Simplified SFT moduli spaces for Legendrian links
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866911056051306496 |
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| author | Avdek, Russell |
| author_facet | Avdek, Russell |
| contents | We study moduli spaces $\mathcal{M}$ of holomorphic maps $U$ from Riemann surfaces to $\mathbb{R}^{4}$ with boundaries on the Lagrangian cylinder over a Legendrian link $Λ\subset (\mathbb{R}^{3}, ξ_{std})$. We allow our domains, $Σ$, to have non-trivial topology in which case $\mathcal{M}$ is the zero locus of an obstruction function $\mathcal{O}$, sending a moduli space of holomorphic maps in $\mathbb{C}$ to $H^{1}(Σ)$.
In general, $\mathcal{O}^{-1}(0)$ is not combinatorially computable. However after a Legendrian isotopy, $Λ$ can be made left-right-simple, implying that any $U$ of index $1$ is a disk with one or two positive punctures for which $π_{\mathbb{C}}\circ U$ is an embedding. Moreover, any $U$ of index $2$ is either a disk or an annulus with $π_{\mathbb{C}} \circ U$ simply covered and without interior critical points. Therefore any SFT invariant of $Λ$ is combinatorially computable using only disks with $\leq 2$ positive punctures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_00505 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Simplified SFT moduli spaces for Legendrian links Avdek, Russell Symplectic Geometry Geometric Topology We study moduli spaces $\mathcal{M}$ of holomorphic maps $U$ from Riemann surfaces to $\mathbb{R}^{4}$ with boundaries on the Lagrangian cylinder over a Legendrian link $Λ\subset (\mathbb{R}^{3}, ξ_{std})$. We allow our domains, $Σ$, to have non-trivial topology in which case $\mathcal{M}$ is the zero locus of an obstruction function $\mathcal{O}$, sending a moduli space of holomorphic maps in $\mathbb{C}$ to $H^{1}(Σ)$. In general, $\mathcal{O}^{-1}(0)$ is not combinatorially computable. However after a Legendrian isotopy, $Λ$ can be made left-right-simple, implying that any $U$ of index $1$ is a disk with one or two positive punctures for which $π_{\mathbb{C}}\circ U$ is an embedding. Moreover, any $U$ of index $2$ is either a disk or an annulus with $π_{\mathbb{C}} \circ U$ simply covered and without interior critical points. Therefore any SFT invariant of $Λ$ is combinatorially computable using only disks with $\leq 2$ positive punctures. |
| title | Simplified SFT moduli spaces for Legendrian links |
| topic | Symplectic Geometry Geometric Topology |
| url | https://arxiv.org/abs/2104.00505 |