Seifert fibrations of lens spaces

  • Hansjörg Geiges
  • Christian Lange


We classify the Seifert fibrations of any given lens space L(pq). Starting from any pair of coprime non-zero integers \(\alpha _1^0,\alpha _2^0\), we give an algorithmic construction of a Seifert fibration \(L(p,q)\rightarrow S^2(\alpha |\alpha _1^0|,\alpha |\alpha _2^0|)\), where the natural number \(\alpha \) is determined by the algorithm. This algorithm produces all possible Seifert fibrations, and the isomorphisms between the resulting Seifert fibrations are described completely. Also, we show that all Seifert fibrations are isomorphic to certain standard models.


Seifert fibration Seifert invariants Lens space 3-manifold 

Mathematics Subject Classification

57M50 55R65 57M10 57M60 



We thank the referee for useful comments that helped to improve the exposition.


  1. 1.
    Brin, M.G.: Seifert fibered spaces, notes for a course given in the spring of 1993. arXiv:0711.1346
  2. 2.
    Frauenfelder, U., Lange, C., and S. Suhr, A Hamiltonian version of a result of Gromoll and Grove. arXiv:1603.05107
  3. 3.
    Geiges, H., Gonzalo Pérez, J.: Transversely holomorphic flows and contact circles on spherical \(3\)-manifolds. Enseign. Math. 62(2), 527–567 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gompf, R.E., Stipsicz, A.I.: 4-Manifolds and Kirby Calculus. American Mathematical Society, Providence, RI (1999)CrossRefzbMATHGoogle Scholar
  5. 5.
    Jankins, M., Neumann, W.D.: Lectures on Seifert manifolds, Brandeis lecture notes 2, Brandeis University, Waltham, MA (1983).
  6. 6.
    Lange, C.: On metrics on 2-orbifolds all of whose geodesics are closed. J. Reine Angew. Math. (to appear)Google Scholar
  7. 7.
    Orlik, P.: Seifert Manifolds. Lecture Notes in Math, vol. 291. Springer-Verlag, Berlin (1972)Google Scholar
  8. 8.
    Pries, C.: Geodesics closed on the projective plane. Geom. Funct. Anal. 18, 1774–1785 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Reidemeister, K.: Homotopieringe und Linsenräume. Abh. Math. Sem. Univ. Hamburg 11, 102–109 (1935)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Scott, P.: The geometries of 3-manifolds. Bull. London Math. Soc. 15, 401–487 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Seifert, H.: Topologie dreidimensionaler gefaserter Räume. Acta Math. 60, 147–238 (1933)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Seifert, H., Threlfall, W.: A Textbook of Topology. Academic Press, New York (1990)zbMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität zu KölnCologneGermany

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