Floer theory for Lagrangian cobordisms
TL;DR: In this paper, intersection Floer homology for exact Lagrangian cobordisms between Legendrian submanifolds in the contactisation of a Liouville manifold is defined, provided that the Chekanov-Eliashberg algebras of the negative ends of the cobordism admit augmentations.
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Abstract: In this article we define intersection Floer homology for exact Lagrangian cobordisms between Legendrian submanifolds in the contactisation of a Liouville manifold, provided that the Chekanov-Eliashberg algebras of the negative ends of the cobordisms admit augmentations. From this theory we derive several long exact sequences relating the Morse homology of an exact Lagrangian cobordism with the bilinearised contact homologies of its ends. These are then used to investigate the topological properties of exact Lagrangian cobordisms.
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Citations
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Floer homology and lagrangian concordance
Baptiste Chantraine,Georgios Dimitroglou Rizell,Paolo Ghiggini,Roman Golovko +3 more
- 02 Jun 2015
TL;DR: In this article, the authors derive constraints on Lagrangian concordances from Legendrian submanifolds of the standard contact sphere admitting exact Lagrangians fillings, and show that such a concordance induces an isomorphism on the level of bilinearised Legendrian contact cohomology.
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On homological rigidity and flexibility of exact Lagrangian endocobordisms
TL;DR: In this paper, an exact Lagrangian cobordism L ⊂ ℝ × P × τ� from a Legendrian submanifold Λ to itself satisfies Hi(L; 𝔽) = Hi(Λ; and#x 1d53D;) for any field and given that the concatenation of any spin-exact Lagrangians filling of Λ and L is also spin.
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Functorial LCH for immersed Lagrangian cobordisms
Yu Pan,Dan Rutherford +1 more
TL;DR: In this article, the authors extend the functorality of the Legendrian contact homology DG-algebra (DGA) from embedded exact Lagrangian cobordisms, as in \cite{EHK}, to a class of immersed Lagrangians by considering their Legendrian lifts as conical Legendrian cobordism.
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The minimal length of a Lagrangian cobordism between Legendrians
Joshua M. Sabloff,Lisa Traynor +1 more
TL;DR: In this article, a set of real-valued capacities for a Lagrangian submanifold is derived from a filtered version of Legendrian contact homology, which yield lower bounds on the length of the cobordism.
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Families of Legendrians and Lagrangians with unbounded spectral norm
TL;DR: In this paper, the authors explore two natural generalisations of the above geometric setting in which the bound of the spectral norm fails: first, passing to Legendrian isotopies in the contactisation of the unit co-disc bundle, and, second, considering Hamiltonians but after modifying the codisc bundle by attaching a critical Weinstein one-handle.
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References
Estimating the number of Reeb chords using a linear representation of the characteristic algebra
TL;DR: In this paper, a lower bound for the number of Reeb chords on a chord-generic horizontally displaceable Legendrian submanifold with a Chekanov-Eliashberg algebra was shown.
On homological rigidity and flexibility of exact Lagrangian endocobordisms
TL;DR: In this paper, an exact Lagrangian cobordism L ⊂ ℝ × P × τ� from a Legendrian submanifold Λ to itself satisfies Hi(L; 𝔽) = Hi(Λ; and#x 1d53D;) for any field and given that the concatenation of any spin-exact Lagrangians filling of Λ and L is also spin.
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Nonorientable Lagrangian cobordisms between Legendrian knots
TL;DR: In this article, it was shown that any Legendrian knot has a non-orientable Lagrangian endocobordism, and that the crosscap genus of such a nonorientable endocombordism must be a positive multiple of $4.
Non-Orientable Lagrangian Cobordisms between Legendrian Knots
TL;DR: In this article, it was shown that any Legendrian knot has a non-orientable Lagrangian endocobordism, and that the crosscap genus of such a nonorientable endocombordism must be a positive multiple of $4.
•Posted Content
Floer's chain complexes for Lagrangian submanifolds in symplectic manifolds with concave ends
TL;DR: Floer's chain complexes for Lagrangian submanifolds in closed symplectic manifolds are generated by intersection points of Lagrangians and whose differentials count pseudo-holomorphic strips as discussed by the authors.
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