Subcritical contact surgeries and the topology of symplectic fillings
[Chirurgies de contact sous-critiques et topologie des remplissages symplectiques]
Journal de l’École polytechnique — Mathématiques, Tome 3 (2016), pp. 163-208.

Un résultat d’Eliashberg affirme que tout remplissage symplectique d’une somme connexe de contact en dimension 3 est obtenu par somme connexe au bord d’un autre remplissage symplectique. Nous montrons une généralisation partielle de ce résultat pour les chirurgies de contact sous-critiques en dimension supérieure : étant donnée une variété de contact obtenue à partir d’une autre par chirurgie sous-critique, la cosphère de l’anse est nulle dans le groupe de bordisme orienté Ω * SO (W) de tout remplissage symplectiquement asphérique W. En dimension 5, elle est même homotope à zéro. Plus généralement, si le remplissage n’est pas asphérique mais semi-positif, alors la cosphère de l’anse est triviale dans H * (W). Nous montrons aussi, en utilisant des méthodes similaires, que la décomposition en somme connexe de contact pour les structures de contact tendues en dimension 3 ne s’étend pas en dimension supérieure. Nous exhibons en particulier des sommes connexes de variétés de dimension au moins 5 qui portent une structure de contact Stein remplissable qui ne peut pas se mettre sous la forme d’une somme connexe de contact. Les démonstrations s’appuient sur les techniques de remplissage par disques holomorphes avec, pour conditions au bord, des familles de livres ouverts legendriens (que l’on abrège par « Lob »).

By a result of Eliashberg, every symplectic filling of a three-dimensional contact connected sum is obtained by performing a boundary connected sum on another symplectic filling. We prove a partial generalization of this result for subcritical contact surgeries in higher dimensions: given any contact manifold that arises from another contact manifold by subcritical surgery, its belt sphere is zero in the oriented bordism group Ω * SO (W) of any symplectically aspherical filling W, and in dimension five, it will even be nullhomotopic. More generally, if the filling is not aspherical but is semipositive, then the belt sphere will be trivial in H * (W). Using the same methods, we show that the contact connected sum decomposition for tight contact structures in dimension three does not extend to higher dimensions: in particular, we exhibit connected sums of manifolds of dimension at least five with Stein fillable contact structures that do not arise as contact connected sums. The proofs are based on holomorphic disk-filling techniques, with families of Legendrian open books (so-called “Lobs”) as boundary conditions.

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Accepté le :
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DOI : 10.5802/jep.31
Classification : 57R17, 53D10, 32Q65, 57R65
Keywords: Contact surgery, symplectic filling, holomorphic disks
Mot clés : Chirurgie de contact, remplissage symplectique, disques holomorphes
Paolo Ghiggini 1 ; Klaus Niederkrüger 2, 3 ; Chris Wendl 4

1 Laboratoire de Mathématiques Jean Leray BP 92208, 2, Rue de la Houssinière, F-44322 Nantes Cedex 03, France
2 Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, POB 127, H-1364 Budapest, Hungary
3 Institut de mathématiques de Toulouse, Université Paul Sabatier – Toulouse III 118 route de Narbonne, F-31062 Toulouse Cedex 9, France
4 Department of Mathematics, University College London Gower Street, London WC1E 6BT, United Kingdom
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Paolo Ghiggini; Klaus Niederkrüger; Chris Wendl. Subcritical contact surgeries and the topology of symplectic fillings. Journal de l’École polytechnique — Mathématiques, Tome 3 (2016), pp. 163-208. doi : 10.5802/jep.31. https://jep.centre-mersenne.org/articles/10.5802/jep.31/

[BCS15] J. Bowden, D. Crowley & A. Stipsicz - “The topology of Stein fillable manifolds in high dimensions II”, Geom. Topol. 19 (2015), p. 2995-3030 | DOI | MR | Zbl

[BEM15] S. Borman, Y. Eliashberg & E. Murphy - “Existence and classification of overtwisted contact structures in all dimensions”, Acta Math. 215 (2015) no. 2, p. 281-361 | DOI | MR | Zbl

[BG83] E. Bedford & B. Gaveau - “Envelopes of holomorphy of certain 2-spheres in 2 , Amer. J. Math. 105 (1983) no. 4, p. 975-1009 | DOI | MR | Zbl

[CE12] K. Cieliebak & Y. Eliashberg - From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds, AMS Colloquium Publications, vol. 59, American Mathematical Society, Providence, RI, 2012 | Zbl

[Col97] V. Colin - “Chirurgies d’indice un et isotopies de sphères dans les variétés de contact tendues”, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) no. 6, p. 659-663 | DOI | MR | Zbl

[Eli90a] Y. Eliashberg - “Filling by holomorphic discs and its applications”, in Geometry of low-dimensional manifolds, 2 (Durham, 1989), London Math. Soc. Lecture Note Ser., vol. 151, Cambridge University Press, Cambridge, 1990, p. 45-67 | DOI | MR | Zbl

[Eli90b] Y. Eliashberg - “Topological characterization of Stein manifolds of dimension >2, Internat. J. Math. 1 (1990) no. 1, p. 29-46 | DOI | MR | Zbl

[Fra08] U. Frauenfelder - “Gromov convergence of pseudoholomorphic disks”, J. Fixed Point Theory Appl. 3 (2008) no. 2, p. 215-271 | DOI | MR | Zbl

[FZ15] U. Frauenfelder & K. Zehmisch - “Gromov compactness for holomorphic discs with totally real boundary conditions”, J. Fixed Point Theory Appl. 17 (2015) no. 3, p. 521-540 | DOI | MR | Zbl

[Gei08] H. Geiges - An introduction to contact topology, Cambridge Studies in Advanced Mathematics, vol. 109, Cambridge University Press, Cambridge, 2008 | DOI | MR | Zbl

[Gro85] M. Gromov - “Pseudoholomorphic curves in symplectic manifolds”, Invent. Math. 82 (1985) no. 2, p. 307-347 | DOI | Zbl

[GZ16] H. Geiges & K. Zehmisch - “The Weinstein conjecture for connected sums”, Internat. Math. Res. Notices 2016 (2016) no. 2, p. 325-342 | DOI | MR | Zbl

[Hae63] A. Haefliger - “Plongements différentiables dans le domaine stable”, Comment. Math. Helv. 37 (1962/1963), p. 155-176 | DOI | Zbl

[Hir94] M. Hirsch - Differential topology, Graduate Texts in Math., vol. 33, Springer-Verlag, New York, 1994 | DOI | MR

[Hof93] H. Hofer - “Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three”, Invent. Math. 114 (1993) no. 3, p. 515-563 | DOI | MR | Zbl

[HWZ11] H. Hofer, K. Wysocki & E. Zehnder - “Applications of polyfold theory I: Gromov–Witten theory” (2011), arXiv:1107.2097 | Zbl

[Laz00] L. Lazzarini - “Existence of a somewhere injective pseudo-holomorphic disc”, Geom. Funct. Anal. 10 (2000) no. 4, p. 829-862 | DOI | MR | Zbl

[McD91] D. McDuff - “Symplectic manifolds with contact type boundaries”, Invent. Math. 103 (1991) no. 3, p. 651-671 | DOI | MR | Zbl

[Mil65] J. Milnor - Lectures on the h-cobordism theorem, Princeton University Press, Princeton, N.J., 1965 | DOI

[MNW13] P. Massot, K. Niederkrüger & C. Wendl - “Weak and strong fillability of higher dimensional contact manifolds”, Invent. Math. 192 (2013) no. 2, p. 287-373 | DOI | MR | Zbl

[MS98] D. McDuff & D. Salamon - Introduction to symplectic topology, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998 | MR | Zbl

[MS12] D. McDuff & D. Salamon - J-holomorphic curves and symplectic topology, AMS Colloquium Publications, vol. 52, American Mathematical Society, Providence, RI, 2012 | Zbl

[Nie06] K. Niederkrüger - “The plastikstufe—a generalization of the overtwisted disk to higher dimensions”, Algebraic Geom. Topol. 6 (2006), p. 2473-2508 | DOI | MR | Zbl

[Nie13] K. Niederkrüger - On fillability of contact manifolds, Université Paul Sabatier - Toulouse III, 2013, Habilitation à diriger des recherches

[NR11] K. Niederkrüger & A. Rechtman - “The Weinstein conjecture in the presence of submanifolds having a Legendrian foliation”, J. Topol. Anal. 3 (2011) no. 4, p. 405-421 | DOI | MR | Zbl

[OV12] A. Oancea & C. Viterbo - “On the topology of fillings of contact manifolds and applications”, Comment. Math. Helv. 87 (2012) no. 1, p. 41-69 | DOI | MR | Zbl

[Rub97] D. Ruberman - “Null-homotopic embedded spheres of codimension one”, in Tight and taut submanifolds (Berkeley, CA, 1994), Math. Sci. Res. Inst. Publ., vol. 32, Cambridge University Press, Cambridge, 1997, p. 229-232 | MR | Zbl

[Sch99] M. Schwarz - “Equivalences for Morse homology”, in Geometry and topology in dynamics (Winston-Salem, NC, 1998/San Antonio, TX, 1999), Contemp. Math., vol. 246, American Mathematical Society, Providence, RI, 1999, p. 197-216 | DOI | MR | Zbl

[Sma62] S. Smale - “On the structure of manifolds”, Amer. J. Math. 84 (1962), p. 387-399 | DOI | Zbl

[vKN05] O. van Koert & K. Niederkrüger - “Open book decompositions for contact structures on Brieskorn manifolds”, Proc. Amer. Math. Soc. 133 (2005) no. 12, p. 3679-3686 | DOI | MR | Zbl

[Wei91] A. Weinstein - “Contact surgery and symplectic handlebodies”, Hokkaido Math. J. 20 (1991) no. 2, p. 241-251 | DOI | MR | Zbl

[Wen05] C. Wendl - Finite energy foliations and surgery on transverse links, New York University, 2005, Ph. D. Thesis | MR

[Yau04] M.-L. Yau - “Cylindrical contact homology of subcritical Stein-fillable contact manifolds”, Geom. Topol. 8 (2004), p. 1243-1280 | DOI | MR | Zbl

[Zin08] A. Zinger - “Pseudocycles and integral homology”, Trans. Amer. Math. Soc. 360 (2008) no. 5, p. 2741-2765 | DOI | MR | Zbl

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