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

\(\SH(X)\) as a limit

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As before, let \((X, \lambda)\) be a Liouville domain. For \(m\not\in \ell(\Gamma)\) not a period of a Reeb orbit, define \[\SH(X)^{< m}:=\HF(\hat X, H^m_t)\] where \(H^m_t\) is a Hamiltonian which on the symplectization agrees with \(H^m\), the linear Hamiltonian of slope \(m\). Over the symplectization \(\RR\times \partial X\) there are no Hamiltonian orbits, as \(H^m\) has no Hamiltonian orbits. The \(< m\) signifies that this version of symplectic cohomology is only supposed to detect those Reeb orbits of period less than \(m\). In order to recover the symplectic cohomology, we would like to understand the limit of the groups \(SH(X)^{< m}\) as we take \(m\to\infty\). Making sense of a limit algebraically requires constructing maps between these groups. When \(m^+< m^-\), the maximum principle arguments applied to families of Hamiltonians dependent on the \(s\)-parameter hold, allowing us to construct chain maps \[\CF(\hat X, H^{m^+}_t)\to \CF(\hat X, H^{m^-}_t)\] The \(\pm\) index on the slope are meant to represent whether they are the incoming or outgoing side of a Floer trajectory, not the relative sizes of the slopes. From the perspective of ($\SH(X)$ via quadratic $H$), the set of Hamiltonian orbits corresponding to Reeb orbits of period less than \(m^+\) is a subcomplex of the set of Reeb orbits of period less than \(m^-\). Intuitively, the Floer trajectory should decrease the action associated to the Reeb vector field, which is the period of the Reeb orbit.
figure 0.0.1:A limit of Hamiltonians of increasing slopes eventually sees all Reeb orbits
Consider now an increasing sequence of slopes \(m_0< m_1< \cdots \) which are not the periods of any Reeb orbits of \(\partial X\). One can form the telescope complex

could not find file dig_symplenicticCohomologyTelescope

where the vertical maps are the identity, and the diagonal maps are continuations.

proposition 0.0.2

The cohomology of the telescope complex \(\bigoplus_{i=0}^\infty C^\bullet_i \oplus C^\bullet_{i-1}\) is \(\lim_{i\to\infty} H(C^\bullet_i)\). We could therefore also define \[SH(X):=\lim_{i\to\infty} \SH(X)^{< m^i}.\]