Seifert Fiber Space - Negative Orbifold Euler Characteristic

Negative Orbifold Euler Characteristic

This is the general case. All such Seifert fibrations are determined up to isomorphism by their fundamental group. The total spaces are aspherical (in other words all higher homotopy groups vanish). They have Thurston geometries of type the universal cover of SL₂(R), unless some finite cover splits as a product, in which case they have Thurston geometries of type H₂×R. This happens if the manifold is non-orientable or b + Σb_i/a_i= 0.