uu.seUppsala University Publications

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Surgeries on Legendrian SubmanifoldsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2012 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Uppsala: Department of Mathematics, 2012. , p. 40
##### Series

Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 77
##### Keywords [en]

Contact manifolds, Legendrian submanifolds, Legendrian contact homology, Augmentations, Exact Lagrangian cobordisms
##### National Category

Geometry
##### Research subject

Mathematics
##### Identifiers

URN: urn:nbn:se:uu:diva-177017ISBN: 978-91-506-2295-9 (print)OAI: oai:DiVA.org:uu-177017DiVA, id: diva2:538851
##### Public defence

2012-09-13, Häggsalen, Lägerhyddsvägen 1, Uppsala, 12:00 (English)
##### Opponent

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##### Supervisors

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#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt477",{id:"formSmash:j_idt477",widgetVar:"widget_formSmash_j_idt477",multiple:true}); Available from: 2012-08-23 Created: 2012-07-02 Last updated: 2012-10-05Bibliographically approved
##### List of papers

This thesis consists of a summary of two papers dealing with questions related to Legendrian submanifolds of contact manifolds together with exact Lagrangian cobordisms between Legendrian submanifolds. The focus is on studying Legendrian submanifolds from the perspective of their handle decompositions. The techniques used are mainly from Symplectic Field Theory.

In Paper I, a series of examples of Legendrian surfaces in standard contact 5-space are studied. For every g > 0, we produce g+1 Legendrian surfaces of genus g, all with g+1 transverse Reeb chords, which lie in distinct Legendrian isotopy classes. For each g, exactly one of the constructed surfaces has a Legendrian contact homology algebra admitting an augmentation. Moreover, it is shown that the same surface is the only one admitting a generating family. Legendrian contact homology with Novikov coefficients is used to classify the different Legendrian surfaces. In particular, we study their augmentation varieties.

In Paper II, the effect of a Legendrian ambient surgery on a Legendrian submanifold is studied. Given a Legendrian submanifold together which certain extra data, a Legendrian ambient surgery produces a Legendrian embedding of the manifold obtained by surgery on the original submanifold. The construction also provides an exact Lagrangian handle-attachment cobordism between the two submanifolds. The Legendrian contact homology of the submanifold produced by the Legendrian ambient surgery is then computed in terms of pseudo-holomorphic disks determined by data on the original submanifold. Also, the cobordism map induced by the exact Lagrangian handle attachment is computed. As a consequence, it is shown that a sub-critical standard Lagrangian handle attachment cobordism induces a one-to-one correspondence between the augmentations of the Legendrian contact homology algebras of its two ends.

1. Knotted Legendrian surfaces with few Reeb chords$(function(){PrimeFaces.cw("OverlayPanel","overlay513911",{id:"formSmash:j_idt526:0:j_idt530",widgetVar:"overlay513911",target:"formSmash:j_idt526:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Legendrian ambient surgery and Legendrian contact homology$(function(){PrimeFaces.cw("OverlayPanel","overlay538794",{id:"formSmash:j_idt526:1:j_idt530",widgetVar:"overlay538794",target:"formSmash:j_idt526:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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