**Fundamental Group**

The fundamental group of *M* fits into the exact sequence

where π_{1}(*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 π_{1}(*S*1) is cyclic, normal, and generated by the element *h* represented by any regular fiber, but the map from π_{1}(*S*1) to π_{1}(*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 *o*_{1}, and is −1 for type *o*_{2}.

*B* non-orientable:

where ε_{i} is 1 or −1 depending on whether the corresponding generator *v*_{i} preserves or reverses orientation of the fiber. (So ε_{i} are all 1 for type *n*_{1}, all −1 for type *n*_{2}, just the first one is one for type *n*_{3}, and just the first two are one for type *n*_{4}.)

Read more about this topic: Seifert Fiber Space

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“The *fundamental* things apply

As time goes by.”

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