Rational SFT, Linearized Legendrian Contact Homology, and Lagrangian Floer Cohomology

By T. Ekholm. 2009

62 citations recorded by the source corpus.

We relate the version of rational symplectic field theory for exact Lagrangian cobordisms introduced in [6] to linearized Legendrian contact homology. More precisely, if L ⊂ Xis an exact Lagrangian submanifold of an exact symplectic manifold with convex end Λ ⊂ Y, where Yis a contact manifold and Λis a Legendrian submanifold, and if Lhas empty concave end, then the linearized Legendrian contact cohomology of Λ, linearized with respect to the augmentation induced by L, equals the rational SFT of (X,L). Following ideas of Seidel [15], this equality in combination with a version of Lagrangian Floer cohomology of Lleads us to a conjectural exact sequence that in particular implies that if \(X = {\mathbb{C}}^{n}\), then the linearized Legendrian contact cohomology of Λ ⊂ S2n − 1is isomorphic to the singular homology of L. We outline a proof of the conjecture and show how to interpret the duality exact sequence for linearized contact homology of [7] in terms of the resulting isomorphism.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/533fe8651d7da027f7fd55aaad383e85992ffd2d

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In Geometry and Topology

1. A. Oancea
2. Frederick J. Bourgeois III
3. Kai Cieliebak
4. Baptiste Chantraine
5. Vivek V. Shende
6. Steven Sivek
7. Lenhard L. Ng
8. T. Ekholm

In Mathematical Physics

1. A. Oancea
2. Frederick J. Bourgeois III
3. Kai Cieliebak
4. Baptiste Chantraine
5. Steven Sivek
6. Vivek V. Shende
7. T. Ekholm
8. Lenhard L. Ng

Citations in the corpus

• Bilinearised Legendrian contact homology and the augmentation category (2012, methodology): “In [13] and [8], it is shown that an exact Lagrangian filling with vanishing Maslov class of a Legendrian knot leads to an augmentation of its Chekanov algebra.”
• Augmentations are sheaves (2015, background): “; L) !H kHom +( L; L) !H k() ! is the standard long exact sequence in relative cohomology. Proof. This result has appeared in various guises and degrees of completeness in [BC14,DR,EHK, Ekh12] (in [Ekh12] as a conjecture); the basic result that linearized contact homology for Lis the homology of Li”
• Legendrian knots and exact Lagrangian cobordisms (2012, unknown): “the corresponding ε-linearized Legendrian contact homology group (resp. chain complex) by HCε(Λ) (resp. CCε(Λ)). The following theorem is essentially due to Seidel. The proof, given in more detail in [E3], uses wrapped Floer homology (cf. [AS, FSS]). Theorem 7.2 (Seidel). HCε(Λ) ≃ H∗(L). Sketch of p”
• Symplectic homology and the Eilenberg-Steenrod axioms (2015, background): “ncides with the wrapped Floer homology group of L introduced in [3, 42]. The Lagrangian symplectic homology group SH>0 ∗ (L) is isomorphic to the linearized Legendrian contact homology group of ∂+L[34, 37]. The Lagrangian symplectic homology group SH=0 ∗ (L) is isomorphic to the singular cohomolo”