To compute orientations of Morse flow trees in Legendrian contact homology
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arXiv
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| Format: | Preprint |
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2017
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| _version_ | 1866911890077122560 |
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| author | Karlsson, Cecilia |
| author_facet | Karlsson, Cecilia |
| contents | Let $Λ$ be a closed, connected Legendrian submanifold of the 1-jet space of a smooth $n$-dimensional manifold. Associated to $Λ$ there is a Legendrian invariant called Legendrian contact homology, which is defined by counting rigid pseudo-holomorphic disks of $Λ$. Moreover, there exists a bijective correspondence between rigid pseudo-holomorphic disks and rigid Morse flow trees of $Λ$, which allows us to compute the Legendrian contact homology of $Λ$ via Morse theory.
If $Λ$ is spin, then the moduli space of the rigid disks can be given a coherent orientation, so that the Legendrian contact homology of $Λ$ can be defined with coefficients in $\mathbb Z$. In this paper we give an algorithm for computing the corresponding orientation of the moduli space of rigid Morse flow trees if the dimension of $Λ$ is greater than 1, and up to 4 signs that depend on data that can be extracted from the vertices of the trees, but which are not given explicitly, in the case $n=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1704_05156 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | To compute orientations of Morse flow trees in Legendrian contact homology Karlsson, Cecilia Symplectic Geometry 57R17, 53D42 Let $Λ$ be a closed, connected Legendrian submanifold of the 1-jet space of a smooth $n$-dimensional manifold. Associated to $Λ$ there is a Legendrian invariant called Legendrian contact homology, which is defined by counting rigid pseudo-holomorphic disks of $Λ$. Moreover, there exists a bijective correspondence between rigid pseudo-holomorphic disks and rigid Morse flow trees of $Λ$, which allows us to compute the Legendrian contact homology of $Λ$ via Morse theory. If $Λ$ is spin, then the moduli space of the rigid disks can be given a coherent orientation, so that the Legendrian contact homology of $Λ$ can be defined with coefficients in $\mathbb Z$. In this paper we give an algorithm for computing the corresponding orientation of the moduli space of rigid Morse flow trees if the dimension of $Λ$ is greater than 1, and up to 4 signs that depend on data that can be extracted from the vertices of the trees, but which are not given explicitly, in the case $n=1$. |
| title | To compute orientations of Morse flow trees in Legendrian contact homology |
| topic | Symplectic Geometry 57R17, 53D42 |
| url | https://arxiv.org/abs/1704.05156 |