Kyoto Journal of Mathematics

Contact structures on plumbed 3-manifolds

Çağrı Karakurt

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We show that the Ozsváth–Szabó contact invariant c+(ξ)HF+(Y) of a contact 3-manifold (Y,ξ) can be calculated combinatorially if Y is the boundary of a certain type of plumbing X and if ξ is induced by a Stein structure on X. Our technique uses an algorithm of Ozsváth and Szabó to determine the Heegaard–Floer homology of such 3-manifolds. We discuss two important applications of this technique in contact topology. First, we show that it simplifies the calculation of the Ozsváth–Stipsicz–Szabó obstruction to admitting a planar open book for a certain class of contact structures. We also define a numerical invariant of contact manifolds that respects a partial ordering induced by Stein cobordisms. Using this technique, we do a sample calculation showing that the invariant can get infinitely many distinct values.

Article information

Kyoto J. Math., Volume 54, Number 2 (2014), 271-294.

First available in Project Euclid: 2 June 2014

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Zentralblatt MATH identifier

Primary: 57R17: Symplectic and contact topology
Secondary: 57R58: Floer homology 57R65: Surgery and handlebodies 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX]


Karakurt, Çağrı. Contact structures on plumbed 3-manifolds. Kyoto J. Math. 54 (2014), no. 2, 271--294. doi:10.1215/21562261-2642395.

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  • [1] J. Baldwin and O. Plamenevskaya, Khovanov homology, open books, and tight contact structures, Adv. Math. 224 (2010), 2544–2582.
  • [2] F. Ding and H. Geiges, A Legendrian surgery presentation of contact 3-manifolds, Math. Proc. Cambridge Philos. Soc. 136 (2004), 583–598.
  • [3] S. Durusoy, Heegaard–Floer homology and a family of Brieskorn spheres, preprint, arXiv:math/0405524v1 [math.GT].
  • [4] Y. Eliashberg, Topological characterization of Stein manifolds of dimension $>2$, Internat. J. Math. 1 (1990) 29–46.
  • [5] J. B. Etnyre, Planar open book decompositions and contact structures, Int. Math. Res. Not. IMRN 2004, no. 79, 4255–4267.
  • [6] J. B. Etnyre and K. Honda, On symplectic cobordisms, Math. Ann. 323 (2002), 31–39.
  • [7] J. B. Etnyre and B. Ozbagci, Open books and plumbings, Int. Math. Res. Not. IMRN 2006, no. 17, art. ID 72710.
  • [8] J. B. Etnyre and B. Ozbagci, Invariants of contact structures from open books, Trans. Amer. Math. Soc. 360 (2008), no. 6, 3133–3151.
  • [9] D. T. Gay, Four-dimensional symplectic cobordisms containing three-handles, Geom. Topol. 10 (2006), 1749–1759.
  • [10] P. Ghiggini, Ozsváth-Szabó invariants and fillability of contact structures, Math. Z. 253 (2006), 159–175.
  • [11] E. Giroux, “Géométrie de contact: de la dimension trois vers les dimensions supérieures” in Proceedings of the International Congress of Mathematicians, Vol. 2 (Beijing, 2002), Higher Ed. Press, Beijing, 2002, 405–414.
  • [12] K. Honda, W. H. Kazez, and G. Matić, On the contact class in Heegaard Floer homology, J. Differential Geom. 83 (2009), 289–311.
  • [13] Ç. Karakurt, Knot Floer homology and contact surgeries, in preparation.
  • [14] J. Latschev and C. Wendl, Algebraic torsion in contact manifolds, with an appendix by M. Hutchings, Geom. Funct. Anal. 21 (2011), 1144–1195.
  • [15] P. Lisca and A. Stipsicz, Ozsváth-Szabó invariants and tight contact 3-manifolds III, J. Symplectic Geom. 5 (2007), 357–384.
  • [16] H. Ohta and K. Ono, Simple singularities and topology of symplectically filling 4-manifold, Comment. Math. Helv. 74 (1999), 575–590.
  • [17] P. Ozsváth, A. Stipsicz, and Z. Szabó, Planar open books and Floer homology, Int. Math. Res. Not. IMRN 2005, no. 54, 3385–3401.
  • [18] P. Ozsváth and Z. Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003), 179–261.
  • [19] P. Ozsváth and Z. Szabó, On the Floer homology of plumbed three-manifolds, Geom. Topol. 7 (2003), 185–224.
  • [20] P.Ozsváth and Z. Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. (2) 159 (2004), 1027–1158.
  • [21] P. Ozsváth and Z. Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. (2) 159 (2004), 1159–1245.
  • [22] P. Ozsváth and Z. Szabó, Heegaard Floer homology and contact structures, Duke Math. J. 129 (2005), 39–61.
  • [23] P. Ozsváth and Z. Szabó, Holomorphic triangles and invariants for smooth four-manifolds, Adv. Math. 202 (2006), 326–400.
  • [24] O. Plamenevskaya, Contact structures with distinct Heegaard Floer invariants, Math. Res. Lett. 11 (2004), 547–561.
  • [25] O. Plamenevskaya, A combinatorial description of the Heegaard Floer contact invariant, Algebr. Geom. Topol. 7 (2007), 1201–1209.
  • [26] R. Rustamov, Calculation of Heegard Floer homology for a class of Brieskorn spheres, preprint, arXiv:math/0312071v1 [math.SG].
  • [27] R. Rustamov, On Heegaard-Floer homology of plumbed three-manifolds with $b_{1}=1$, preprint, arXiv:math/0405118v1 [math.SG].