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

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application 0.0.1

We look at an application of (cohomology includes into SH). Let \(X\) be a Liouville domain, and suppose that \((\partial X, \alpha)\) has no Reeb orbits. Then the map from (cohomology includes into SH) is an isomorphism \[H^\bullet(X)\to \SH(X).\] Therefore, we can compute \(\SH(X)\) to show that \((\partial X)\) has a Reeb orbit.

example 0.0.2

A key example is the standard contact structure on the sphere. One can compute that the standard symplectic ball \(B^{2n}:=\{(z_1, \ldots, z_n)\st \sum_{i=1}^n |z_i|^2=1 \}\subset \CC^n\) is a Liouville domain, and that \(\SH(B^{2n})=0\). We can conclude that \((S^{2n-1}, \alpha)\) has a Reeb orbit. A generalization of this result (due to [Oan03]) states that whenever \(X\) is a subcritical Stein domain (so the Morse indexes of the critical points of \(\phi: X\to \RR\) are all less than \(n\)) then \(\SH(X)=0\).

References

[Oan03]Alexandru Oancea. La suite spectrale de Leray-Serre en homologie de Floer des variétés symplectiques compactes à bord de type contact. PhD thesis, Université Paris Sud-Paris XI, 2003.