On nonseparating contact hypersurfaces in symplectic 4–manifolds

Albers, Peter¹ ; Bramham, Barney² ; Wendl, Chris³

¹Department of Mathematics, Purdue University, 150 N University Street, West Lafayette, IN 47907-2067, USA
²Max-Planck Institut, Inselstrasse-22, 04103 Leipzig, Germany
³Institut für Mathematik, Humboldt-Universität zu Berlin, 10099 Berlin, Germany

Algebraic and Geometric Topology, Tome 10 (2010) no. 2, pp. 697-737

Résumé

We show that certain classes of contact 3–manifolds do not admit nonseparating contact type embeddings into any closed symplectic 4–manifold, eg this is the case for all contact manifolds that are (partially) planar or have Giroux torsion. The latter implies that manifolds with Giroux torsion do not admit contact type embeddings into any closed symplectic 4–manifold. Similarly, there are symplectic 4–manifolds that can admit smoothly embedded nonseparating hypersurfaces, but not of contact type: we observe that this is the case for all symplectic ruled surfaces.

Détail

DOI : 10.2140/agt.2010.10.697

Keywords: symplectic manifold, contact manifold, pseudoholomorphic curve, separating hypersurface

Bibliographie

[1] C Abbas, Holomorphic open book decompositions

[2] M Bhupal, K Ono, Symplectic fillings of links of quotient surface singularities

[3] F Bourgeois, Y Eliashberg, H Hofer, K Wysocki, E Zehnder, Compactness results in symplectic field theory, Geom. Topol. 7 (2003) 799

[4] Y Eliashberg, Filling by holomorphic discs and its applications, from: "Geometry of low-dimensional manifolds, 2 (Durham, 1989)" (editors S K Donaldson, C B Thomas), London Math. Soc. Lecture Note Ser. 151, Cambridge Univ. Press (1990) 45

[5] Y Eliashberg, A few remarks about symplectic filling, Geom. Topol. 8 (2004) 277

[6] J B Etnyre, Planar open book decompositions and contact structures, Int. Math. Res. Not. (2004) 4255

[7] J B Etnyre, private communication (2008)

[8] J B Etnyre, K Honda, On symplectic cobordisms, Math. Ann. 323 (2002) 31

[9] J B Etnyre, J Van Horn-Morris, Fibered transverse knots and the Bennequin bound

[10] D T Gay, Four-dimensional symplectic cobordisms containing three-handles, Geom. Topol. 10 (2006) 1749

[11] H Geiges, Examples of symplectic $4$–manifolds with disconnected boundary of contact type, Bull. London Math. Soc. 27 (1995) 278

[12] H Geiges, Constructions of contact manifolds, Math. Proc. Cambridge Philos. Soc. 121 (1997) 455

[13] H Geiges, An introduction to contact topology, Cambridge Studies in Advanced Math. 109, Cambridge Univ. Press (2008)

[14] E Giroux, Links and contact structures, Lecture notes, Georgia Topology Conference (2001)

[15] M Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307

[16] M Gromov, Partial differential relations, Ergebnisse der Math. und ihrer Grenzgebiete (3) 9, Springer (1986)

[17] H Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math. 114 (1993) 515

[18] H Hofer, V Lizan, J C Sikorav, On genericity for holomorphic curves in four-dimensional almost-complex manifolds, J. Geom. Anal. 7 (1997) 149

[19] H Hofer, K Wysocki, E Zehnder, Properties of pseudo-holomorphic curves in symplectisations. II. Embedding controls and algebraic invariants, Geom. Funct. Anal. 5 (1995) 270

[20] H Hofer, K Wysocki, E Zehnder, Properties of pseudoholomorphic curves in symplectisation. IV. Asymptotics with degeneracies, from: "Contact and symplectic geometry (Cambridge, 1994)" (editor C B Thomas), Publ. Newton Inst. 8, Cambridge Univ. Press (1996) 78

[21] H Hofer, K Wysocki, E Zehnder, Properties of pseudoholomorphic curves in symplectisations. I. Asymptotics, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996) 337

[22] H Hofer, K Wysocki, E Zehnder, Properties of pseudoholomorphic curves in symplectizations. III. Fredholm theory, from: "Topics in nonlinear analysis" (editors J Escher, G Simonett), Progr. Nonlinear Differential Equations Appl. 35, Birkhäuser (1999) 381

[23] C Hummel, Gromov's compactness theorem for pseudo-holomorphic curves, Progress in Math. 151, Birkhäuser Verlag (1997)

[24] Y Kanda, The classification of tight contact structures on the $3$–torus, Comm. Anal. Geom. 5 (1997) 413

[25] R Lutz, Structures de contact sur les fibrés principaux en cercles de dimension trois, Ann. Inst. Fourier (Grenoble) 27 (1977) 1

[26] D Mcduff, The structure of rational and ruled symplectic $4$–manifolds, J. Amer. Math. Soc. 3 (1990) 679

[27] D Mcduff, Symplectic manifolds with contact type boundaries, Invent. Math. 103 (1991) 651

[28] D Mcduff, D Salamon, $J$–holomorphic curves and symplectic topology, Amer. Math. Soc. Coll. Publ. 52, Amer. Math. Soc. (2004)

[29] E Mora, Pseudoholomorphic cylinders in symplectisations, PhD thesis, New York University (2003)

[30] H Ohta, K Ono, Simple singularities and symplectic fillings, J. Differential Geom. 69 (2005) 1

[31] R Siefring, Intersection theory of punctured pseudoholomorphic curves

[32] R Siefring, C Wendl, Pseudoholomorphic curves, intersections and Morse–Bott asymptotics, in preparation

[33] J Y Welschinger, Effective classes and Lagrangian tori in symplectic four-manifolds, J. Symplectic Geom. 5 (2007) 9

[34] C Wendl, Automatic transversality and orbifolds of punctured holomorphic curves in dimension four, to appear in Comment. Math. Helv.

[35] C Wendl, Compactness for embedded pseudoholomorphic curves in $3$–manifolds, to appear in J. Eur. Math. Soc. (JEMS)

[36] C Wendl, Contact fiber sums, monodromy maps and symplectic fillings, in preparation

[37] C Wendl, Holomorphic curves in blown up open books

[38] C Wendl, Open book decompositions and stable Hamiltonian structures, to appear in Expos. Math.

[39] C Wendl, Punctured holomorphic curves with boundary in $3$–manifolds: Fredholm theory and embededdness, in preparation

[40] C Wendl, Strongly fillable contact manifolds and $J$–holomorphic foliations, to appear in Duke Math. J.

[41] C Wendl, Finite energy foliations and surgery on transverse links, PhD thesis, New York University (2005)

[42] K Zehmisch, The Eliashberg–Gromov tightness theorem, Diplom Thesis, Universität Leipzig (2003)