Tight contact structures via dynamics

Etnyre, John; Ghrist, Robert

doi:10.1090/S0002-9939-99-05377-0

Tight contact structures via dynamics
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by John Etnyre and Robert Ghrist PDF

Proc. Amer. Math. Soc. 127 (1999), 3697-3706 Request permission

Abstract:

We consider the problem of realizing tight contact structures on closed orientable three-manifolds. By applying the theorems of Hofer et al., one may deduce tightness from dynamical properties of (Reeb) flows transverse to the contact structure. We detail how two classical constructions, Dehn surgery and branched covering, may be performed on dynamically-constrained links in such a way as to preserve a transverse tight contact structure.

References

B. Aebisher, Symplectic geometry, Progress in Math., no. 124, Birkhaüser, Berlin, 1994.
D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature. , Proceedings of the Steklov Institute of Mathematics, No. 90 (1967), American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by S. Feder. MR 0242194
Daniel Bennequin, Entrelacements et équations de Pfaff, Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982) Astérisque, vol. 107, Soc. Math. France, Paris, 1983, pp. 87–161 (French). MR 753131
Yves Benoist, Patrick Foulon, and François Labourie, Flots d’Anosov à distributions stable et instable différentiables, J. Amer. Math. Soc. 5 (1992), no. 1, 33–74 (French, with English summary). MR 1124979, DOI 10.1090/S0894-0347-1992-1124979-1
T. Dombre, U. Frisch, J. M. Greene, M. Hénon, A. Mehr, and A. M. Soward, Chaotic streamlines in the ABC flows, J. Fluid Mech. 167 (1986), 353–391. MR 851673, DOI 10.1017/S0022112086002859
Y. Eliashberg, Classification of overtwisted contact structures on $3$-manifolds, Invent. Math. 98 (1989), no. 3, 623–637. MR 1022310, DOI 10.1007/BF01393840
Yakov Eliashberg, Topological characterization of Stein manifolds of dimension $>2$, Internat. J. Math. 1 (1990), no. 1, 29–46. MR 1044658, DOI 10.1142/S0129167X90000034
Yakov Eliashberg, Contact $3$-manifolds twenty years since J. Martinet’s work, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 1-2, 165–192 (English, with French summary). MR 1162559
Yakov M. Eliashberg and William P. Thurston, Confoliations, University Lecture Series, vol. 13, American Mathematical Society, Providence, RI, 1998. MR 1483314, DOI 10.1090/ulect/013
J. Etnyre, Tight contact structures on lens spaces, Preprint, 1997.
A. Fathi, F. Laudenbach, and V. Poenaru et al., Travaux de Thurston sur les surfaces, Astérique 66-67 (1979).
David Gabai, Foliations and the topology of $3$-manifolds, J. Differential Geom. 18 (1983), no. 3, 445–503. MR 723813
Emmanuel Giroux, Une structure de contact, même tendue, est plus ou moins tordue, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 6, 697–705 (French, with English summary). MR 1307678
R. Gompf, Handlebody construction of Stein surfaces, Preprint, 1996.
Jesús Gonzalo, Branched covers and contact structures, Proc. Amer. Math. Soc. 101 (1987), no. 2, 347–352. MR 902554, DOI 10.1090/S0002-9939-1987-0902554-9
M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347. MR 809718, DOI 10.1007/BF01388806
H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math. 114 (1993), no. 3, 515–563. MR 1244912, DOI 10.1007/BF01232679
H. Hofer and M. Kriener, Holomorphic curves in contact dynamics, Notes from 1997 Park City Mathematics Institute, July 1997.
Helmut Hofer and Eduard Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 1994. MR 1306732, DOI 10.1007/978-3-0348-8540-9
Yutaka Kanda, The classification of tight contact structures on the $3$-torus, Comm. Anal. Geom. 5 (1997), no. 3, 413–438. MR 1487723, DOI 10.4310/CAG.1997.v5.n3.a2
W. B. R. Lickorish, A representation of orientable combinatorial $3$-manifolds, Ann. of Math. (2) 76 (1962), 531–540. MR 151948, DOI 10.2307/1970373
J. Martinet, Formes de contact sur les variétés de dimension $3$, Proceedings of Liverpool Singularities Symposium, II (1969/1970), Lecture Notes in Math., Vol. 209, Springer, Berlin, 1971, pp. 142–163 (French). MR 0350771
Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. Oxford Science Publications. MR 1373431
Joseph F. Plante, Anosov flows, Amer. J. Math. 94 (1972), 729–754. MR 377930, DOI 10.2307/2373755
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
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
Andrew H. Wallace, Modifications and cobounding manifolds, Canadian J. Math. 12 (1960), 503–528. MR 125588, DOI 10.4153/CJM-1960-045-7
Alan Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J. 20 (1991), no. 2, 241–251. MR 1114405, DOI 10.14492/hokmj/1381413841