Zero Orbifold Euler Characteristic
The normalized symbols of Seifert fibrations with zero orbifold Euler characteristic are given in the list below. The manifolds have Euclidean Thurston geometry if they are non-orientable or if b + Σbi/ai= 0, and nil geometry otherwise. Equivalently, the manifold has Euclidean geometry if and only if its fundamental group has an abelian group of finite index. There are 10 Euclidean manifolds, but four of them have two different Seifert fibrations. All surface bundles associated to automorphisms of the 2-torus of trace 2, 1, 0, −1, or −2 are Seifert fibrations with zero orbifold Euler characteristic (the ones for other (Anosov) automorphisms are not Seifert fiber spaces, but have sol geometry). The manifolds with nil geometry all have a unique Seifert fibration, and are characterized by their fundamental groups. The total spaces are all acyclic.
{b; (o1, 0); (3, b1), (3, b2), (3, b3)} (b integral, bi is 1 or 2) For b + Σbi/ai= 0 this is an oriented Euclidean 2-torus bundle over the circle, and is the surface bundle associated to an order 3 (trace −1) rotation of the 2-torus.
{b; (o1, 0); (2,1), (4, b2), (4, b3)} (b integral, bi is 1 or 3) For b + Σbi/ai= 0 this is an oriented Euclidean 2-torus bundle over the circle, and is the surface bundle associated to an order 4 (trace 0) rotation of the 2-torus.
{b; (o1, 0); (2, 1), (3, b2), (6, b3)} (b integral, b2 is 1 or 2, b3 is 1 or 5) For b + Σbi/ai= 0 this is an oriented Euclidean 2-torus bundle over the circle, and is the surface bundle associated to an order 6 (trace 1) rotation of the 2-torus.
{b; (o1, 0); (2, 1), (2, 1), (2, 1), (2, 1)} (b integral) These are oriented 2-torus bundles for trace −2 automorphisms of the 2-torus. For b=−2 this is an oriented Euclidean 2-torus bundle over the circle (the surface bundle associated to an order 2 rotation of the 2-torus) and is homeomorphic to {0; (n2, 2);}.
{b; (o1, 1); } (b integral) This is an oriented 2-torus bundle over the circle, given as the surface bundle associated to a trace 2 automorphism of the 2-torus. For b=0 this is Euclidean, and is the 3-torus (the surface bundle associated to the identity map of the 2-torus).
{b; (o2, 1); } (b is 0 or 1) Two non-orientable Euclidean Klein bottle bundles over the circle. The first homology is Z+Z+Z/2Z if b=0, and Z+Z if b=1. The first is the Klein bottle times S1 and other is the surface bundle associated to a Dehn twist of the Klein bottle. They are homeomorphic to the torus bundles {b; (n1, 2);}.
{0; (n1, 1); (2, 1), (2, 1)} Homeomorphic to the non-orientable Euclidean Klein bottle bundle {1; (n3, 2);}, with first homology Z + Z/4Z.
{b; (n1, 2); } (b is 0 or 1) These are the non-orientable Euclidean surface bundles associated with orientation reversing order 2 automorphisms of a 2-torus with no fixed points. The first homology is Z+Z+Z/2Z if b=0, and Z+Z if b=1. They are homeomorphic to the Klein bottle bundles {b; (o2, 1);}.
{b; (n2, 1); (2, 1), (2, 1)} (b integral) For b=−1 this is oriented Euclidean.
{b; (n2, 2); } (b integral) For b=0 this is an oriented Euclidean manifold, homeomorphic to the 2-torus bundle {−2; (o1, 0); (2, 1), (2, 1), (2, 1), (2, 1)} over the cicle associated to an order 2 rotation of the 2-torus.
{b; (n3, 2); } (b is 0 or 1) The other two non-orientable Euclidean Klein bottle bundles. The one with b = 1 is homeomorphic to {0; (n1, 1); (2, 1), (2, 1)}. The first homology is Z+Z/2Z+Z/2Z if b=0, and Z+Z/4Z if b=1. These two Klein bottle bundle are surface bundles associated to the y-homeomorphism and the product of this and the twist.
Read more about this topic: Seifert Fiber Space