Is there any readily available reference for a balanced presentation of the fundamental group in terms of the classifying invariants of an arbitrary Seifert fibered space? I can't find it, and the usual presentation coming from the classification resists GAP (checked on four fibrations with three exceptional fibers over the sphere, got two generators and three relations each time).
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$\begingroup$ I'm a little confused. In general, Seifert fibered spaces can't be classified by their fundamental group -- for example, see Lens spaces. $\endgroup$– Ryan BudneyCommented Jul 2, 2021 at 21:00
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1$\begingroup$ @RyanBudney, I meant the classifying invariants of the Seifert space (it's symbols, if you will). They do classify the Seifert spaces, and therefore encode balanced presentations of their fundamental groups - so I ask about these presentations expressed in terms of these invariants. $\endgroup$– lemon314Commented Jul 2, 2021 at 21:12
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$\begingroup$ You can't derive the Seifert invariants from the fundamental group. At least, not if you are considering it as only a group. If you use presentations you could give the group additional structure -- marked subgroups, etc. $\endgroup$– Ryan BudneyCommented Jul 2, 2021 at 21:22
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1$\begingroup$ @RyanBudney, you misunderstand: the input is the Seifert fibration, say $\{g, e, (a_{i}/b_{i})\}$. The output is to be a balanced presentation of its fundamental group. Not the other way round. $\endgroup$– lemon314Commented Jul 2, 2021 at 21:41
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1 Answer
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You can get a balanced presentation of the group from a Heegaard diagram. The paper of Boileau-Zieschang, Heegaard genus of closed orientable Seifert 3-manifolds. Invent. Math. 76 (1984), no. 3, 455–468 shows how to find such a diagram. Following their discussion (too involved to reproduce here) you get a presentation that is determined by the Seifert invariants.
If you only care about the group, then their lemma 1.5 shows how to go from the `usual' presentation (which is not balanced) to a balanced one.
Except for some small examples, I think that these are in fact minimal (with respect to the number of generators) presentations of the fundamental group.
answered Jul 2, 2021 at 21:46
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$\begingroup$ Thank you very much for your answer. I wonder if there's a more explicit reference, as the paper doesn't spell out the presentation. If none shows up, I will accept this answer in couple of days. I'll look into Proposition 1.5, because although it's only about a very specific group, the method seems applicable in greater generality. $\endgroup$– lemon314Commented Jul 3, 2021 at 0:07
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$\begingroup$ @lemon314: the presentation will depend on the precise Heegaard splitting chosen. You get the vertical Heegaard splittings by tubing together neighbourhoods of singular fibers. So these presentations will be just a few Tietze moves away from the standard textbook presentations you get from thinking of them as Seifert fibered spaces. $\endgroup$ Commented Jul 3, 2021 at 0:30