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Surgeries on Legendrian Submanifolds
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry. (Topologi)
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 [en]
Contact manifolds, Legendrian submanifolds, Legendrian contact homology, Augmentations, Exact Lagrangian cobordisms
Geometry
Mathematics
Identifiers
ISBN: 978-91-506-2295-9OAI: oai:DiVA.org:uu-177017DiVA: diva2:538851
Public defence
2012-09-13, Häggsalen, Lägerhyddsvägen 1, Uppsala, 12:00 (English)
Supervisors
Available from: 2012-08-23 Created: 2012-07-02 Last updated: 2012-10-05Bibliographically approved
List of papers
1. Knotted Legendrian surfaces with few Reeb chords
Open this publication in new window or tab >>Knotted Legendrian surfaces with few Reeb chords
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.

Mathematics
Identifiers
urn:nbn:se:uu:diva-172252 (URN)10.2140/agt.2011.11.2903 (DOI)000299576600013 ()
Available from: 2012-04-04 Created: 2012-04-03 Last updated: 2012-10-05Bibliographically approved
2. Legendrian ambient surgery and Legendrian contact homology
Open this publication in new window or tab >>Legendrian ambient surgery and Legendrian contact homology
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Let $L$ be a Legendrian submanifold of a contact manifold $Y$ and let $S \subset L$ be a framed sphere bounding a subcritical isotropic disk in $Y \setminus L$. We may perform an ambient surgery on $L$ along $S$, obtaining a Legendrian submanifold $L_S \subset Y$ which is Lagrangian cobordant to $L$. We produce an isomorphism of the Legendrian contact homology algebra of $L_S$ with an algebra obtained from the algebra of $L$ after twisting the differential by a count of holomorphic disks with boundary points mapping to $S$. This isomorphism induces a bijection between the sets of augmentations for the algebras of $L$ and $L_S$.

Geometry
Mathematics
Identifiers
Available from: 2012-07-02 Created: 2012-07-02 Last updated: 2012-10-05

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