Invariants of contact structures from open books

Etnyre, John; Ozbagci, Burak

doi:10.1090/S0002-9947-08-04459-0

Invariants of contact structures from open books
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by John B. Etnyre and Burak Ozbagci PDF

Trans. Amer. Math. Soc. 360 (2008), 3133-3151 Request permission

Abstract:

In this note we define three invariants of contact structures in terms of open books supporting the contact structures. These invariants are the support genus (which is the minimal genus of a page of a supporting open book for the contact structure), the binding number (which is the minimal number of binding components of a supporting open book for the contact structure with minimal genus pages) and the norm (which is minus the maximal Euler characteristic of a page of a supporting open book).

References

Fan Ding, Hansjörg Geiges, and András I. Stipsicz, Surgery diagrams for contact 3-manifolds, Turkish J. Math. 28 (2004), no. 1, 41–74. MR 2056760
Yakov Eliashberg, Filling by holomorphic discs and its applications, Geometry of low-dimensional manifolds, 2 (Durham, 1989) London Math. Soc. Lecture Note Ser., vol. 151, Cambridge Univ. Press, Cambridge, 1990, pp. 45–67. MR 1171908
John B. Etnyre, Planar open book decompositions and contact structures, Int. Math. Res. Not. 79 (2004), 4255–4267. MR 2126827, DOI 10.1155/S1073792804142207
John B. Etnyre, Lectures on open book decompositions and contact structures, Floer homology, gauge theory, and low-dimensional topology, Clay Math. Proc., vol. 5, Amer. Math. Soc., Providence, RI, 2006, pp. 103–141. MR 2249250
John B. Etnyre and Terry Fuller, Realizing 4-manifolds as achiral Lefschetz fibrations, Int. Math. Res. Not. , posted on (2006), Art. ID 70272, 21. MR 2219214, DOI 10.1155/IMRN/2006/70272
John B. Etnyre and Burak Ozbagci, Open books and plumbings, Int. Math. Res. Not. , posted on (2006), Art. ID 72710, 17. MR 2272094, DOI 10.1155/IMRN/2006/72710
Hansjörg Geiges, Constructions of contact manifolds, Math. Proc. Cambridge Philos. Soc. 121 (1997), no. 3, 455–464. MR 1434654, DOI 10.1017/S0305004196001260
Emmanuel Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 405–414 (French, with French summary). MR 1957051
Robert E. Gompf, Handlebody construction of Stein surfaces, Ann. of Math. (2) 148 (1998), no. 2, 619–693. MR 1668563, DOI 10.2307/121005
Robert E. Gompf and András I. Stipsicz, $4$-manifolds and Kirby calculus, Graduate Studies in Mathematics, vol. 20, American Mathematical Society, Providence, RI, 1999. MR 1707327, DOI 10.1090/gsm/020
U. Hirsch and W. D. Neumann, On cyclic branched coverings of spheres, Math. Ann. 215 (1975), 289–291. MR 375321, DOI 10.1007/BF01343895
K. Honda, W. Kazez and G. Matic, Right-veering diffeomorphisms of compact surfaces with boundary I, preprint 2005.
Walter D. Neumann and Lee Rudolph, Difference index of vectorfields and the enhanced Milnor number, Topology 29 (1990), no. 1, 83–100. MR 1046626, DOI 10.1016/0040-9383(90)90026-G
Burak Ozbagci and András I. Stipsicz, Surgery on contact 3-manifolds and Stein surfaces, Bolyai Society Mathematical Studies, vol. 13, Springer-Verlag, Berlin; János Bolyai Mathematical Society, Budapest, 2004. MR 2114165, DOI 10.1007/978-3-662-10167-4
Peter Ozsváth, András Stipsicz, and Zoltán Szabó, Planar open books and Floer homology, Int. Math. Res. Not. 54 (2005), 3385–3401. MR 2200085, DOI 10.1155/IMRN.2005.3385
Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 0515288
S. Schönenberger, Planar open books and symplectic fillings, Ph.D. Dissertation, the University of Pennsylvania, 2005.
W. P. Thurston and H. E. Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc. 52 (1975), 345–347. MR 375366, DOI 10.1090/S0002-9939-1975-0375366-7
J. Van Horn, Ph.D. Dissertation, the University of Texas at Austin, in preparation.