Symplectomorphisms of exotic discs

Roger Casals; Ailsa Keating; Ivan Smith; Sylvain Courte

doi:10.5802/jep.71

Symplectomorphisms of exotic discs
[Symplectomorphismes de disques exotiques]

Roger Casals ¹ ; Ailsa Keating ² ; Ivan Smith ² ; Sylvain Courte ³

¹ Massachusetts Institute of Technology, Department of Mathematics 77 Massachusetts Avenue Cambridge, MA 02139, United States of America
² Centre for Mathematical Sciences, University of Cambridge Wilberforce Road, Cambridge CB3 0WB, United Kingdom
³ Institut Fourier, Université Grenoble Alpes et CNRS 100 rue des maths, 38610, Gières, France

Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 289-316.

Résumé
Abstract

Le but principal de cet article est la construction d’une structure symplectique sur un disque avec un symplectomorphisme à support compact qui n’est pas isotope à l’identité. Cette structure symplectique a un bord concave donné par la symplectification d’une structure de contact vrillée. La construction du symplectomorphisme est basée sur une version unitaire de l’accouplement de Milnor-Munkres. En chemin, nous introduisons un analogue symplectique de la filtration de Gromoll. Dans l’appendice, S. Courte montre que, pour notre structure symplectique, l’application qui associe à une classe d’applications symplectiques à support compact une classe d’applications lisses à support compact est surjective.

We construct a symplectic structure on a disc that admits a compactly supported symplectomorphism which is not smoothly isotopic to the identity. The symplectic structure has an overtwisted concave end; the construction of the symplectomorphism is based on a unitary version of the Milnor–Munkres pairing. En route, we introduce a symplectic analogue of the Gromoll filtration. The Appendix by S. Courte shows that for our symplectic structure the map from compactly supported symplectic mapping classes to compactly supported smooth mapping classes is in fact surjective.

Reçu le : 2017-04-10
Accepté le : 2018-03-19
Publié le : 2018-04-04

MR Zbl

DOI : 10.5802/jep.71

Classification : 57R17, 53D10, 53D15
Keywords: Symplectomorphism, overtwisted contact structure, Milnor–Munkres pairing, Gromoll filtration
Mot clés : Symplectomorphisme, structure de contact vrillée, accouplement de Milnor-Munkres, filtration de Gromoll

Affiliations des auteurs :

Roger Casals ¹ ; Ailsa Keating ² ; Ivan Smith ² ; Sylvain Courte ³

¹ Massachusetts Institute of Technology, Department of Mathematics 77 Massachusetts Avenue Cambridge, MA 02139, United States of America
² Centre for Mathematical Sciences, University of Cambridge Wilberforce Road, Cambridge CB3 0WB, United Kingdom
³ Institut Fourier, Université Grenoble Alpes et CNRS 100 rue des maths, 38610, Gières, France

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Roger Casals and Ailsa Keating and Ivan Smith and Sylvain Courte},
     title = {Symplectomorphisms of exotic discs},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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     publisher = {\'Ecole polytechnique},
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Roger Casals; Ailsa Keating; Ivan Smith; Sylvain Courte. Symplectomorphisms of exotic discs. Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 289-316. doi : 10.5802/jep.71. https://jep.centre-mersenne.org/articles/10.5802/jep.71/

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