To compute orientations of Morse flow trees in Legendrian contact homology

Abstract

Let $\Lambda$ be a Legendrian submanifold of the 1-jet space of a smooth manifold. Associated to $\Lambda$ there is a Legendrian invariant called Legendrian contact homology, which is defined by counting rigid pseudo-holomorphic disks of $\Lambda$. Moreover, there exists a bijective correspondence between rigid pseudo-holomorphic disks and rigid Morse flow trees of $\Lambda$, which allows us to compute the Legendrian contact homology of $\Lambda$ via Morse theory. If $\Lambda$ is spin, then the moduli space of the rigid disks can be given a coherent orientation, so that the Legendrian contact homology of $\Lambda$ can be defined with coefficients in $\mathbb Z$. In this paper we give an explicit algorithm for computing the corresponding orientation of the moduli space of rigid Morse flow trees, using the Morse theoretical framework.Comment: 31 pages, 17 figures. Several changes, in particular in Section 3 and some changes in the formulas in Section 6. Included a section with an example. arXiv admin note: text overlap with arXiv:1601.0734