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 S2×R Thurston geometry if the fundamental group is infinite. Equivalently, the geometry is S2×R if the manifold is non-orientable or if b + Σb_i/a_i= 0, and spherical geometry otherwise.

{b; (o₁, 0);} (b integral) is S2×S1 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 S2×S1 if ba₁a₂+a₁b₂+a₂b₁ = 0, otherwise the lens space L(ba₁a₂+a₁b₂+a₂b₁, ma₂+nb₂) where ma₁ − n(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 S2×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; (n₁, 1);(a₁, b₁)} (b is 0 or 1.) These are the non-orientable 3-manifolds with S2×R 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.