Hamiltonian Floer theory on surfaces: linking, positively transverse foliations and spectral invariants (Q6585700)

For a collection of capped 1-periodic orbits of a Hamiltonian \(H\) on a surface \(\Sigma\) which are ``maximally unlinked relative to the Morse range,'' the author associates a singular foliation \(S^1 \times \Sigma\) which is positively transverse to the vector field \(\partial_t \oplus X^H\). A topological characterization of the Floer chains which represent the fundamental class in \(CF_\ast(H,J)\) and lie in the image of some chain-level PSS map is given, which leads to a new family of spectral invariants with properties similar to the Oh-Schwarz spectral invariants. The results in this paper significantly extend those in [\textit{V. Humilière} et al., Geom. Topol. 20, No. 4, 2253--2334 (2016; Zbl 1356.53082)].\N\NLet \((\Sigma,\omega)\) be a symplectic surface, \(H:S^1 \times \Sigma \rightarrow \mathbb{R}\) a periodic time-dependent Hamiltonian on the surface, and \(\phi^H:[0,1] \times \Sigma \rightarrow \Sigma\) the associated isotopy given by the flow of the time dependent Hamiltonian vector field \(X_t^H\) defined by \(\omega(X_t^H, \cdot) = - dH_t(\cdot)\). The isotopy \(\phi^H\) and a family \((J_t)_{t \in S^1}\) of \(\omega\)-compatible almost complex structures determine a Floer chain complex \(CF_\ast(H,J)\), under certain generic conditions, which is generated by the set \(\widetilde{\mathrm{Per}}_0(H)\) of capped 1-periodic orbits \(\hat{x}\) of \(\phi^H\). The boundary operator for \(CF_\ast(H,J)\) is given by counting \(J\)-holomorphic Floer cylinders in some appropriate moduli spaces \(\widetilde{\mathcal{M}}(\hat{x},\hat{y};H,J)\).\N\NThere is a \(\mathbb{Z}\)-grading \(\mu_{CZ}\) on \(CF_\ast(H,J)\) and a real-valued filtration induced by the Hamiltonian action functional \(\mathcal{A}_H\) associated to \(H\). A collection of capped orbits \(\hat{X} \in \widetilde{\mathrm{Per}}_0(H)\) that are pairwise distinct and the grading \(\mu_{CZ}\) form a geometric object that the author calls an indexed capped braid \((\hat{X},\mu_{CZ})\). The author considers the topology of the indexed capped braids in \(\widetilde{\mathrm{Per}}_0(H)\) as ``encoding the topological structure of 1-periodic orbits of \((\phi_t^H)_{t \in [0,1]}\) along with the coarse local structure of the isotopy near these orbits.'' The overall question addressed in the paper is: ``What relations does the topological structure of \(\widetilde{\mathrm{Per}}_0(H)\) impose on the (filtered) algebraic structure of \(CF_\ast(H,J)\) and vice versa?''\N\NThe author says that a capped braid \(\hat{X}\) is unlinked if the capping disks of the strands may be chosen so that their graphs in \(D^2 \times \Sigma\) are disjoint, and \(\hat{X}\) is called positive (negative) if the capping disks may be chosen so that their graphs are transverse with only positive (negative) intersections. An indexed capped braid \(\hat{X}\) is called maximally unlinked relative to the Morse range, \(\hat{X} \in murm(H)\), if \(\hat{X}\) is unlinked, every capped strand of \(\hat{X}\) has index -1, 0, or 1, and \(\hat{X}\) is maximal among all collections of capped orbits in \(\widetilde{\mathrm{Per}}_0(H)\) with these properties. We can now state the first two main theorems in the paper.\N\NTheorem A. Let \((\Sigma,\omega)\) be a closed symplectic surface. \(H \in C^\infty(S^1 \times \Sigma)\) be a nondegenerate Hamiltonian, and let \(J \in C^\infty(S^1;\mathcal{J}_\omega(\Sigma))\) be such that \((H,J)\) is Floer regular. For any capped braid \(\hat{X} \in murm(H)\), we may construct an oriented singular foliation \(\mathcal{F}^{\hat{X}}\) of \(S^1 \times \Sigma\) with the following properties\N\begin{itemize}\N\item[1.] The singular leaves of \(\mathcal{F}^{\hat{X}}\) are precisely the graphs of the orbits in \(\hat{X}\). That is, they are parameterized by the maps \(t \mapsto (t,x(t))\) for \(\hat{x} = [x,w_x] \in \hat{X}\).\N\item[2.] The regular leaves are precisely the annuli parameterized by the maps \N\begin{align*} \N\check{u}:\mathbb{R} \times S^1 & \rightarrow S^1 \times \Sigma\\\N(s,t) & \mapsto (t,u(s,t)), \N\end{align*} \Nfor some \(u \in \widetilde{\mathcal{M}}(\hat{x},\hat{y}; H,J)\) and some \(\hat{x}\), \(\hat{y} \in \hat{X}\).\N\item[3.] The vector field \(\check{X}^H(t,z) = \partial_t \oplus H_t^H(z)\) is positively transverse to every regular leaf of \(\mathcal{F}^{\hat{X}}\). That is, \(\check{X}^H \wedge \check{u}_\ast(\partial_s \wedge \partial_t)\) is a positively oriented volume element on each regular leaf, where \(\check{u}\) is as above.\N\end{itemize}\NLet \(\mathcal{F}_t^{\hat{X}}\) be the singular foliation on \(\Sigma\) given by intersecting \(\mathcal{F}^{\hat{X}}\) with the fiber \(\{t\} \times \Sigma\), where \(t \in S^1\). There is a loop of diffeomorphisms \((\psi_t^{\hat{X}})_{t \in S^1}\) such that \(\psi_t^{\hat{X}}\) sends \(\mathcal{F}_0^{\hat{X}}\) to \(\mathcal{F}_1^{\hat{X}}\) given by \(t \mapsto (t,u_s(t))\).\N\NTheorem B. For every \(t \in S^1\), \(\mathcal{F}_t^{\hat{X}}\) is a singular foliation of Morse type (ie. there is a choice of Morse function and metric on \(\Sigma\) for which \(\mathcal{F}_t^{\hat{X}}\) is the singular foliation associated to the negative gradient flow). Moreover, the loop \((\psi_t^{\hat{X}})_{t \in S^1}\) is a contractible loop of diffeomorphisms such that the orbits of \((\psi^{\hat{X}})^{-1} \circ \phi^H\) are positively transverse to the foliation \(\mathcal{F}_0^{\hat{X}}\).\N\NNow recall that there is a PSS isomorphism between Floer homology and quantum homology defined by a chain level map\N\[\N\Phi_{\mathcal{D}}^{PSS}: QC_\ast(f,g) \rightarrow CF_{\ast - n}(H,J)\N\]\Nusing a Morse-Smale pair \((f,g)\) and some ancillary data \(\mathcal{D}\). The author proves the following.\N\NTheorem C. Let \(\sigma \in CF_1(H,J)\). \(\sigma\) is a non-trivial cycle such that \(\sigma \in im\ \Phi_{\mathcal{D}}^{PSS}\) for some regular PSS data \(\mathcal{D}\) if and only if supp \(\sigma\) is a maximal positive capped braid relative index 1.\N\NLet \(c_{im}(\alpha;H)\) be the infimal action level required to represent a non-zero quantum homology class \(\alpha\) in the filtered Floer complex of \(H\) via some chain-level PSS map. These quantities define spectral invariants, which the author calls PSS-image spectral invariants. Theorem C has the following corollary.\N\NCorollary. Let \(H\) be non-degenerate\N\[\Nc_{im}([\Sigma];H) = \max_{\hat{X} \in mp_{(1)}(H)} \max_{\hat{x}\in \hat{X}} \mathcal{A}_H(\hat{x}).\N\]\N\NThe notation \(\hat{X} \in mp_{(1)}(H)\) means that \(\hat{X} \subset \widetilde{Per}_0(H)\) is maximally positive relative index \(1\). That is, \(\hat{X}\) is positive, every capped strand of \(\hat{X}\) has index 1, and \(\hat{X}\) is maximal among all collections of capped orbits with these properties.\N\NThe PSS-image spectral invariants have many of the same formal properties of the Oh-Schwarz spectral invariants. In particular, the PSS-image spectral invariants can be used to define a conjugation invariant norm on \(\mathrm{Ham}(M,\omega)\):\N\[\N\gamma_{im} := \inf \left( c_{im}([M];H) + c_{im}([M];\bar{H}) \right),\N\]\Nwhere the infimum is taken over the Hamiltonians \(H\) such that \(\phi_1^H = \phi\). The author obtains the following result.\N\NTheorem D. On surfaces, the conjugation invariant norm \(\gamma_{im}\) is both \(C^0\)-continuous and Hofer-continuous. Moreover, if \(\phi\) is non-degenerate and \(\Sigma \neq S^2\), then\N\[\N\gamma_{im}(\phi) = \min_{\hat{X} \in mp_{(1)}(H)} \max_{\hat{x}\in \hat{X}} \mathcal{A}_H(\hat{x}) - \max_{\hat{X} \in mn_{(-1)}(H)} \min_{\hat{x}\in \hat{X}} \mathcal{A}_H(\hat{x}),\N\]\Nfor \(H\) any Hamiltonian such that \(\phi_1^H = \phi\). (Here \(mn_{(-1)}(H)\) denotes the set of all capped braids \(\hat{X} \subseteq \widetilde{Per}_0(H)\) which are `maximally negative relative index \(-1\)'.)\N\NAs a final consequence of Theorem C the author proves the following.\N\NTheorem E Assume that \(H \in C^\infty(S^1 \times S^2)\) is non-degenerate and normalized so that \(\int H_t \omega = 0\) for all \(t \in S^1\), then\N\[\N\min \left\{\min_{\hat{X} \in mp_{(1)}(H)} \max_{\hat{x}\in \hat{X}} \mathcal{A}_H(\hat{x}), - \max_{\hat{X} \in mn_{(-1)}(H)} \min_{\hat{x}\in \hat{X}} \mathcal{A}_H(\hat{x}) \right\} < -k\ \mathrm{Area}(S^2,\omega)\N\]\Nonly if the commutator length of \(\tilde{\phi}^H\) in \(\widetilde{\mathrm{Ham}}(S^2)\) is strictly greater than \(2k+1\).