Tight small Seifert fibered manifolds with e0 = −2

Tosun, Bülent¹

¹Department of Mathematics, University of Alabama, Tuscaloosa, AL, United States

Algebraic and Geometric Topology, Tome 20 (2020) no. 1, pp. 1-27

Résumé

We provide the classification of tight contact structures on some small Seifert fibered manifolds. As an application of this classification, combined with work of Lekili, we obtain infinitely many counterexamples to a question of Honda, Kazez and Matić that asks whether a right-veering, nondestabilizable open book necessarily supports a tight contact structure.

Détail

DOI : 10.2140/agt.2020.20.1

Classification : 57M50, 53D10
Keywords: small Seifert fibered spaces, tight contact structures, classification of tight structures

Bibliographie

[1] V Colin, E Giroux, K Honda, Finitude homotopique et isotopique des structures de contact tendues, Publ. Math. Inst. Hautes Études Sci. 109 (2009) 245 | DOI

[2] F Ding, H Geiges, Symplectic fillability of tight contact structures on torus bundles, Algebr. Geom. Topol. 1 (2001) 153 | DOI

[3] J B Etnyre, K Honda, On the nonexistence of tight contact structures, Ann. of Math. 153 (2001) 749 | DOI

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

[5] P Ghiggini, Strongly fillable contact 3–manifolds without Stein fillings, Geom. Topol. 9 (2005) 1677 | DOI

[6] P Ghiggini, Ozsváth–Szabó invariants and fillability of contact structures, Math. Z. 253 (2006) 159 | DOI

[7] P Ghiggini, On tight contact structures with negative maximal twisting number on small Seifert manifolds, Algebr. Geom. Topol. 8 (2008) 381 | DOI

[8] P Ghiggini, K Honda, Giroux torsion and twisted coefficients, (2008)

[9] P Ghiggini, K Honda, J Van Horn-Morris, The vanishing of the contact invariant in the presence of torsion, preprint (2007)

[10] P Ghiggini, P Lisca, A I Stipsicz, Classification of tight contact structures on small Seifert 3–manifolds with e0 ≥ 0, Proc. Amer. Math. Soc. 134 (2006) 909 | DOI

[11] P Ghiggini, P Lisca, A I Stipsicz, Tight contact structures on some small Seifert fibered 3–manifolds, Amer. J. Math. 129 (2007) 1403 | DOI

[12] P Ghiggini, S Schönenberger, On the classification of tight contact structures, from: "Topology and geometry of manifolds" (editors G Matić, C McCrory), Proc. Sympos. Pure Math. 71, Amer. Math. Soc. (2003) 121 | DOI

[13] P Ghiggini, J Van Horn-Morris, Tight contact structures on the Brieskorn spheres −Σ(2,3,6n − 1) and contact invariants, J. Reine Angew. Math. 718 (2016) 1 | DOI

[14] E Giroux, Convexité en topologie de contact, Comment. Math. Helv. 66 (1991) 637 | DOI

[15] E Giroux, Une infinité de structures de contact tendues sur une infinité de variétés, Invent. Math. 135 (1999) 789 | DOI

[16] E Giroux, Structures de contact en dimension trois et bifurcations des feuilletages de surfaces, Invent. Math. 141 (2000) 615 | DOI

[17] R E Gompf, Handlebody construction of Stein surfaces, Ann. of Math. 148 (1998) 619 | DOI

[18] A Hatcher, Notes on basic of 3–manifold topology, book project (2007)

[19] K Honda, On the classification of tight contact structures, I, Geom. Topol. 4 (2000) 309 | DOI

[20] K Honda, On the classification of tight contact structures, II, J. Differential Geom. 55 (2000) 83 | DOI

[21] K Honda, W H Kazez, G Matić, Right-veering diffeomorphisms of compact surfaces with boundary, Invent. Math. 169 (2007) 427 | DOI

[22] T Ito, K Kawamuro, Visualizing overtwisted discs in open books, Publ. Res. Inst. Math. Sci. 50 (2014) 169 | DOI

[23] S Jabuka, T E Mark, Product formulae for Ozsváth–Szabó 4–manifold invariants, Geom. Topol. 12 (2008) 1557 | DOI

[24] M Jankins, W D Neumann, Rotation numbers of products of circle homeomorphisms, Math. Ann. 271 (1985) 381 | DOI

[25] W H Kazez, R Roberts, Fractional Dehn twists in knot theory and contact topology, Algebr. Geom. Topol. 13 (2013) 3603 | DOI

[26] Y Lekili, Planar open books with four binding components, Algebr. Geom. Topol. 11 (2011) 909 | DOI

[27] P Lisca, On overtwisted, right-veering open books, Pacific J. Math. 257 (2012) 219 | DOI

[28] P Lisca, G Matić, Tight contact structures and Seiberg–Witten invariants, Invent. Math. 129 (1997) 509 | DOI

[29] P Lisca, A I Stipsicz, Ozsváth–Szabó invariants and tight contact 3–manifolds, III, J. Symplectic Geom. 5 (2007) 357 | DOI

[30] P Lisca, A I Stipsicz, On the existence of tight contact structures on Seifert fibered 3–manifolds, Duke Math. J. 148 (2009) 175 | DOI

[31] P Massot, Geodesible contact structures on 3–manifolds, Geom. Topol. 12 (2008) 1729 | DOI

[32] P Ozsváth, Z Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003) 179 | DOI

[33] P Ozsváth, Z Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. 159 (2004) 1159 | DOI

[34] P Ozsváth, Z Szabó, Heegaard Floer homology and contact structures, Duke Math. J. 129 (2005) 39 | DOI

[35] P Ozsváth, Z Szabó, Holomorphic triangles and invariants for smooth four-manifolds, Adv. Math. 202 (2006) 326 | DOI

[36] O Plamenevskaya, Contact structures with distinct Heegaard Floer invariants, Math. Res. Lett. 11 (2004) 547 | DOI

[37] H Wu, Legendrian vertical circles in small Seifert spaces, Commun. Contemp. Math. 8 (2006) 219 | DOI

[38] H Wu, On Legendrian surgeries, Math. Res. Lett. 14 (2007) 513 | DOI