# Seifert Fiber Space - Positive Orbifold Euler Characteristic

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 SR Thurston geometry if the fundamental group is infinite. Equivalently, the geometry is SR if the manifold is non-orientable or if b + Σbi/ai= 0, and spherical geometry otherwise.

{b; (o1, 0);} (b integral) is SS1 for b=0, otherwise a lens space L(b,1). ({1; (o1, 0);} =L(1,1) is the 3-sphere.)

{b; (o1, 0);(a1, b1)} (b integral) is the Lens space L(ba1+b1,a1).

{b; (o1, 0);(a1, b1), (a2, b2)} (b integral) is SS1 if ba1a2+a1b2+a2b1 = 0, otherwise the lens space L(ba1a2+a1b2+a2b1, ma2+nb2) where ma1n(ba1 +b1) = 1.

{b; (o1, 0);(2, 1), (2, 1), (a3, b3)} (b integral) This is the Prism manifold with fundamental group of order 4a3|(b+1)a3+b3| and first homology group of order 4|(b+1)a3+b3|.

{b; (o1, 0);(2, 1), (3, b2), (3, b3)} (b integral) The fundamental group is a central extension of the tetrahedral group of order 12 by a cyclic group.

{b; (o1, 0);(2, 1), (3, b2), (4, b3)} (b integral) The fundamental group is the product of a cyclic group of order |12b+6+4b2 + 3b3| and a double cover of order 48 of the octahedral group of order 24.

{b; (o1, 0);(2, 1), (3, b2), (5, b3)} (b integral) The fundamental group is the product of a cyclic group of order m=|30b+15+10b2 +6b3| 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; (o1, 0);(2, 1), (3, 1), (5, 1)} is the Poincaré sphere.

{b; (n1, 1);} (b is 0 or 1.) These are the non-orientable 3-manifolds with SR 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; (n1, 1);(a1, b1)} (b is 0 or 1.) These are the non-orientable 3-manifolds with SR geometry. If ba1+b1 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; (n2, 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; (n2, 1);(a1, b1)} (b integral.) This is the (unique) Prism manifold with fundamental group of order 4a1|ba1 + b1| and first homology group of order 4a1.