Hostname: page-component-669899f699-ggqkh Total loading time: 0 Render date: 2025-04-29T05:22:14.265Z Has data issue: false hasContentIssue false
  • English
  • Français

On the contact mapping class group of Legendrian circle bundles

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

Published online by Cambridge University Press:  01 February 2017

Emmanuel Giroux  and
Patrick Massot
Emmanuel Giroux
Affiliation:
Unité de Mathématiques Pures et Appliquées, École Normale Supérieure de Lyon, CNRS, 46 allée d’Italie, 69364 Lyon Cedex 07, France email [email protected]
Patrick Massot
Affiliation:
CMLS, École polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, France email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we determine the group of contact transformations modulo contact isotopies for Legendrian circle bundles over closed surfaces of non-positive Euler characteristic. These results extend and correct those presented by the first author in a former work. The main ingredient we use is connectedness of certain spaces of embeddings of surfaces into contact 3-manifolds. This connectedness question is also studied for itself with a number of (hopefully instructive) examples.

Keywords

contact structuresmapping class groups

MSC classification

Primary: 57M50: Geometric structures on low-dimensional manifolds 57R17: Symplectic and contact topology
Secondary: 53D35: Global theory of symplectic and contact manifolds 53D10: Contact manifolds, general
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 2 , February 2017 , pp. 294 - 312
Copyright
© The Authors 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Bennequin, D., Entrelacements et équations de Pfaff , in Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982), Astérisque, vol. 107 (Société Mathématique de France, Paris, 1983), 87–161 (in French).Google Scholar
Bourgeois, F., Contact homology and homotopy groups of the space of contact structures , Math. Res. Lett. 13 (2006), 71–85.CrossRefGoogle Scholar
Cerf, J., Topologie de certains espaces de plongements , Bull. Soc. Math. France 89 (1961), 227–380 (French).CrossRefGoogle Scholar
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 (French, with English and French summaries), doi:10.1016/S0764-4442(97)86985-6.CrossRefGoogle Scholar
Colin, V., Recollement de variétés de contact tendues , Bull. Soc. Math. France 127 (1999), 43–69 (French, with English and French summaries).CrossRefGoogle Scholar
Eliashberg, Y., Contact 3-manifolds twenty years since J. Martinet’s work , Ann. Inst. Fourier (Grenoble) 42 (1992), 165–192 (English, with French summary).CrossRefGoogle Scholar
Eliashberg, Y. and Fraser, M., Classification of Topologically Trivial Legendrian Knots , in Geometry, topology, and dynamics (Montreal, PQ, 1995), CRM Proc. Lecture Notes, vol. 15 (American Mathematical Society, Providence, RI, 1998), 17–51.CrossRefGoogle Scholar
Eliashberg, Y., Hofer, H. and Salamon, D., Lagrangian intersections in contact geometry , Geom. Funct. Anal. 5 (1995), 244–269, doi:10.1007/BF01895668.CrossRefGoogle Scholar
Eliashberg, Y. and Polterovich, L., New applications of Luttinger’s surgery , Comment. Math. Helv. 69 (1994), 512–522, doi:10.1007/BF02564502.CrossRefGoogle Scholar
Geiges, H. and Gonzalo Pérez, J., On the topology of the space of contact structures on torus bundles , Bull. Lond. Math. Soc. 36 (2004), 640–646, doi:10.1112/S0024609304003376.CrossRefGoogle Scholar
Geiges, H. and Klukas, M., The fundamental group of the space of contact structures on the 3-torus , Math. Res. Lett. 21 (2014), 1257–1262.CrossRefGoogle Scholar
Ghiggini, P., Tight contact structures on Seifert manifolds over T 2 with one singular fibre , Algebr. Geom. Topol. 5 (2005), 785–833, doi:10.2140/agt.2005.5.785.CrossRefGoogle Scholar
Ghiggini, P., Linear Legendrian curves in T 3 , Math. Proc. Cambridge Philos. Soc. 140 (2006), 451–473, doi:10.1017/S0305004105009151.CrossRefGoogle Scholar
Giroux, E., Convexité en topologie de contact , Comment. Math. Helv. 66 (1991), 637–677, doi:10.1007/BF02566670 (French).CrossRefGoogle Scholar
Giroux, E., Une structure de contact, même tendue, est plus ou moins tordue , Ann. Sci. Éc. Norm. Supér. (4) 27 (1994), 697–705 (French, with English summary).CrossRefGoogle Scholar
Giroux, E., Une infinité de structures de contact tendues sur une infinité de variétés , Invent. Math. 135 (1999), 789–802, doi:10.1007/s002220050301 (French).CrossRefGoogle Scholar
Giroux, E., Structures de contact en dimension trois et bifurcations des feuilletages de surfaces , Invent. Math. 141 (2000), 615–689, doi:10.1007/s002220000082 (French).CrossRefGoogle Scholar
Giroux, E., Structures de contact sur les variétés fibrées en cercles au-dessus d’une surface , Comment. Math. Helv. 76 (2001), 218–262, doi:10.1007/PL00000378 (French, with French summary).CrossRefGoogle Scholar
Giroux, E., Sur les transformations de contact au-dessus des surfaces , in Essays on geometry and related topics, Vols. 1, 2, Monogr. Enseign. Math., vol. 38 (Enseignement Math., Geneva, 2001), 329–350 (French).Google Scholar
Hatcher, A., Homeomorphisms of sufficiently large P 2 -irreducible 3-manifolds , Topology 15 (1976), 343–347.CrossRefGoogle Scholar
Honda, K., On the classification of tight contact structures. I , Geom. Topol. 4 (2000), 309–368, doi:10.2140/gt.2000.4.309.CrossRefGoogle Scholar
Laudenbach, F., Topologie de la dimension trois: homotopie et isotopie, Astérisque, vol. 12 (Société Mathématique de France, Paris, 1974), with an English summary and table of contents.Google Scholar
Libermann, P., Sur les automorphismes infinitésimaux des structures symplectiques et des structures de contact, Colloque Géom. Diff. Globale (Bruxelles, 1958), 37–59 (Centre Belge Rech. Math., Louvain, 1959) (French).Google Scholar
Lutz, R., Structures de contact sur les fibrés principaux en cercles de dimension trois , Ann. Inst. Fourier (Grenoble) 27 (1977), 1–15 (French, with English summary).CrossRefGoogle Scholar
Lutz, R., Structures de contact et systèmes de Pfaff à pivot , in Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982), Astérisque, vol. 107 (Société Mathématique de France, Paris, 1983), 175–187 (French).Google Scholar
Massot, P., Geodesible contact structures on 3-manifolds , Geom. Topol. 12 (2008), 1729–1776, doi:10.2140/gt.2008.12.1729.CrossRefGoogle Scholar
Palais, R. S., Local triviality of the restriction map for embeddings , Comment. Math. Helv. 34 (1960), 305–312.CrossRefGoogle Scholar
Pick, G., Geometrisches zur Zahlenlehre , Sitzungberichte Lotos Naturwissen Zeitschrift (Prague) 19 (1899), 311–319.Google Scholar
Waldhausen, F., Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I, II , Invent. Math. 3 (1967), 308–333; Invent. Math. 4 (1967), 87–117 (German).CrossRefGoogle Scholar