A note on coherent orientations for exact Lagrangian cobordisms
Abstract
Let $L \subset \mathbb R \times J^1(M)$ be a spin, exact Lagrangian cobordism in the symplectization of the 1jet 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 pseudoholomorphic disks in the cotangent bundle of $M$. It is also known that this count can be lifted to a mod 2 count of pseudoholomorphic disks in the symplectization $\mathbb R \times J^1(M)$, and that $L$ induces a morphism between the $\mathbb Z_2$valued DGA:s of the ends $\Lambda_\pm$ in a functorial way. We prove that this hold with integer coefficients as well. The proofs are built on the technique of orienting the moduli spaces of pseudoholomorphic disks using capping operators at the Reeb chords. We give an expression for how the DGA:s change if we change the capping operators.
 Publication:

arXiv eprints
 Pub Date:
 July 2017
 arXiv:
 arXiv:1707.04219
 Bibcode:
 2017arXiv170704219K
 Keywords:

 Mathematics  Symplectic Geometry
 EPrint:
 41 pages, final version, accepted for publication in Quantum Topology. More details have been added to some of the proofs