**Positive Orbifold Euler Characteristic**

The normalized symbols of Seifert fibrations with positive orbifold Euler characteristic are given in the list below. The Seifert manifolds often have many different Seifert fibrations. They have a spherical Thurston geometry if the fundamental group is finite, and an *S*2×**R** Thurston geometry if the fundamental group is infinite. Equivalently, the geometry is *S*2×**R** if the manifold is non-orientable or if *b* + Σ*b*_{i}/*a*_{i}= 0, and spherical geometry otherwise.

**{ b; (o_{1}, 0);} (b integral)** is

*S*2×

*S*1 for

*b*=0, otherwise a lens space

*L*(

*b*,1). ({1; (

*o*

_{1}, 0);} =

*L*(1,1) is the 3-sphere.)

**{ b; (o_{1}, 0);(a_{1}, b_{1})} (b integral)** is the Lens space

*L*(

*ba*

_{1}+

*b*

_{1},

*a*

_{1}).

**{ b; (o_{1}, 0);(a_{1}, b_{1}), (a_{2}, b_{2})} (b integral)** is

*S*2×

*S*1 if

*ba*

_{1}

*a*

_{2}+

*a*

_{1}

*b*

_{2}+

*a*

_{2}

*b*

_{1}= 0, otherwise the lens space

*L*(

*ba*

_{1}

*a*

_{2}+

*a*

_{1}

*b*

_{2}+

*a*

_{2}

*b*

_{1},

*ma*

_{2}+

*nb*

_{2}) where

*ma*

_{1}−

*n*(

*ba*

_{1}+

*b*

_{1}) = 1.

**{ b; (o_{1}, 0);(2, 1), (2, 1), (a_{3}, b_{3})} (b integral)** This is the Prism manifold with fundamental group of order 4

*a*

_{3}|(

*b*+1)

*a*

_{3}+

*b*

_{3}| and first homology group of order 4|(

*b*+1)

*a*

_{3}+

*b*

_{3}|.

**{ b; (o_{1}, 0);(2, 1), (3, b_{2}), (3, b_{3})} (b integral)** The fundamental group is a central extension of the tetrahedral group of order 12 by a cyclic group.

**{ b; (o_{1}, 0);(2, 1), (3, b_{2}), (4, b_{3})} (b integral)** The fundamental group is the product of a cyclic group of order |12

*b*+6+4

*b*

_{2}+ 3

*b*

_{3}| and a double cover of order 48 of the octahedral group of order 24.

**{ b; (o_{1}, 0);(2, 1), (3, b_{2}), (5, b_{3})} (b integral)** The fundamental group is the product of a cyclic group of order

*m*=|30

*b*+15+10

*b*

_{2}+6

*b*

_{3}| and the order 120 perfect double cover of the icosahedral group. The manifolds are quotients of the Poincaré sphere by cyclic groups of order

*m*. In particular {−1; (

*o*

_{1}, 0);(2, 1), (3, 1), (5, 1)} is the Poincaré sphere.

**{ b; (n_{1}, 1);} (b is 0 or 1.)** These are the non-orientable 3-manifolds with

*S*2×

**R**geometry. If

*b*is even this is homeomorphic to the projective plane times the circle, otherwise it is homeomorphic to a surface bundle associated to an orientation reversing automorphism of the 2-sphere.

**{ b; (n_{1}, 1);(a_{1}, b_{1})} (b is 0 or 1.)** These are the non-orientable 3-manifolds with

*S*2×

**R**geometry. If

*ba*

_{1}+

*b*

_{1}is even this is homeomorphic to the projective plane times the circle, otherwise it is homeomorphic to a surface bundle associated to an orientation reversing automorphism of the 2-sphere.

**{ b; (n_{2}, 1);} (b integral.)** This is the Prism manifold with fundamental group of order 4|

*b*| and first homology group of order 4, except for

*b*=0 when it is a sum of two copies of real projective space, and |

*b*|=1 when it is the lens space with fundamental group of order 4.

**{ b; (n_{2}, 1);(a_{1}, b_{1})} (b integral.)** This is the (unique) Prism manifold with fundamental group of order 4

*a*

_{1}|

*ba*

_{1}+

*b*

_{1}| and first homology group of order 4

*a*

_{1}.

Read more about this topic: Seifert Fiber Space

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