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Strong symplectic fillability of contact torus bundles

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Abstract

In this paper, we study strong symplectic fillability and Stein fillability of some tight contact structures on negative parabolic and negative hyperbolic torus bundles over the circle. For the universally tight contact structure with twisting \(\pi \) in \(S^1\)-direction on a negative parabolic torus bundle, we completely determine its strong symplectic fillability and Stein fillability. For the universally tight contact structure with twisting \(\pi \) in \(S^1\)-direction on a negative hyperbolic torus bundle, we give a necessary condition for it being strongly symplectically fillable. For the virtually overtwisted tight contact structure on the negative parabolic torus bundle with monodromy \(-\,T^n\) (\(n<0\)), we prove that it is Stein fillable. In addition, we give a partial answer to a conjecture of Golla and Lisca.

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References

  1. Bhupal, M., Ozbagci, B.: Canonical contact structures on some singularity links. Bull. Lond. Math. Soc. 46, 576–586 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ding, F., Geiges, H.: Symplectic fillability of tight contact structures on torus bundles. Algebr. Geom. Topol. 1, 153–172 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Eliashberg, Y.: Unique holomorphically fillable contact structure on the 3-torus. Int. Math. Res. Not. 2, 77–82 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Etnyre, J.B.: Private communications (2015)

  5. Etnyre, J.B., Honda, K.: Tight contact structures with no symplectic fillings. Invent. Math. 148(3), 609–626 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Etnyre, J.B., Honda, K.: On symplectic cobordisms. Math. Ann. 323(1), 31–39 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gay, D.T.: Open books and configurations of symplectic surfaces. Algebr. Geom. Topol. 3, 569–586 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gay, D.T.: Correction to: open books and configurations of symplectic surfaces. Algebr. Geom. Topol. 3, 1275–1276 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gay, D.T.: Four-dimensional symplectic cobordisms containing three-handles. Geom. Topol. 10, 1749–1759 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Giroux, E.: Structures de contact en dimension trois et bifurcations des feuilletages de surfaces. Invent. Math. 141(3), 615–689 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Golla, M., Lisca, P.: On Stein fillings of contact torus bundles. Bull. Lond. Math. Soc. 48(1), 19–37 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Golla, M., Lisca, P.: On Stein fillings of contact torus bundles—erratum

  13. Gompf, R.E.: Handlebody construction of Stein surfaces. Ann. Math. (2) 148, 619–693 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gompf, R.E., Stipsicz, A.I.: 4-Manifolds and Kirby Calculus, Graduate Studies in Mathematics, vol. 20. American Mathematical Society, Providence (1999)

    MATH  Google Scholar 

  15. Honda, K.: On the classification of tight contact structures II. J. Differ. Geom. 55(1), 83–143 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kirby, R., Melvin, P.: Dedekind sums, \(\mu \)-invariants and the signature cocycle. Math. Ann. 299(2), 231–267 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, T.J., Mak, C.Y.: Symplectic Divisorial Capping in Dimension 4. arXiv:1407.0564v3

  18. Lisca, P., Stipsicz, A.I.: Tight, not semi-fillable contact circle bundles. Math. Ann. 328(1–2), 285–298 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mark, T.E., Tosun, B.: Naturality of Heegaard Floer invariants under positive rational contact surgery. arXiv:1509.01511v2

  20. McDuff, D.: The structure of rational and ruled symplectic \(4\)-manifolds. J. Am. Math. Soc. 3(3), 679–712 (1990)

    MathSciNet  MATH  Google Scholar 

  21. 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 

  22. Niederkrüger, K., Wendl, C.: Weak symplectic fillings and holomorphic curves. Ann. Sci. Éc. Norm. Supér. (4) 44(5), 801–853 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Taubes, C.H.: SW \(\Rightarrow \)Gr: from the Seiberg–Witten equations to pseudo-holomorphic curves. J. Am. Math. Soc. 9(3), 845–918 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  24. Van Horn-Morris, J.: Constructions of open book decompositions. Ph.D. thesis, The University of Texas at Austin (2007)

Download references

Acknowledgements

Authors would like to thank John Etnyre and Paolo Lisca for useful email correspondence. We are also grateful to the referee(s) for valuable suggestions. The first author is partially supported by Grant No. 11371033 of the National Natural Science Foundation of China. The second author is partially supported by Grant No. 11471212 of the National Natural Science Foundation of China.

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Authors and Affiliations

  1. School of Mathematical Sciences and LMAM, Peking University, Beijing, 100871, China

    Fan Ding

  2. School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China

    Youlin Li

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  1. Fan Ding
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  2. Youlin Li
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Correspondence to Youlin Li.

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Ding, F., Li, Y. Strong symplectic fillability of contact torus bundles. Geom Dedicata 195, 403–415 (2018). https://doi.org/10.1007/s10711-017-0299-9

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