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Knotted Legendrian surfaces with few Reeb chords
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
2011 (English)In: Algebraic and Geometric Topology, ISSN 1472-2747, E-ISSN 1472-2739, Vol. 11, no 5, 2903-2936 p.Article in journal (Refereed) Published
Abstract [en]

For g > 0, we construct g + 1 Legendrian embeddings of a surface of genus g into J(1)(R-2) = R-5 which lie in pairwise distinct Legendrian isotopy classes and which all have g + 1 transverse Reeb chords (g + 1 is the conjecturally minimal number of chords). Furthermore, for g of the g + 1 embeddings the Legendrian contact homology DGA does not admit any augmentation over Z(2), and hence cannot be linearized. We also investigate these surfaces from the point of view of the theory of generating families. Finally, we consider Legendrian spheres and planes in J(1)(S-2) from a similar perspective.

Place, publisher, year, edition, pages
2011. Vol. 11, no 5, 2903-2936 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-172252DOI: 10.2140/agt.2011.11.2903ISI: 000299576600013OAI: oai:DiVA.org:uu-172252DiVA: diva2:513911
Available from: 2012-04-04 Created: 2012-04-03 Last updated: 2017-12-07Bibliographically approved
In thesis
1. Surgeries on Legendrian Submanifolds
Open this publication in new window or tab >>Surgeries on Legendrian Submanifolds
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

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.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics, 2012. 40 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 77
Keyword
Contact manifolds, Legendrian submanifolds, Legendrian contact homology, Augmentations, Exact Lagrangian cobordisms
National Category
Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-177017 (URN)978-91-506-2295-9 (ISBN)
Public defence
2012-09-13, Häggsalen, Lägerhyddsvägen 1, Uppsala, 12:00 (English)
Opponent
Supervisors
Available from: 2012-08-23 Created: 2012-07-02 Last updated: 2012-10-05Bibliographically approved

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Dimitroglou Rizell, Georgios

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