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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie


IMPACT FACTOR 2018: 1.859

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1435-5345
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Volume 2016, Issue 718

Issues

Tight contact structures on the Brieskorn spheres -Σ(2,3,6n-1) and contact invariants

Paolo Ghiggini / Jeremy Van Horn-Morris
Published Online: 2014-06-11 | DOI: https://doi.org/10.1515/crelle-2014-0038

Abstract

We compute the Ozsváth–Szabó contact invariants for all tight contact structures on the manifolds -Σ(2,3,6n-1) using twisted coefficients and a previous computation by the first author and Ko Honda. This computation completes the classification of the tight contact structures in this family of 3-manifolds.

References

  • [1]

    Dehn M., Die Gruppe der Abbildungsklassen, Acta Math. 69 (1938), no. 1, 135–206. Google Scholar

  • [2]

    Etnyre J. and Honda K., On the nonexistence of tight contact structures, Ann. of Math. (2) 153 (2001), no. 3, 749–766. Google Scholar

  • [3]

    Ghiggini P., Strongly fillable contact 3-manifolds without Stein fillings, Geom. Topol. 9 (2005), 1677–1687. Google Scholar

  • [4]

    Ghiggini P., Infinitely many universally tight contact manifolds with trivial Ozsváth–Szabó contact invariants, Geom. Topol. 10 (2006), 335–357. Google Scholar

  • [5]

    Ghiggini P., Ozsváth–Szabó invariants and fillability of contact structures, Math. Z. 253 (2006), no. 1, 159–175. Google Scholar

  • [6]

    Ghiggini P. and Honda K., Giroux torsion and twisted coefficients, preprint 2008, http://arxiv.org/abs/0804.1568.

  • [7]

    Ghiggini P., Honda K. and Van Horn-Morris J., The vanishing of the contact invariant in the presence of torsion, preprint 2008, http://arxiv.org/abs/0706.1602.

  • [8]

    Ghiggini P. and Schönenberger S., On the classification of tight contact structures, Topology and geometry of manifolds, Proc. Sympos. Pure Math. 71, American Mathematical Society, Providence (2003), 121–151. Google Scholar

  • [9]

    Giroux E., Une infinité de structures de contact tendues sur une infinité de variétés, Invent. Math. 135 (1999), no. 3, 789–802. Google Scholar

  • [10]

    Giroux E., Structures de contact en dimension trois et bifurcations des feuilletages de surfaces, Invent. Math. 141 (2000), no. 3, 615–689. Google Scholar

  • [11]

    Gompf R., Handlebody construction of Stein surfaces, Ann. of Math. (2) 148 (1998), no. 2, 619–693. Google Scholar

  • [12]

    Greenberg M. and Harper J., Algebraic topology. A first course, Math. Lecture Note Ser. 58, Benjamin/ Cummings Publishing, Reading 1981. Google Scholar

  • [13]

    Honda K., On the classification of tight contact structures I, Geom. Topol. 4 (2000), 309–368. Google Scholar

  • [14]

    Honda K., On the classification of tight contact structures II, J. Differential Geom. 55 (2000), no. 1, 83–143. Google Scholar

  • [15]

    Jabuka S. and Mark T., Product formulae for Ozsváth–Szabó 4-manifold invariants, Geom. Topol. 12 (2008), no. 3, 1557–1651. Google Scholar

  • [16]

    Johnson D., Homeomorphisms of a surface which act trivially on homology, Proc. Amer. Math. Soc. 75 (1979), no. 1, 119–125. Google Scholar

  • [17]

    Lisca P. and Matić G., Tight contact structures and Seiberg–Witten invariants, Invent. Math. 129 (1997), no. 3, 509–525. Google Scholar

  • [18]

    Lisca P. and Stipsicz A., Ozsváth-Szabó invariants and tight contact three-manifolds. I, Geom. Topol. 8 (2004), 925–945. Google Scholar

  • [19]

    Onaran S., Legendrian knots and open book decompositions, Ph.D. thesis, Middle East Technical University 2009. Google Scholar

  • [20]

    Ozsváth P. and Szabó Z., Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003), no. 2, 179–261. Google Scholar

  • [21]

    Ozsváth P. and Szabó Z., Holomorphic disks and genus bounds, Geom. Topol. 8 (2004), 311–334. Google Scholar

  • [22]

    Ozsváth P. and Szabó Z., Holomorphic disks and three-manifold invariants: Properties and applications, Ann. of Math. (2) 159 (2004), no. 3, 1159–1245. Google Scholar

  • [23]

    Ozsváth P. and Szabó Z., Heegaard Floer homology and contact structures, Duke Math. J. 129 (2005), no. 1, 39–61. Google Scholar

  • [24]

    Ozsváth P. and Szabó Z., Holomorphic triangles and invariants for smooth four-manifolds, Adv. Math. 202 (2006), no. 2, 326–400. Google Scholar

  • [25]

    Plamenevskaya O., Contact structures with distinct Heegaard Floer invariants, Math. Res. Lett. 11 (2004), no. 4, 547–561. Google Scholar

  • [26]

    Van Horn-Morris J., Contructions of open book decompositions, Ph.D. dissertation, University of Texas at Austin 2007. Google Scholar

  • [27]

    Wu H., On Legendrian surgeries, Math. Res. Lett. 14 (2007), no. 3, 513–530. Google Scholar

About the article

Received: 2011-01-26

Revised: 2014-01-16

Published Online: 2014-06-11

Published in Print: 2016-09-01


Funding Source: Agence Nationale de la Recherche

Award identifier / Grant number: ‘Floer Power’

The first author acknowledges partial support from the ANR project ‘Floer Power.’


Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2016, Issue 718, Pages 1–24, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle-2014-0038.

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