Hamiltonian Floer theory on surfaces
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866929367150493696 |
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| author | Connery-Grigg, Dustin |
| author_facet | Connery-Grigg, Dustin |
| contents | We develop connections between the qualitative dynamics of Hamiltonian isotopies on a surface $Σ$ and their chain-level Floer theory using ideas drawn from Hofer-Wysocki-Zehnder's theory of finite energy foliations. We associate to every collection of capped $1$-periodic orbits which is `maximally unlinked relative the Morse range' a singular foliation on $S^1 \times Σ$ which is positively transverse to the vector field $\partial_t \oplus X^H$ and which is assembled in a straight-forward way from the relevant Floer moduli spaces. This provides a Floer-theoretic method for producing foliations of the type which appear in Le Calvez's theory of positively transverse foliations for surface homeomorphisms. Additionally, we provide a purely topological characterization of those Floer chains which both represent the fundamental class in $CF_*(H,J)$, and which lie in the image of some chain-level PSS map. This leads to the definition of a novel family of spectral invariants which share many of the same formal properties as the Oh-Schwarz spectral invariants, and we compute the novel spectral invariant associated to the fundamental class in entirely dynamical terms. This significantly extends a project initiated by Humilière-Le Roux-Seyfaddini in arXiv:1502.03834. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_11231 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Hamiltonian Floer theory on surfaces Connery-Grigg, Dustin Symplectic Geometry Differential Geometry Dynamical Systems 53D40 (Primary) 37J65, 37J39 (Secondary) We develop connections between the qualitative dynamics of Hamiltonian isotopies on a surface $Σ$ and their chain-level Floer theory using ideas drawn from Hofer-Wysocki-Zehnder's theory of finite energy foliations. We associate to every collection of capped $1$-periodic orbits which is `maximally unlinked relative the Morse range' a singular foliation on $S^1 \times Σ$ which is positively transverse to the vector field $\partial_t \oplus X^H$ and which is assembled in a straight-forward way from the relevant Floer moduli spaces. This provides a Floer-theoretic method for producing foliations of the type which appear in Le Calvez's theory of positively transverse foliations for surface homeomorphisms. Additionally, we provide a purely topological characterization of those Floer chains which both represent the fundamental class in $CF_*(H,J)$, and which lie in the image of some chain-level PSS map. This leads to the definition of a novel family of spectral invariants which share many of the same formal properties as the Oh-Schwarz spectral invariants, and we compute the novel spectral invariant associated to the fundamental class in entirely dynamical terms. This significantly extends a project initiated by Humilière-Le Roux-Seyfaddini in arXiv:1502.03834. |
| title | Hamiltonian Floer theory on surfaces |
| topic | Symplectic Geometry Differential Geometry Dynamical Systems 53D40 (Primary) 37J65, 37J39 (Secondary) |
| url | https://arxiv.org/abs/2102.11231 |