# A note on coherent orientations for exact Lagrangian cobordisms

@article{Karlsson2019ANO, title={A note on coherent orientations for exact Lagrangian cobordisms}, author={Cecilia Karlsson}, journal={Quantum Topology}, year={2019} }

Let $L \subset \mathbb R \times J^1(M)$ be a spin, exact Lagrangian cobordism in the symplectization of the 1-jet space of a smooth manifold $M$. Assume that $L$ has cylindrical Legendrian ends $\Lambda_\pm \subset J^1(M)$. It is well known that the Legendrian contact homology of $\Lambda_\pm$ can be defined with integer coefficients, via a signed count of pseudo-holomorphic disks in the cotangent bundle of $M$. It is also known that this count can be lifted to a mod 2 count of pseudo…

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## References

SHOWING 1-10 OF 25 REFERENCES

Rational symplectic field theory over Z2 for exact Lagrangian cobordisms

- Mathematics
- 2006

We construct a version of rational Symplectic Field Theory for pairs $(X,L)$, where $X$ is an exact symplectic manifold, where $L\subset X$ is an exact Lagrangian submanifold with components…

Legendrian knots and exact Lagrangian cobordisms

- Mathematics
- 2012

We introduce constructions of exact Lagrangian cobordisms with cylindrical Legendrian ends and study their invariants which arise from Symplectic Field Theory. A pair $(X,L)$ consisting of an exact…

Duality between Lagrangian and Legendrian invariants

- Mathematics
- 2017

Consider a pair $(X,L)$, of a Weinstein manifold $X$ with an exact Lagrangian submanifold $L$, with ideal contact boundary $(Y,\Lambda)$, where $Y$ is a contact manifold and $\Lambda\subset Y$ is a…

On homological rigidity and flexibility of exact Lagrangian endocobordisms

- Mathematics
- 2014

We show that an exact Lagrangian cobordism L ⊂ ℝ × P × ℝ from a Legendrian submanifold Λ ⊂ P × ℝ to itself satisfies Hi(L; 𝔽) = Hi(Λ; 𝔽) for any field 𝔽, given that Λ admits a spin exact…

ORIENTATIONS IN LEGENDRIAN CONTACT HOMOLOGY AND EXACT LAGRANGIAN IMMERSIONS

- Mathematics
- 2004

We show how to orient moduli spaces of holomorphic disks with boundary on an exact Lagrangian immersion of a spin manifold into complex n-space in a coherent manner. This allows us to lift the…

LIFTING PSEUDO-HOLOMORPHIC POLYGONS TO THE SYMPLECTISATION OF P × R AND APPLICATIONS

- Mathematics
- 2018

Let R × (P × R) be the symplectisation of the contactisation of an exact symplectic manifold P , and let R × Λ be a cylinder over a Legendrian submanifold of the contactisation. We show that a…

The contact homology of Legendrian submanifolds in R2n+1

- Mathematics
- 2005

We define the contact homology for Legendrian submanifolds in standard contact (2n + 1)-space using moduli spaces of holomorphic disks with Lagrangian boundary conditions in complex n-space. This…

Legendrian contact homology in $P \times \mathbb{R}$

- Mathematics
- 2007

A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form P x R, where P is an exact symplectic manifold, is established. The class of such contact…

Differential algebra of Legendrian links

- Mathematics
- 2002

Let the space R = {(q, p, u)} be equipped with the standard contact form α = du − pdq. A link L ⊂ R3 is called Legendrian if the restriction of α to L vanishes. Two Legendrian links are said to be…

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

- Mathematics
- 2007

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…