Annali di Matematica Pura ed Applicata

, Volume 178, Issue 1, pp 81–102 | Cite as

Branched spines and contact structures on 3-manifolds

  • Riccardo Benedetti
  • Carlo Petronio


We introduce and analyze the characteristic foliation induced by a contact structure on a branched surface, in particular a branched standard spine of a 3-manifold. We extend to (fairly general) singular foliations of branched surfaces the local existence and uniqueness results which hold for genuine surfaces. Moreover we show that global uniqueness holds when restricting to tight structures. We establish branched versions of the elimination lemma. We prove a smooth version of the Gillman-Rolfsen PL-embedding theorem, deducing that branched spines can be used to construct contact structures in a given homotopy class of plane fields.

Mathematics Subject Classification (1991)

57N10 (primary) 57R15 57R25 (secondary) 


  1. [1]
    B. Aebischer et al.,Simplectic Topology: an Introduction Based on the Seminar in Bern, 1992, Progr. in Math. 124, Birkhäuser Verlag, Basel, 1994.Google Scholar
  2. [2]
    R. BenedettiC. Petronio,Branched Standard Spines of 3-manifolds, Lecture Notes in Math. 1653, Springer-Verlag, Berlin, 1997.zbMATHGoogle Scholar
  3. [3]
    R. BenedettiC. Petronio,A finite graphic calculus for 3-manifolds, Manuscripta Math.,88 (1995), pp. 291–310.zbMATHMathSciNetGoogle Scholar
  4. [4]
    J. Christy,Branched surfaces and attractors I, Trans. Amer. Math. Soc.,336 (1993), pp. 759–784.zbMATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    Ya. Eliashberg,Classification of overtwisted contact structures, Invent. Math.,98 (1989). pp. 623–637.zbMATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    Ya. Eliashberg,Contact 3-manifolds twenty years since J. Martinet's work, Ann. Inst. Fourier (Grenoble),42, (1992), pp. 165–192.zbMATHMathSciNetGoogle Scholar
  7. [7]
    D. GillmanD. Rolfsen,The Zeeman conjecture for standard spines is equivalent to the Poincaré conjecture, Topology,22 (1983), pp. 315–323.zbMATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    D. GillmanD. Rolfsen,Three-manifolds embed in small 3-complexes, Int. J. Math.,3 (1992), pp. 179–183.zbMATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    E. Giorgi,Characteristic foliations of spheres embedded in the standard overtwisted structure (ℝ3, ζ1), To appear in Geom. Dedicata.Google Scholar
  10. [10]
    E. Giroux,Convexité en topologie de contact, Comm. Math. Helv.,66 (1991), pp. 637–677.zbMATHMathSciNetGoogle Scholar
  11. [11]
    E. Giroux,Topologie de contact en dimension 3, Sém. Bourbaki,760 (1992–93) pp. 7–33.Google Scholar
  12. [12]
    E. Giroux, Seminars given in Pisa in April 1997, paper in preparation.Google Scholar
  13. [13]
    I. Ishii,Moves for flow-spines and topological invariants of 3-manifolds. Tokyo J. Math.15 (1992), pp. 297–312.zbMATHMathSciNetCrossRefGoogle Scholar
  14. [14]
    P. Lisca,Symplectic fillings and positive scalar curvature, preprint, Pisa, 1998.Google Scholar

Copyright information

© Fondazione Annali di Matematica Pure ed Applicata 2000

Authors and Affiliations

  • Riccardo Benedetti
    • 1
  • Carlo Petronio
    • 1
  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly

Personalised recommendations