Archiv der Mathematik

, Volume 109, Issue 3, pp 273–283 | Cite as

The \(L^2\)-Alexander torsion for Seifert fiber spaces

  • Gerrit HerrmannEmail author


We calculate the \(L^2\)-Alexander torsion for Seifert fiber spaces and graph manifolds in terms of the Thurston norm.


Seifert fiber space \(L^2\)-Alexander torsion Thurston norm 

Mathematics Subject Classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Aschenbrenner, S. Friedl, and H. Wilton, 3–Manifold Groups, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2015.Google Scholar
  2. 2.
    F. Ben Aribi, The \(L^2\)–Alexander invariant detects the unknot, C. R. Math. Acad. Sci. Paris 351 (2013), 215–219.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    F. Ben Aribi, Gluing formulas for the \(L^2\)-Alexander torsions, preprint, (2016), arXiv:1603.00367.
  4. 4.
    G.E. Bredon, Topology and Geometry, Graduate Texts in Mathematics, Springer, New York, 1993.Google Scholar
  5. 5.
    J. Dubois, S. Friedl, and W. Lück, The \(L^2\)-Alexander torsion is symmetric, Algebr. Geom. Topol. 15 (2015), 3599–3612.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    J. Dubois, S. Friedl, and W. Lück, Three Flavors of Twisted Invariants of Knots, Introduction to Modern Mathematics, 143–170, Adv. Lect. Math. 33, Int. Press, Sommerville, MA, 2015.Google Scholar
  7. 7.
    J. Dubois, S. Friedl, and W. Lück, \({L}^2\)-Alexander torsion of 3-manifolds, J. Topol. 9 (2016), 889–926.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    J. Dubois and C. Wegner, \(L^2\)-Alexander invariant for torus knots, C. R. Math. Acad. Sci. Paris 348 (2010), 1185–1189.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    J. Dubois and C. Wegner, Weighted \(L^2\)-invariants and applications to knot theory, Commun. Contemp. Math. 17 (2015), 1450010, 29pp.Google Scholar
  10. 10.
    D. Eisenbud and W.D. Neumann, Three-dimensional Link Theory and Invariants of Plane Curve Singularities, Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1985.Google Scholar
  11. 11.
    S. Friedl and W. Lück, The \(L^2\)-torsion function and the Thurston norm of 3-manifolds, preprint, 2015, arXiv:1510.00264v1.
  12. 12.
    D. Gabai, Foliations and the topology of 3-manifolds, J. Differential Geom. 18 (1983), 445–503.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994.CrossRefGoogle Scholar
  14. 14.
    Y. Liu, Degree of \({L}^2\)-Alexander torsion for 3-manifolds, Invent. Math. 207 (2017), 981–1030.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    W. Lück, \(L^2\)-Invariants: Theory and Applications to Geometry and K-Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, A Series of Modern Surveys in Mathematics, 44, Springer-Verlag, Berlin, 2002.Google Scholar
  16. 16.
    W. Lück and T. Schick, \(L^2\)-torsion of hyperbolic manifolds of finite volume, Geom. Funct. Anal. 9 (1999), 518–567.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    W. Li and W. Zhang, An \(L^2\)-Alexander invariant for knots, Commun. Contemp. Math. 8 (2006), 167–187.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    W. Li and W. Zhang, An \(L^2\)-Alexander–Conway invariant for knots and the volume conjecture, In: Differential Geometry and Physics, 303–312, Nankai Tracts Math., 10, World Sci. Publ., Hackensack, NJ, 2006.Google Scholar
  19. 19.
    W. Li and W. Zhang, Twisted \(L^2\)-Alexander–Conway invariants for knots, In: Topology and Physics, 236–259, Nankai Tracts Math., 12, World Sci. Publ., Hackensack, NJ, 2008.Google Scholar
  20. 20.
    P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), 401–487.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    W. P. Thurston, A norm for the homology of 3-manifolds, Mem. Amer. Math. Soc. 59 (1986), i-vi and 99–130.Google Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität RegensburgRegensburgGermany

Personalised recommendations