Seifert fibrations of lens spaces
type: Item
instance of: scholarly article
title: Seifert fibrations of lens spaces
zbMATH Open document ID: 1404.57034
author: Hansjörg Geiges
author: Christian Lange
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2018-06-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1608.06844
review text: 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.
reviewed by: Toshio Saito
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 55R65
Mathematics Subject Classification ID: 57M10
Mathematics Subject Classification ID: 57M60
zbMATH DE Number: 6888208
zbMATH Keywords: Seifert fibration
zbMATH Keywords: Seifert invariants
zbMATH Keywords: lens space
zbMATH Keywords: 3-manifold
MaRDI profile type: Publication
OpenAlex ID: W2964330066
arXiv ID: 1608.06844
cites work: Transversely holomorphic flows and contact circles on spherical 3-manifolds
cites work: A Hamiltonian version of a result of Gromoll and Grove
cites work: Q4700561
cites work: Geodesics closed on the projective plane
cites work: Seifert manifolds
cites work: The Geometries of 3-Manifolds
cites work: Q3923266
DOI: 10.1007/S12188-017-0188-Z
description: scientific article