Seifert fibrations of lens spaces
This paper introduces an algorithm to produce all possible Seifert fibrations for a given lens space and describes isomorphisms between the resulting Seifert fibrations. For a given lens space \(L(p,q)\), say \(p>0\), there are infinitely many distinct Seifert fibrations denoted by \(M(0;(\alpha_1,\beta_1),(\alpha_2,\beta_2))\). To find a Seifert fibration, we first take a coprime pair of non-zero integers \(\alpha_1^0\), \(\alpha_2^0\). Choose integers \(r\), \(s\) so that \(pr+qs=1\). Set \(\alpha=\dfrac{p}{\gcd (p,s\alpha_1^0-\alpha_2^0)}\), \(\alpha_1=\alpha \alpha_1^0\) and \(\alpha_2=\alpha \alpha_2^0\). Set \(\alpha_1'=\dfrac{s\alpha_1^0-\alpha_2^0}{\gcd (p,s\alpha_1^0-\alpha_2^0)}\) and choose integers \(\beta_1\), \(\beta'_1\) so that \(\alpha_1\beta'_1-\alpha'_1\beta_1=1\). Set \(\beta_2=-s\beta_1+p\beta'_1\). The main theorem shows that this gives a Seifert fibration \(M(0;(\alpha_1,\beta_1),(\alpha_2,\beta_2))\) of \(L(p,q)\); this construction is independent of the choice of \(r\), \(s\), \(\beta_1\), \(\beta'_1\), and any Seifert fibration of \(L(p,q)\) can be obtained in this way. Isomorphisms between these Seifert fibrations are given in the latter half of Section 4.
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