Subcritical contact surgeries and the topology of symplectic fillings

Paolo Ghiggini; Klaus Niederkrüger; Chris Wendl

doi:10.5802/jep.31

Subcritical contact surgeries and the topology of symplectic fillings
[Chirurgies de contact sous-critiques et topologie des remplissages symplectiques]

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

Journal de l’École polytechnique — Mathématiques, Tome 3 (2016), pp. 163-208.

Résumé
Abstract

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 $Ω_{*}^{S O} (W)$ 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 $Ω_{*}^{S O} (W)$ 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.

Reçu le : 2014-12-14
Accepté le : 2016-02-10
Publié le : 2016-03-07

MR Zbl

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

Affiliations des auteurs :

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/

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