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 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-holomorphic 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 pseudo-holomorphic 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:
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arXiv e-prints
- Pub Date:
- July 2017
- DOI:
- 10.48550/arXiv.1707.04219
- arXiv:
- arXiv:1707.04219
- Bibcode:
- 2017arXiv170704219K
- Keywords:
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- Mathematics - Symplectic Geometry
- E-Print:
- 41 pages, final version, accepted for publication in Quantum Topology. More details have been added to some of the proofs