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 + Σbi/ai= 0.
Read more about this topic: Seifert Fiber Space
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