Hamiltonian Floer theory on surfaces: linking, positively transverse foliations and spectral invariants

• 成果类型: Article
• 署名作者: Connery-Grigg, Dustin
• 署名单位: Sorbonne Universite
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-024-01274-0
• 发表日期: 2024
• 页码: 1377-1468
• 关键词: pseudo-holomorphic-curves; pseudoholomorphic curves; symplectisations; Orbits

摘要

We develop connections between the qualitative dynamics of Hamiltonian isotopies on a surface Sigma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Sigma $\end{document} and their chain-level Floer theory using ideas drawn from Hofer-Wysocki-Zehnder's theory of finite energy foliations. We associate to every collection of capped 1-periodic orbits which is 'maximally unlinked relative the Morse range' a singular foliation on S1x Sigma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S<^>{1} \times \Sigma $\end{document} which is positively transverse to the vector field partial derivative t circle plus XH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\partial _{t} \oplus X<^>{H}$\end{document} and which is assembled in a straight-forward way from the relevant Floer moduli spaces. Additionally, we provide a purely topological characterization of those Floer chains which both represent the fundamental class in CF & lowast;(H,J)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$CF_{*}(H,J)$\end{document}, 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 Humili & egrave;re-Le Roux-Seyfaddini in (Humili & egrave;re et al. in Geom. Topol. 20(4):2253-2334, 2016).