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| In [[mathematics]], a '''spherical 3-manifold''' ''M'' is a [[3-manifold]] of the form
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| :<math>M=S^3/\Gamma</math> | |
| where <math>\Gamma</math> is a [[Finite group|finite]] [[subgroup]] of [[Special orthogonal group|SO(4)]] [[Group action|acting freely]] by rotations on the [[3-sphere]] <math>S^3</math>. All such manifolds are [[prime decomposition (3-manifold)|prime]], [[orientable]], and [[closed manifold|closed]]. Spherical 3-manifolds are sometimes called '''elliptic 3-manifolds''' or Clifford-Klein manifolds.
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| ==Properties==
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| A spherical 3-manifold has a finite [[fundamental group]] [[isomorphic]] to Γ itself. The [[Thurston elliptization conjecture| elliptization conjecture]], proved by [[Grigori Perelman]], states that conversely all 3-manifolds with finite fundamental group are spherical manifolds.
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| The fundamental group is either [[Cyclic group|cyclic]], or is a central extension of a [[Dihedral group|dihedral]], [[Tetrahedral group|tetrahedral]], [[Octahedral group|octahedral]], or [[Icosahedral group|icosahedral]] group by a cyclic group of even order. This divides the set of such manifolds into 5 classes, described in the following sections.
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| The spherical manifolds are exactly the manifolds with spherical geometry, one of the 8 geometries of Thurston's [[geometrization conjecture]].
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| ==Cyclic case (lens spaces)==
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| The manifolds <math>S^3/\Gamma</math> with Γ [[cyclic group|cyclic]] are precisely the 3-dimensional [[lens space]]s. 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.
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| Three-dimensional lens spaces arise as quotients of <math>S^3 \subset \mathbb{C}^2</math> by
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| the action of the group that is generated by elements of the form
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| : <math>\begin{pmatrix}\omega &0\\0&\omega^q\end{pmatrix}.</math> | |
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| where <math>\omega=e^{2\pi i/p}</math>. Such a lens space <math>L(p;q)</math> has fundamental group <math>\mathbb{Z}/p\mathbb{Z}</math> for all <math>q</math>, so spaces with different <math>p</math> are not homotopy equivalent.
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| Moreover, classifications up to homeomorphism and homotopy equivalence are known, as follows. The three-dimensional spaces <math>L(p;q_1)</math> and
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| <math>L(p;q_2)</math> are:
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| #homotopy equivalent if and only if <math>q_1 q_2 \equiv \pm n^2 \pmod{p}</math> for some <math>n \in \mathbb{N};</math>
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| #homeomorphic if and only if <math>q_1 \equiv \pm q_2^{\pm1} \pmod{p}.</math>
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| 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.
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| The lens space ''L''(1,0) is the 3-sphere, and the lens space ''L''(2,1) is 3 dimensional real projective space.
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| Lens spaces can be represented as [[Seifert fiber space]]s 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.
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| ==Dihedral case (prism manifolds)==
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| A '''prism manifold''' is a closed [[3-manifold|3-dimensional manifold]] ''M'' whose fundamental group
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| is a central extension of a dihedral group.
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| The fundamental group π<sub>1</sub>(''M'') of ''M'' is a product of a cyclic group of order ''m'' with a group having presentation
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| :<math>\langle x,y\mid xyx^{-1}=y^{-1}, x^{2^k}=y^n\rangle</math>
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| for integers ''k'', ''m'', ''n'' with ''k'' ≥ 1, ''m'' ≥ 1, ''n''
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| ≥ 2 and ''m'' coprime to 2''n''.
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| Alternatively, the fundamental group has presentation
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| :<math>\langle x,y \mid xyx^{-1}=y^{-1}, x^{2m}=y^n\rangle</math>
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| for coprime integers ''m'', ''n'' with ''m'' ≥ 1, ''n'' ≥ 2. (The ''n'' here equals the previous ''n'', and the ''m'' here is 2<sup>''k''-1</sup> times the previous ''m''.)
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| We continue with the latter presentation. This group is a [[metacyclic group]] of order 4''mn'' with [[abelianization]] of order 4''m'' (so ''m'' and ''n'' are both determined by this group).
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| The element ''y'' generates a [[cyclic group|cyclic]] [[normal subgroup]] of order 2''n'', and the element ''x'' has order 4''m''. The [[center (group theory)|center]] is cyclic of order 2''m'' and is generated by ''x''<sup>2</sup>, and the quotient by the center is the [[dihedral group]] of order 2''n''.
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| When ''m'' = 1 this group is a binary dihedral or [[dicyclic group]]. The simplest example is ''m'' = 1, ''n'' = 2, when π<sub>1</sub>(''M'') is the [[quaternion group]] of order 8.
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| 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 [[homeomorphism|homeomorphic]] to ''M''.
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| Prism manifolds can be represented as [[Seifert fiber space]]s in two ways.
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| ==Tetrahedral case==
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| The fundamental group is a product of a cyclic group of order ''m'' with a group having presentation
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| :<math>\langle x,y,z \mid (xy)^2=x^2=y^2, zxz^{-1}=y,zyz^{-1}=xy, z^{3^k}=1\rangle</math>
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| for integers ''k'', ''m'' with ''k'' ≥ 1, ''m'' ≥ 1 and ''m'' coprime to 6.
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| Alternatively, the fundamental group has presentation
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| :<math>\langle x,y,z \mid (xy)^2=x^2=y^2, zxz^{-1}=y,zyz^{-1}=xy, z^{3m}=1\rangle</math> | |
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| for an odd integer ''m'' ≥ 1. (The ''m'' here is 3<sup>''k''-1</sup> times the previous ''m''.)
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| We continue with the latter presentation. This group has order 24''m''. The elements ''x'' and ''y'' generate a normal subgroup isomorphic to the [[quaternion group]] of order 8. The [[Center (geometry)|center]] is cyclic of order 2''m''. It is generated by the elements ''z''<sup>3</sup> and ''x''<sup>2</sup> = ''y''<sup>2</sup>, and the quotient by the center is the tetrahedral group, equivalently, the [[alternating group]] ''A''<sub>4</sub>.
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| When ''m'' = 1 this group is the [[binary tetrahedral group]].
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| These manifolds are uniquely determined by their fundamental groups. They can all be represented in an essentially unique way as [[Seifert fiber space]]s: the quotient manifold is a sphere and there are 3 exceptional fibers of orders 2, 3, and 3.
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| ==Octahedral case==
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| 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
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| :<math>\langle x,y \mid (xy)^2=x^3=y^4\rangle.</math>
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| These manifolds are uniquely determined by their fundamental groups. They can all be represented in an essentially unique way as [[Seifert fiber space]]s: the quotient manifold is a sphere and there are 3 exceptional fibers of orders 2, 3, and 4.
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| ==Icosahedral case==
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| 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
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| :<math>\langle x,y \mid (xy)^2=x^3=y^5\rangle.</math>
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| When ''m'' is 1, the manifold is the [[Poincaré homology sphere]].
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| 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.
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| ==References==
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| *[[Peter Orlik]], ''Seifert manifolds'', Lecture Notes in Mathematics, vol. 291, [[Springer-Verlag]] (1972). ISBN 0-387-06014-6
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| *[[William Jaco]], ''Lectures on 3-manifold topology'' ISBN 0-8218-1693-4
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| * [[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
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| [[Category:Geometric topology]]
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| [[Category:Riemannian geometry]]
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| [[Category:Group theory]]
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| [[Category:3-manifolds]]
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