Skip to main content
Log in

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

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

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

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. M. Aschenbrenner, S. Friedl, and H. Wilton, 3–Manifold Groups, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2015.

  2. F. Ben Aribi, The \(L^2\)–Alexander invariant detects the unknot, C. R. Math. Acad. Sci. Paris 351 (2013), 215–219.

    Article  MathSciNet  MATH  Google Scholar 

  3. F. Ben Aribi, Gluing formulas for the \(L^2\)-Alexander torsions, preprint, (2016), arXiv:1603.00367.

  4. G.E. Bredon, Topology and Geometry, Graduate Texts in Mathematics, Springer, New York, 1993.

  5. J. Dubois, S. Friedl, and W. Lück, The \(L^2\)-Alexander torsion is symmetric, Algebr. Geom. Topol. 15 (2015), 3599–3612.

    Article  MathSciNet  MATH  Google Scholar 

  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.

  7. J. Dubois, S. Friedl, and W. Lück, \({L}^2\)-Alexander torsion of 3-manifolds, J. Topol. 9 (2016), 889–926.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Dubois and C. Wegner, \(L^2\)-Alexander invariant for torus knots, C. R. Math. Acad. Sci. Paris 348 (2010), 1185–1189.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Dubois and C. Wegner, Weighted \(L^2\)-invariants and applications to knot theory, Commun. Contemp. Math. 17 (2015), 1450010, 29pp.

  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.

  11. S. Friedl and W. Lück, The \(L^2\)-torsion function and the Thurston norm of 3-manifolds, preprint, 2015, arXiv:1510.00264v1.

  12. D. Gabai, Foliations and the topology of 3-manifolds, J. Differential Geom. 18 (1983), 445–503.

    Article  MathSciNet  MATH  Google Scholar 

  13. P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994.

    Book  Google Scholar 

  14. Y. Liu, Degree of \({L}^2\)-Alexander torsion for 3-manifolds, Invent. Math. 207 (2017), 981–1030.

    Article  MathSciNet  MATH  Google Scholar 

  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.

  16. W. Lück and T. Schick, \(L^2\)-torsion of hyperbolic manifolds of finite volume, Geom. Funct. Anal. 9 (1999), 518–567.

    Article  MathSciNet  MATH  Google Scholar 

  17. W. Li and W. Zhang, An \(L^2\)-Alexander invariant for knots, Commun. Contemp. Math. 8 (2006), 167–187.

    Article  MathSciNet  MATH  Google Scholar 

  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.

  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.

  20. P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), 401–487.

    Article  MathSciNet  MATH  Google Scholar 

  21. W. P. Thurston, A norm for the homology of 3-manifolds, Mem. Amer. Math. Soc. 59 (1986), i-vi and 99–130.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Gerrit Herrmann.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Herrmann, G. The \(L^2\)-Alexander torsion for Seifert fiber spaces. Arch. Math. 109, 273–283 (2017). https://doi.org/10.1007/s00013-017-1062-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-017-1062-z

Keywords

Mathematics Subject Classification

Navigation