Tightness is preserved by Legendrian surgery

Abstract

This paper describes a characterization of tightness of closed contact 3-manifolds in terms of supporting open book decompositions. The main result is that tightness of a closed contact 3-manifold is preserved under Legendrian surgery.

  • [d] Go to document S. K. Donaldson, "Polynomial invariants for smooth four-manifolds," Topology, vol. 29, iss. 3, pp. 257-315, 1990.
    @article{d, mrkey = {1066174},
      author = {Donaldson, S. K.},
      title = {Polynomial invariants for smooth four-manifolds},
      journal = {Topology},
      fjournal = {Topology. An International Journal of Mathematics},
      volume = {29},
      year = {1990},
      number = {3},
      pages = {257--315},
      issn = {0040-9383},
      coden = {TPLGAF},
      mrclass = {57R57 (57N13 57R55 58D27)},
      mrnumber = {1066174},
      mrreviewer = {Ronald J. Stern},
      doi = {10.1016/0040-9383(90)90001-Z},
      zblnumber = {0715.57007},
      }
  • [el] Go to document Y. Eliashberg, "Classification of overtwisted contact structures on $3$-manifolds," Invent. Math., vol. 98, iss. 3, pp. 623-637, 1989.
    @article{el, mrkey = {1022310},
      author = {Eliashberg, Y.},
      title = {Classification of overtwisted contact structures on {$3$}-manifolds},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {98},
      year = {1989},
      number = {3},
      pages = {623--637},
      issn = {0020-9910},
      coden = {INVMBH},
      mrclass = {53C15 (53C57 58A15 58C27)},
      mrnumber = {1022310},
      mrreviewer = {Yong-Geun Oh},
      doi = {10.1007/BF01393840},
      zblnumber = {0684.57012},
      }
  • [el2] Go to document Y. Eliashberg, "Topological characterization of Stein manifolds of dimension $>2$," Internat. J. Math., vol. 1, iss. 1, pp. 29-46, 1990.
    @article{el2, mrkey = {1044658},
      author = {Eliashberg, Yakov},
      title = {Topological characterization of {S}tein manifolds of dimension {$>2$}},
      journal = {Internat. J. Math.},
      fjournal = {International Journal of Mathematics},
      volume = {1},
      year = {1990},
      number = {1},
      pages = {29--46},
      issn = {0129-167X},
      mrclass = {32E10 (32F05 53C15 57R45)},
      mrnumber = {1044658},
      mrreviewer = {R. R. Simha},
      doi = {10.1142/S0129167X90000034},
      zblnumber = {0699.58002},
      }
  • [el3] Y. Eliashberg, "Filling by holomorphic discs and its applications," in Geometry of Low-Dimensional Manifolds, 2, Cambridge: Cambridge Univ. Press, 1990, vol. 151, pp. 45-67.
    @incollection{el3, mrkey = {1171908},
      author = {Eliashberg, Yakov},
      title = {Filling by holomorphic discs and its applications},
      booktitle = {Geometry of Low-Dimensional Manifolds, 2},
      venue = {{D}urham, 1989},
      series = {London Math. Soc. Lecture Note Ser.},
      volume = {151},
      pages = {45--67},
      publisher = {Cambridge Univ. Press},
      address = {Cambridge},
      year = {1990},
      mrclass = {53C23 (32D10 53C15 58E99)},
      mrnumber = {1171908},
      mrreviewer = {Nikolai K. Smolentsev},
      zblnumber = {0731.53036},
      }
  • [e] J. B. Etnyre, "Lectures on open book decompositions and contact structures," in Floer Homology, Gauge Theory, and Low-Dimensional Topology, Providence, RI: Amer. Math. Soc., 2006, vol. 5, pp. 103-141.
    @incollection{e, mrkey = {2249250},
      author = {Etnyre, John B.},
      title = {Lectures on open book decompositions and contact structures},
      booktitle = {Floer Homology, Gauge Theory, and Low-Dimensional Topology},
      series = {Clay Math. Proc.},
      volume = {5},
      pages = {103--141},
      publisher = {Amer. Math. Soc.},
      address = {Providence, RI},
      year = {2006},
      mrclass = {57R17 (53D35)},
      mrnumber = {2249250},
      mrreviewer = {Paolo Ghiggini},
      zblnumber = {1108.53050},
      }
  • [gi] E. Giroux, "Géométrie de contact: de la dimension trois vers les dimensions supérieures," in Proceedings of the International Congress of Mathematicians, Vol. II, Beijing, 2002, pp. 405-414.
    @inproceedings{gi, mrkey = {1957051},
      author = {Giroux, Emmanuel},
      title = {Géométrie de contact: de la dimension trois vers les dimensions supérieures},
      booktitle = {Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. {II}},
      venue = {{B}eijing, 2002},
      pages = {405--414},
      publisher = {Higher Ed. Press},
      address = {Beijing},
      year = {2002},
      mrclass = {53D35 (57M50 57R17)},
      mrnumber = {1957051},
      mrreviewer = {Hansj{ö}rg Geiges},
      zblnumber = {1015.53049},
      }
  • [go] Go to document N. Goodman, "Overtwisted open books from sobering arcs," Algebr. Geom. Topol., vol. 5, pp. 1173-1195, 2005.
    @article{go, mrkey = {2171807},
      author = {Goodman, Noah},
      title = {Overtwisted open books from sobering arcs},
      journal = {Algebr. Geom. Topol.},
      fjournal = {Algebraic \& Geometric Topology},
      volume = {5},
      year = {2005},
      pages = {1173--1195},
      issn = {1472-2747},
      mrclass = {57R17 (57M50 57N10)},
      mrnumber = {2171807},
      mrreviewer = {Hans U. Boden},
      doi = {10.2140/agt.2005.5.1173},
      zblnumber = {1090.57020},
      }
  • [gr] Go to document M. Gromov, "Pseudo holomorphic curves in symplectic manifolds," Invent. Math., vol. 82, pp. 307-347, 1985.
    @article{gr,
      author = {Gromov, M.},
      title = {Pseudo holomorphic curves in symplectic manifolds},
      journal = {Invent. Math.},
      volume = {82},
      year = {1985},
      pages = {307--347},
      zblnumber = {0592.53025},
      mrnumber = {0809718},
      doi = {10.1007/BF01388806},
      }
  • [ho1] Go to document K. Honda, "On the classification of tight contact structures. I," Geom. Topol., vol. 4, pp. 309-368, 2000.
    @article{ho1, mrkey = {1786111},
      author = {Honda, Ko},
      title = {On the classification of tight contact structures. {I}},
      journal = {Geom. Topol.},
      fjournal = {Geometry and Topology},
      volume = {4},
      year = {2000},
      pages = {309--368},
      issn = {1465-3060},
      mrclass = {53D35 (57M50 57R17)},
      mrnumber = {1786111},
      mrreviewer = {Hansj{ö}rg Geiges},
      doi = {10.2140/gt.2000.4.309},
      zblnumber = {0980.57010},
      }
  • [ho2] Go to document K. Honda, "Gluing tight contact structures," Duke Math. J., vol. 115, iss. 3, pp. 435-478, 2002.
    @article{ho2, mrkey = {1940409},
      author = {Honda, Ko},
      title = {Gluing tight contact structures},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {115},
      year = {2002},
      number = {3},
      pages = {435--478},
      issn = {0012-7094},
      coden = {DUMJAO},
      mrclass = {53D35 (57R17)},
      mrnumber = {1940409},
      mrreviewer = {Vincent Colin},
      doi = {10.1215/S0012-7094-02-11532-4},
      zblnumber = {1026.53049},
      }
  • [hkm] Go to document K. Honda, W. H. Kazez, and G. Matić, "Right-veering diffeomorphisms of compact surfaces with boundary," Invent. Math., vol. 169, iss. 2, pp. 427-449, 2007.
    @article{hkm, mrkey = {2318562},
      author = {Honda, Ko and Kazez, William H. and Mati{ć},
      Gordana},
      title = {Right-veering diffeomorphisms of compact surfaces with boundary},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {169},
      year = {2007},
      number = {2},
      pages = {427--449},
      issn = {0020-9910},
      coden = {INVMBH},
      mrclass = {57R17 (53D35)},
      mrnumber = {2318562},
      mrreviewer = {John B. Etnyre},
      doi = {10.1007/s00222-007-0051-4},
      zblnumber = {1167.57008},
      }
  • [hkm2] Go to document K. Honda, W. H. Kazez, and G. Matić, "On the contact class in Heegaard Floer homology," J. Differential Geom., vol. 83, iss. 2, pp. 289-311, 2009.
    @article{hkm2, mrkey = {2577470},
      author = {Honda, Ko and Kazez, William H. and Mati{ć},
      Gordana},
      title = {On the contact class in {H}eegaard {F}loer homology},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {83},
      year = {2009},
      number = {2},
      pages = {289--311},
      issn = {0022-040X},
      coden = {JDGEAS},
      mrclass = {57R17 (53D35 57R58)},
      mrnumber = {2577470},
      mrreviewer = {Margaret F. Symington},
      url = {http://projecteuclid.org/euclid.jdg/1261495333},
      zblnumber = {1186.53098},
      }
  • [t] Go to document C. H. Taubes, "The Seiberg-Witten and Gromov invariants," Math. Res. Lett., vol. 2, iss. 2, pp. 221-238, 1995.
    @article{t, mrkey = {1324704},
      author = {Taubes, Clifford Henry},
      title = {The {S}eiberg-{W}itten and {G}romov invariants},
      journal = {Math. Res. Lett.},
      fjournal = {Mathematical Research Letters},
      volume = {2},
      year = {1995},
      number = {2},
      pages = {221--238},
      issn = {1073-2780},
      mrclass = {57R57 (57R15 58D15 58D29)},
      mrnumber = {1324704},
      mrreviewer = {Jim A. Bryan},
      doi = {10.4310/MRL.1995.v2.n2.a10},
      zblnumber = {0854.57020},
      }
  • [w2] A. Wand, Tightness and open book decompositions.
    @misc{w2,
      author = {Wand, Andrew},
      title = {Tightness and open book decompositions},
      note = {in preparation},
      }
  • [w3] Go to document A. Wand, Detecting tightness via open book decompositions.
    @misc{w3,
      author = {Wand, Andrew},
      title = {Detecting tightness via open book decompositions},
      note = {preprint},
      url = {http://www.andywand.com/bonn_lectures_current_version.pdf},
      }
  • [we] Go to document A. Weinstein, "Contact surgery and symplectic handlebodies," Hokkaido Math. J., vol. 20, iss. 2, pp. 241-251, 1991.
    @article{we, mrkey = {1114405},
      author = {Weinstein, Alan},
      title = {Contact surgery and symplectic handlebodies},
      journal = {Hokkaido Math. J.},
      fjournal = {Hokkaido Mathematical Journal},
      volume = {20},
      year = {1991},
      number = {2},
      pages = {241--251},
      issn = {0385-4035},
      coden = {HMAJDN},
      mrclass = {53C15 (57R65)},
      mrnumber = {1114405},
      mrreviewer = {Yong-Geun Oh},
      doi = {10.14492/hokmj/1381413841},
      zblnumber = {0737.57012},
      }

Authors

Andy Wand

Université de Nantes, Nantes, France