Morse flow trees and Legendrian contact homology in 1-jet spaces

By T. Ekholm. 2005

89 citations recorded by the source corpus.

Let L ⊂ J 1 (M) be a Legendrian submanifold of the 1-jet space of a Riemannian n-manifold M. A correspondence is established between rigid flow trees in M determined by L and boundary punctured rigid pseudo-holomorphic disks in TM, with boundary on the projection of L and asymptotic to the double points of this projection at punctures, provided n ≤ 2, or provided n > 2 and the front of L has only cusp edge singularities. This result, in particular, shows how to compute the Legendrian contact homology of L in terms of Morse theory.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/459142951efcea5f5dd46ae1f08829d2a4b1fde9

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. Lenhard L. Ng
2. T. Ekholm
3. Y. Eliashberg
4. John B. Etnyre
5. M. Abouzaid
6. Michael C. Sullivan
7. Emmy Murphy
8. Roger Casals

In Geometry and Topology

1. T. Ekholm
2. John B. Etnyre
3. Lenhard L. Ng
4. Michael C. Sullivan
5. Roger Casals
6. Frederick J. Bourgeois III
7. Y. Eliashberg
8. Emmy Murphy

In Mathematical Physics

1. T. Ekholm
2. John B. Etnyre
3. Lenhard L. Ng
4. Michael C. Sullivan
5. Roger Casals
6. Y. Eliashberg
7. Yankı Lekili
8. Frederick J. Bourgeois III

Citations in the corpus

• Nearby Lagrangians with vanishing Maslov class are homotopy equivalent (2010, methodology): “The comparison between the product on Floer complexes and Morse complexes can be done using Fukaya and Oh’s degeneration argument [11], which shows that for a specifically chosen almost complex structure, the counts of holomorphic triangles and of Morse trees agree (see also [10, 14]).”
• Legendrian fronts for affine varieties (2016, methodology): “In general, the computation requires T. Ekholm’s description of the pseudoholomorphic invariants in terms of Morse flow trees [27], combined with a high-dimensional version of [29].”
• Knot contact homology (2011, methodology): “In this subsection we give a slight modification of results from [8] in the case of a Legendrian surface Λ in the 1-jet space of a surface S, Λ ⊂ J(1)(S).”
• Rational symplectic field theory over Z2 for exact Lagrangian cobordisms (2006, unknown): “…there is a formal disk map which realizes this word then the homotopy classes of formal disk maps form a principal homogeneous space over the product of the kernel of a map π1(L̄′1) × · · · × π1(L̄ ′ m)→ π1(X̄) determined by the Reeb chord endpoints and the quotient π2(X̄)/im(π2(L̄′) → π2(X̄)).”
• A Duality Exact Sequence for Legendrian Contact Homology (2008, methodology): “This will allow us to apply the techniques of [3] during our analysis of the correspondence between rigid generalized disks and Jholomorphic disks with boundary on the perturbed 2L.”
• Morse homology, tropical geometry, and homological mirror symmetry for toric varieties (2006, result): “These results have been reworked and amplified in [9, 37].”
• Effect of Legendrian surgery (2009, methodology): “The authors acknowledge the support of: F. Bourgeois:The Fonds National de la Recherche Scientifique (Belgium) T. Ekholm: The G̈oran Gustafsson Foundation for Research in Natural Sciences and Medicine Y. Eliashberg:NSF grants DMS-0707103 and DMS 0244663”
• A topological introduction to knot contact homology (2012, methodology): “These disk counts use gradient flow trees in the manner of [Ekh07].”
• Duality between Lagrangian and Legendrian invariants (2017, methodology): “Proof This is a consequence of the main results in [22], [22, Theorem 1.”