Colin, Vincent Recollement de variétés de contact tendues. (Gluing tight contact manifolds.). (French) Zbl 0930.53053 Bull. Soc. Math. Fr. 127, No. 1, 43-69 (1999); addendum ibid. 127, 623 (1999). Author’s abstract: “We study the behaviour of tight contact structures under surgery operations along disks and tori. The main result says that if one glues two tight contact manifolds along incompressible tori, the resulting manifold is tight provided that the original structures are universally tight and that the tori are quasi pre-Lagrangian (for instance, this is the case if \(\xi\) induces on the tori a foliation by circles). Moreover, we construct an example which shows that without this last assumption, the new manifold can be overtwisted. As an application of these techniques, using a recent theorem of Y. M. Eliashberg and W. P. Thurston [‘Confoliations’ (Univ. Lect. Ser. 13, AMS, Providence) (1998; Zbl 0893.53001)], we construct a tight contact structure on “almost” every graph manifold and on a new class of toroidal homology spheres”. Reviewer: D.Perrone (Lecce) Cited in 3 ReviewsCited in 16 Documents MSC: 53D35 Global theory of symplectic and contact manifolds 57R17 Symplectic and contact topology in high or arbitrary dimension 53D10 Contact manifolds (general theory) 57R65 Surgery and handlebodies Keywords:universal tightness; incompressible torus; surgery along disks and tori; tight contact structures; graph manifold; toroidal homology spheres Citations:Zbl 0893.53001 × Cite Format Result Cite Review PDF Full Text: DOI Numdam Numdam EuDML Link References: [1] BENNEQUIN (D.) . - Entrelacements et équations de Pfaff , Astérisque 107-108, 1983 , p. 87-161. MR 86e:58070 | Zbl 0573.58022 · Zbl 0573.58022 [2] COLIN (V.) . - Chirurgies d’indice un et isotopies de sphères dans les variétés de contact tendues , C.R. Acad. Sci. Paris, t. 324, série 1, 1997 , p. 659-663. MR 98e:57036 | Zbl 0876.57051 · Zbl 0876.57051 · doi:10.1016/S0764-4442(97)86985-6 [3] COLIN (V.) . - Sur la stabilité, l’existence et l’unicité des structures de contact en dimension 3 . - Thèse n^\circ 96 de l’ENS-Lyon, 1998 . [4] ELIASHBERG (Y.) . - Classification of over-twisted contact structures on 3-manifolds , Inv. Math., t. 98, 1989 , p. 623-637. MR 90k:53064 | Zbl 0684.57012 · Zbl 0684.57012 · doi:10.1007/BF01393840 [5] ELIASHBERG (Y.) . - Contact 3-manifolds, twenty years since J. Martinet’s work , Ann. Inst. Fourier, t. 42, 1992 , p. 165-192. Numdam | MR 93k:57029 | Zbl 0756.53017 · Zbl 0756.53017 · doi:10.5802/aif.1288 [6] ELIASHBERG (Y.) . - Filling by holomorphic discs and its applications , London Math. Soc. Lect. Notes Ser., t. 151, 1991 , p. 45-67. MR 93g:53060 | Zbl 0731.53036 · Zbl 0731.53036 [7] ELIASHBERG (Y.) , THURSTON (W.) . - Confoliations . - Univ. Lect. Ser., vol. 13, 1998 . MR 98m:53042 | Zbl 0893.53001 · Zbl 0893.53001 [8] GABAI (D.) . - Foliations on the topology of 3-manifolds , III, J. Diff. Geom., t. 26, 1987 , p. 479-536. MR 89a:57014b | Zbl 0639.57008 · Zbl 0639.57008 [9] GIROUX (E.) . - Convexité en topologie de contact , Comment. Math. Helvetici, t. 66, 1991 , p. 18-33. MR 93b:57029 | Zbl 0766.53028 · Zbl 0766.53028 · doi:10.1007/BF02566670 [10] GIROUX (E.) . - Topologie de contact en dimension 3 , Sém. Bourbaki, t. 760, 1992 - 1993 , p. 1-27. Numdam [11] GIROUX (E.) . - Une structure de contact, même tendue est plus ou moins tordue , Ann. Scient. École Norm. Sup., t. 27, 1995 , p. 697-705. Numdam | MR 96b:57034 | Zbl 0819.53018 · Zbl 0819.53018 [12] GIROUX (E.) . - Structures de contact en dimension trois et bifurcations des feuilletages des surfaces , en préparation. · Zbl 1186.53097 [13] GOMPF (R.) . - Handlebody construction of Stein surfaces , Ann. Of Math., à paraître. arXiv | Zbl 0919.57012 · Zbl 0919.57012 · doi:10.2307/121005 [14] HEMPEL (J.) . - Three manifolds , Ann. Math. Stu., Princeton Univ. Press., t. 86, 1973 . MR 54 #3702 [15] JACO (W.) . - Lectures on three manifolds topology , A.M.S., t. 43, 1977 . MR 81k:57009 | Zbl 0433.57001 · Zbl 0433.57001 [16] KANDA (Y.) . - The classification of tight contact structures on the 3-torus , Comm. Anal. Geom., t. 5, 1997 , p. 413-438. MR 99c:57054 | Zbl 0899.53028 · Zbl 0899.53028 [17] LUTZ (R.) . - Structures de contact sur les fibrés principaux en cercles de dimension 3 , Ann. Inst. Fourier, t. 27, 1977 , p. 1-15. Numdam | MR 57 #17668 | Zbl 0328.53024 · Zbl 0328.53024 · doi:10.5802/aif.658 [18] MAKAR-LIMANOV (S.) . - Morse surgeries of index 0 and 1 on tight manifolds . - preprint, 1996 . · Zbl 0896.14021 [19] MAKAR-LIMANOV (S.) . - Tight contact structure on solid tori , Trans. Amer. Math. Soc., t. 1013-1044, 1998 ; CMP 96 : 16. MR 98e:53046 | Zbl 0894.53036 · Zbl 0894.53036 · doi:10.1090/S0002-9947-98-01822-4 [20] MARTINET (J.) . - Formes de contact sur les variétés de dimension 3 , Springer Lect. Notes in Math., t. 209, 1971 , p. 142-163. MR 50 #3263 | Zbl 0215.23003 · Zbl 0215.23003 [21] WALDHAUSEN (F.) . - On irreducible 3-manifolds which are sufficiently large , Ann. of Math., t. 87, 1968 , p. 56-88. MR 36 #7146 | Zbl 0157.30603 · Zbl 0157.30603 · doi:10.2307/1970594 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.