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The diffeotopy group of S1×S2 via contact topology

Part of: Symplectic geometry, contact geometry Differential topology Low-dimensional topology

Published online by Cambridge University Press:  18 March 2010

Fan Ding  and
Hansjörg Geiges
Fan Ding
Affiliation:
Department of Mathematics, Peking University, Beijing 100871, PR China (email: [email protected])
Hansjörg Geiges
Affiliation:
Mathematisches Institut, Universität zu Köln, Weyertal 86–90, 50931 Köln, Germany (email: [email protected])
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Abstract

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As shown by Gluck in 1962, the diffeotopy group of S1×S2 is isomorphic to ℤ2⊕ℤ2 ⊕ℤ2. Here an alternative proof of this result is given, relying on contact topology. We then discuss two applications to contact topology: (i) it is shown that the fundamental group of the space of contact structures on S1×S2, based at the standard tight contact structure, is isomorphic to ℤ; (ii) inspired by previous work of Fraser, an example is given of an integer family of Legendrian knots in S1×S2#S1×S2 (with its standard tight contact structure) that can be distinguished with the help of contact surgery, but not by the classical invariants (topological knot type, Thurston–Bennequin invariant, and rotation number).

Keywords

diffeotopy groupcontact topologyLegendrian knot

MSC classification

Secondary: 57R50: Diffeomorphisms 53D10: Contact manifolds, general 57M25: Knots and links in $S^3$ 57R52: Isotopy
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 4 , July 2010 , pp. 1096 - 1112
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Bonahon, F., Difféotopies des espaces lenticulaires, Topology 22 (1983), 305–314.CrossRefGoogle Scholar
[2]Bourgeois, F., Contact homology and homotopy groups of the space of contact structures, Math. Res. Lett. 13 (2006), 71–85.CrossRefGoogle Scholar
[3]Cerf, J., Sur les difféomorphismes de la sphère de dimension trois (Γ4=0), Lecture Notes in Mathematics, vol. 53 (Springer, Berlin, 1968).CrossRefGoogle Scholar
[4]Chekanov, Yu., Differential algebra of Legendrian links, Invent. Math. 150 (2002), 441–483.CrossRefGoogle Scholar
[5]Chekanov, Yu. V. and Pushkar, P. E., Combinatorics of fronts of Legendrian links, and Arnold’s 4-conjectures, Russ. Math. Surveys 60 (2005), 95–149.CrossRefGoogle Scholar
[6]Chekanov, Yu., van Koert, O. and Schlenk, F., Minimal atlases of closed contact manifolds, in New perspectives and challenges in symplectic field theory, CRM Proceedings and Lecture Notes, vol. 49, eds Abreu, M.et al. (American Mathematical Society, Providence, RI, 2009), 73–112.CrossRefGoogle Scholar
[7]Colin, V., Chirurgies d’indice un et isotopies de sphères dans les variétés de contact tendues, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), 659–663.CrossRefGoogle Scholar
[8]Ding, F. and Geiges, H., A Legendrian surgery presentation of contact 3-manifolds, Math. Proc. Camb. Philos. Soc. 136 (2004), 583–598.CrossRefGoogle Scholar
[9]Ding, F. and Geiges, H., Legendrian knots and links classified by classical invariants, Commun. Contemp. Math. 9 (2007), 135–162.CrossRefGoogle Scholar
[10]Ding, F. and Geiges, H., Handle moves in contact surgery diagrams, J. Topol. 2 (2009), 105–122.CrossRefGoogle Scholar
[11]Ding, F. and Geiges, H., Legendrian helix and cable links, Commun. Contemp. Math., to appear.Google Scholar
[12]Eliashberg, Ya., Contact 3-manifolds twenty years since J. Martinet’s work, Ann. Inst. Fourier (Grenoble) 42 (1992), 165–192.CrossRefGoogle Scholar
[13]Eliashberg, Ya., Unique holomorphically fillable contact structure on the 3-torus, Int. Math. Res. Not. 1996 (1996), 77–82.CrossRefGoogle Scholar
[14]Fraser, M., Example of nonisotopic Legendrian curves not distinguished by the invariants tb and r, Int. Math. Res. Not. 1996 (1996), 923–928.CrossRefGoogle Scholar
[15]Fuchs, D., Chekanov–Eliashberg invariant of Legendrian knots: existence of augmentations, J. Geom. Phys. 47 (2003), 43–65.CrossRefGoogle Scholar
[16]Geiges, H., An introduction to contact topology, Cambridge Studies in Advanced Mathematics, vol. 109 (Cambridge University Press, Cambridge, 2008).CrossRefGoogle Scholar
[17]Geiges, H. and Gonzalo, J., On the topology of the space of contact structures on torus bundles, Bull. Lond. Math. Soc. 36 (2004), 640–646.CrossRefGoogle Scholar
[18]Ghiggini, P., Linear Legendrian curves in T 3, Math. Proc. Camb. Philos. Soc. 140 (2006), 451–473.CrossRefGoogle Scholar
[19]Giroux, E., Sur les transformations de contact au-dessus des surfaces, in Essays on geometry and related topics, Monographies de L’Enseignement Mathématique, vol. 38, part 2, eds Ghys, É.et al. (Genève, 2001), 329–350.Google Scholar
[20]Gluck, H., The embedding of two-spheres in the four-sphere, Trans. Amer. Math. Soc. 104 (1962), 308–333.CrossRefGoogle Scholar
[21]Gompf, R. E., Handlebody construction of Stein surfaces, Ann. of Math. (2) 148 (1998), 619–693.CrossRefGoogle Scholar
[22]Gompf, R. E. and Stipsicz, A. I., 4-manifolds and Kirby calculus, Graduate Studies in Mathematics, vol. 20 (American Mathematical Society, Providence, RI, 1999).CrossRefGoogle Scholar
[23]Hatcher, A., On the diffeomorphism group of S 1×S 2, Proc. Amer. Math. Soc. 83 (1981), 427–430. Update available at http://www.math.cornell.edu/∼hatcher/.Google Scholar
[24]Hodgson, C. and Rubinstein, J. H., Involutions and isotopies of lens spaces, in Knot theory and manifolds (Vancouver, 1983), Lecture Notes in Mathematics, vol. 1144, ed. Rolfsen, D. (Springer, Berlin, 1985), 60–96.CrossRefGoogle Scholar
[25]Honda, K., On the classification of tight contact structures I, Geom. Topol. 4 (2000), 309–368 (Erratum: Factoring nonrotative T 2×I layers, Geom. Topol. 5 (2001), 925–938).CrossRefGoogle Scholar
[26]Kirby, R., Problems in low-dimensional topology, in Geometric topology (Athens, GA, 1993), AMS/IP Studies in Advanced Mathematics, vol. 2, part 2 (American Mathematical Society, Providence, RI; International Press, Cambridge, MA, 1997), 35–473.Google Scholar
[27]Laudenbach, F., Sur les 2-sphères d’une variété de dimension 3, Ann. of Math. (2) 97 (1973), 57–81.CrossRefGoogle Scholar
[28]Lisca, P., Ozsváth, P., Stipsicz, A. I. and Szabó, Z., Heegaard Floer invariants of Legendrian knots in contact three-manifolds, J. Eur. Math. Soc. (JEMS) 11 (2009), 1307–1363.Google Scholar
[29]Massot, P., Sur quelques propriétés riemanniennes des structures de contact en dimension trois, Thèse de doctorat, ENS Lyon (2008).Google Scholar
[30]Özbağcı, B. and Stipsicz, A. I., Surgery on contact 3-manifolds and Stein surfaces, Bolyai Society Mathematical Studies, vol. 13, (Springer, Berlin; János Bolyai Mathematical Society, Budapest, 2004).CrossRefGoogle Scholar
[31]Ozsváth, P., Szabó, Z. and Thurston, D., Legendrian knots, transverse knots and combinatorial Floer homology, Geom. Topol. 12 (2008), 941–980.CrossRefGoogle Scholar
[32]Stillwell, J. C., Classical topology and combinatorial group theory, Graduate Texts in Mathematics, vol. 72, second edition (Springer, Berlin, 1993).CrossRefGoogle Scholar
[33]Stipsicz, A. I., Gauge theory and Stein fillings of certain 3-manifolds, Turkish J. Math. 26 (2002), 115–130.Google Scholar