\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\QQ{{\mathbb Q}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\CritVal{\operatorname{CritVal}} \def\FS{\operatorname{FS}} \def\Sing{\operatorname{Sing}} \def\Coh{\operatorname{Coh}} \def\Vect{\operatorname{Vect}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \)

Symplectic snippets

\(\SH(X)\) via quadratic \(H\)

Let \((X, \lambda)\) be a Liouville domain, and let \(\hat X\) be its completion. The goal is to construct a Floer cohomology which witnesses all of the Reeb orbits of \(\partial X\). With that in mind, it is natural to study the Hamiltonian Floer cohomology of Hamiltonians \(H_t\) with the property that over the symplectization they are of the form \[H|_{\RR\times \partial X}=h(\exp(r))\] where \(\lim_{r\to\infty} h'(\exp(r))=\infty\). This Hamiltonian witnesses every Reeb orbit with sufficiently large period.
quadratic Hamiltonian
figure 0.0.1:An increasing Hamiltonian over the symplectization witnesses all Reeb orbits
A problem with this definition is that the Hamiltonian we have chosen is time independent, and so whenever \((\partial X, \alpha)\) has any Reeb orbits, the time-1 orbits of \(H\) will necessarily be degenerate (all orbits come in \(S^1\) families from reparameterization). The standard work-around is to introduce a small time-dependent perturbation to \(H\) in such a way that we can still apply our maximum principle argument. We therefore look at time-dependent Hamiltonians \(H_t\) which, outside of a compact set, are of the form \(h_t(\exp(r))\), with \(\lim_{r\to\infty} h_t(\exp(r))=\infty\) for all \(t\). The maximum principle argument from (Liouville manifolds are geometrically bounded) can be made to hold in this setting (Proposition 4.1 [Wen]). One then takes the definition of the symplectic cohomology to be \[\SH(X):=\CF(\hat X, H_t)=\bigoplus_{\gamma\st \dot \gamma= V_{H_t}} \ZZ\langle \gamma\rangle.\] with differential given by structure coefficients \(\langle d(\gamma_+), \gamma_\rangle\) counting \(V_{H_t}\)-perturbed pseudoholomorphic cylinders with ends limiting to \(\gamma_\pm\).

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References

[Wen]Chris Wendl. A beginner’s overview of symplectic homology. Preprint. www. mathematik. hu-berlin. de/wendl/pub/SH. pdf.