On the contact mapping class group of the contactization of the Am -Milnor fiber

Sergei Lanzat; Frol Zapolsky

doi:10.1007/s40316-017-0085-y

On the contact mapping class group of the contactization of the A_m -Milnor fiber

Sergei Lanzat, Frol Zapolsky

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We construct an embedding of the full braid group on m+ 1 strands B_m ₊ ₁, m≥ 1 , into the contact mapping class group of the contactization Q× S¹ of the A_m-Milnor fiber Q. The construction uses the embedding of B_m ₊ ₁ into the symplectic mapping class group of Q due to Khovanov and Seidel, and a natural lifting homomorphism. In order to show that the composed homomorphism is still injective, we use a partially linearized variant of the Chekanov–Eliashberg dga for Legendrians which lie above one another in Q× R, reducing the proof to Floer homology. As corollaries we obtain a contribution to the contact isotopy problem for Q× S¹, as well as the fact that in dimension 4, the lifting homomorphism embeds the symplectic mapping class group of Q into the contact mapping class group of Q× S¹.

Original language	English
Pages (from-to)	79-94
Number of pages	16
Journal	Annales Mathematiques du Quebec
Volume	42
Issue number	1
DOIs	https://doi.org/10.1007/s40316-017-0085-y
State	Published - 1 Apr 2018

Bibliographical note

Funding Information:
Acknowledgements We wish to thank Mohammed Abouzaid, Frédéric Bourgeois, Baptiste Chantraine, Yasha Eliashberg, Ailsa Keating, Patrick Massot, Dusa McDuff, Cedric Membrez, and Leonid Polterovich for discussions of the results of this paper and for their interest. We are partially supported by grant number 1281 from the GIF, the German–Israeli Foundation for Scientific Research and Development. FZ is also partially supported by grant number 1825/14 from the Israel Science Foundation. Specials thanks are due to Peter Albers, out of a discussion with whom an idea was born which led to this paper; it is part of an ongoing joint research project, which is funded by the aforementioned grant from the GIF. Finally we are grateful to the anonymous referee whose recommendations and remarks allowed us to significantly improve the exposition.

Publisher Copyright:
© 2017, Fondation Carl-Herz and Springer International Publishing AG.

Keywords

Braid group
Contact isotopy problem
Generalized Dehn twist
Legendrian contact homology
Milnor fiber
dg bimodule of a Legendrian link

ASJC Scopus subject areas

Mathematics (all)

Access to Document

10.1007/s40316-017-0085-y

Cite this

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title = "On the contact mapping class group of the contactization of the Am -Milnor fiber",

abstract = "We construct an embedding of the full braid group on m+ 1 strands Bm + 1, m≥ 1 , into the contact mapping class group of the contactization Q× S1 of the Am-Milnor fiber Q. The construction uses the embedding of Bm + 1 into the symplectic mapping class group of Q due to Khovanov and Seidel, and a natural lifting homomorphism. In order to show that the composed homomorphism is still injective, we use a partially linearized variant of the Chekanov–Eliashberg dga for Legendrians which lie above one another in Q× R, reducing the proof to Floer homology. As corollaries we obtain a contribution to the contact isotopy problem for Q× S1, as well as the fact that in dimension 4, the lifting homomorphism embeds the symplectic mapping class group of Q into the contact mapping class group of Q× S1.",

keywords = "Braid group, Contact isotopy problem, Generalized Dehn twist, Legendrian contact homology, Milnor fiber, dg bimodule of a Legendrian link",

author = "Sergei Lanzat and Frol Zapolsky",

note = "Funding Information: Acknowledgements We wish to thank Mohammed Abouzaid, Fr{\'e}d{\'e}ric Bourgeois, Baptiste Chantraine, Yasha Eliashberg, Ailsa Keating, Patrick Massot, Dusa McDuff, Cedric Membrez, and Leonid Polterovich for discussions of the results of this paper and for their interest. We are partially supported by grant number 1281 from the GIF, the German–Israeli Foundation for Scientific Research and Development. FZ is also partially supported by grant number 1825/14 from the Israel Science Foundation. Specials thanks are due to Peter Albers, out of a discussion with whom an idea was born which led to this paper; it is part of an ongoing joint research project, which is funded by the aforementioned grant from the GIF. Finally we are grateful to the anonymous referee whose recommendations and remarks allowed us to significantly improve the exposition. Publisher Copyright: {\textcopyright} 2017, Fondation Carl-Herz and Springer International Publishing AG.",

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N2 - We construct an embedding of the full braid group on m+ 1 strands Bm + 1, m≥ 1 , into the contact mapping class group of the contactization Q× S1 of the Am-Milnor fiber Q. The construction uses the embedding of Bm + 1 into the symplectic mapping class group of Q due to Khovanov and Seidel, and a natural lifting homomorphism. In order to show that the composed homomorphism is still injective, we use a partially linearized variant of the Chekanov–Eliashberg dga for Legendrians which lie above one another in Q× R, reducing the proof to Floer homology. As corollaries we obtain a contribution to the contact isotopy problem for Q× S1, as well as the fact that in dimension 4, the lifting homomorphism embeds the symplectic mapping class group of Q into the contact mapping class group of Q× S1.

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