.

References

• Peter Orlik, Seifert manifolds, Lecture Notes in Mathematics, vol. 291, Springer-Verlag (1972). ISBN 0-387-06014-6
• William Jaco, Lectures on 3-manifold topology ISBN 0-8218-1693-4
• William Thurston, Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, New Jersey, 1997. ISBN 0-691-08304-5