Skip to main content
SpringerLink
Log in

Links of rational singularities, L-spaces and LO fundamental groups

  • Published:
Inventiones mathematicae Aims and scope

Abstract

We prove that the link of a complex normal surface singularity is an L-space if and only if the singularity is rational. This via a result of Hanselman et al. (Taut foliations on graph manifolds, 2015. arXiv:1508.0591), proving the conjecture of Boyer et al. (Math Ann 356(4):1213–1245, 2013), shows that a singularity link is not rational if and only if its fundamental group is left-orderable if and only if it admits a coorientable taut foliation.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Artin, M.: Some numerical criteria for contractability of curves on algebraic surfaces. Am. J. Math. 84, 485–496 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  2. Artin, M.: On isolated rational singularities of surfaces. Am. J. Math. 88, 129–136 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  3. Boileau, M., Boyer, S.: Graph manifolds \({\mathbb{Z}}\)-homology 3-spheres and taut foliations. J. Topol. 8(2), 571–585 (1915)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boyer, S., Rolfsen, D., Wiest, D.: Orderable 3-manifold groups. Ann. Inst. Fourier (Grenoble) 55(1), 243–288 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brittenham, M.: Tautly foliated 3-manifolds with no \({\mathbb{R}}\)-covered foliations. In: Walczak, P., Conlon, L., Langevin, R., Tsuboi, T. (eds.) Foliations: Geometry and Dynamics (Warsaw 2000), pp. 213–224. World Scientific Publishing, River edge (2002)

    Chapter  Google Scholar 

  6. Bowden, J.: Approximating \(C^0\)-foliations by contact structures. arXiv:1509.07709 (2015)

  7. Boyer, S., Gordon, CMcA, Watson, L.: On L-spaces and left-orderable fundamental groups. Math. Ann. 356(4), 1213–1245 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Boyer, S., Clay, A.: Foliations, orders, representations, L-spaces and graph manifolds. arXiv:1401.7726 (2014)

  9. Boyer, S., Clay, A.: Slope detections, foliations in graph manifolds, and L-spaces. arXiv:1510.02378 (2015)

  10. Calegari, D., Walker, A.: Ziggurats and rotation numbers. J. Mod. Dyn. 5(4), 711–746 (2011)

    MathSciNet  MATH  Google Scholar 

  11. Chowdhury, S.: Ziggurat fringes are self-similar. arXiv:1503.0422 (2015)

  12. Clay, A., Lidman, T., Watson, L.: Graph manifolds, left-orderability and amalgamation. Algebr. Geom. Topol. 13(4), 2347–2368 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Eisenbud, D., Hirsch, U., Neumann, W.: Transverse foliations of Seifert bundles and self homeomorphisms of the circle. Comment. Math. Helv. 56, 638–660 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gabai, D.: Foliations and genera of links. Topology 23, 381–394 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gordenko, A.S.: Self-similarity of Jankins–Neumann ziggurat. arXiv:1503.0311 (2015)

  16. Gordon, C., Lidman, T.: Taut foliations, left-orderability, and cyclic branched covers. Acta Math. Vietnam. 39, 599–635 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Grauert, H.: Über Modifikationen und exzeptionelle analytische Mengen. Math. Ann. 146, 331–368 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  18. Grauert, H., Remmert, R.: Komplexe Räume. Math. Ann. 136, 245–318 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hanselman, J., Rasmussen, J., Rasmussen, S.D., Watson, L.: Taut foliations on graph manifolds. arXiv:1508.0591 (2015)

  20. Hu, Y.: Left-orderability and cyclic branched coverings. Algebr. Geom. Topol. 15, 399–413 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jankins, M., Neumann, W.D.: Rotation numbers of products of circle homeomorphisms. Math. Ann. 271, 381–400 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kazez, W.H., Roberts, R.: \(C^0\) approximations of foliations. arXiv:1509.08382 (2015)

  23. Kollár, J., Mori, S.: Birational geometry of algebraic varieties, vol. 134. Cambridge University Press, Cambridge (1998)

    Book  MATH  Google Scholar 

  24. Laufer, H.B.: On rational singularities. Am. J. Math. 94, 597–608 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lisca, P., Matić, G.: Transverse contact structures on Seifert 3-manifolds. Algebr. Geom. Topol. 4, 1125–1144 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mauro, M.: On lattice cohomology and left-orderability. arXiv:1308.1890 (2013)

  27. Naimi, R.: Foliations transverse to fibers of Seifert manifolds. Comment. Math. Helv. 69(1), 155–162 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  28. Némethi, A.: Five lectures on normal surface singularities; lectures delivered at the Summer School in Low dimensional topology, Budapest, Hungary 1998. Bolyai Soc. Math. Stud. 8, 269–351 (1999)

    Google Scholar 

  29. Némethi, A.: On the Ozsváth–Szabó invariant of negative definite plumbed 3-manifolds. Geom. Topol. 9, 991–1042 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Némethi, A.: Graded roots and singularities. Singularities in geometry and topology. World Scientific Publishing, Hackensack (2007)

    MATH  Google Scholar 

  31. Némethi, A.: Lattice cohomology of normal surface singularities. Publ. RIMS Kyoto Univ. 44, 507–543 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Némethi, A.: Two exact sequences for lattice cohomology. In: Proceedings of the conference to honor H. Moscovici’s 65th birthday, Contemporary Math, vol. 546, pp. 249–269. (2011)

  33. Neumann, W.D.: A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves. Trans. Am. Math. Soc. 268(2), 299–344 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  34. Ozsváth, P.S., Szabó, Z.: On the Floer homology of plumbed three-manifolds. Geom. Topol. 7, 185–224 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ozsváth, P.S., Szabó, Z.: Holomorphic discs and topological invariants for closed three-manifold. Ann. Math. 159, 1027–1158 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ozsváth, P.S., Szabó, Z.: Holomorphic discs and three-manifold invariants: properties and applications. Ann. Math. 159, 1159–1245 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ozsváth, P.S., Szabó, Z.: Holomorphic discs and genus bounds. Geom. Topol. 8, 311–334 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ozsváth, P.S., Stipsicz, A.I., Szabó, Z.: A spectral sequence on lattice cohomology. Quantum Topol. 5(4), 487–521 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  39. Pinkham, H.: Normal surface singularities with \({\mathbb{C}}^{*}\) action. Math. Ann. 227, 183–193 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  40. Stein, K.: Analytische Zerlegungen komplexer Räume. Math. Ann. 132, 63–93 (1956)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda utca 13-15, 1053, Budapest, Hungary

    András Némethi

  2. ELTE - University of Budapest, Budapest, Hungary

    András Némethi

  3. BCAM - Basque Center for Applied Mathematics, Bilbao, Spain

    András Némethi

Authors
  1. András Némethi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to András Némethi.

Additional information

AN was partially supported by NKFIH Grant 112735 and ERC Adv. Grant LDTBud of A. Stipsicz at Rényi Institute of Math., Budapest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Némethi, A. Links of rational singularities, L-spaces and LO fundamental groups. Invent. math. 210, 69–83 (2017). https://doi.org/10.1007/s00222-017-0724-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-017-0724-6

Mathematics Subject Classification

Search

Navigation