Seifert Fiber Space - Fundamental Group

Fundamental Group

The fundamental group of M fits into the exact sequence

where π1(B) is the orbifold fundamental group of B (which is not the same as the fundamental group of the underlying topological manifold). The image of group π1(S1) is cyclic, normal, and generated by the element h represented by any regular fiber, but the map from π1(S1) to π1(M) is not always injective.

The fundamental group of M has the following presentation by generators and relations:

B orientable:

where ε is 1 for type o1, and is −1 for type o2.

B non-orientable:

where εi is 1 or −1 depending on whether the corresponding generator vi preserves or reverses orientation of the fiber. (So εi are all 1 for type n1, all −1 for type n2, just the first one is one for type n3, and just the first two are one for type n4.)

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