Everything about Spherical 3-manifold totally explained
In
mathematics, a
spherical 3-manifold M is a
3-manifold of the form:
where Γ is a
finite subgroup of
SO(4) acting freely by rotations on the
3-sphere . 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
icoshedral 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
with Γ
cyclic are precisely the 3-dimensional
lens spaces. A lens space isn't 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
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
»
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
»
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.
Further Information
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