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

applications of \(\SH(X)\)

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We look at some applications and theorems from symplectic cohomology.

proposition 0.0.1

Let \(X\) be a Liouville domain. There is a map \[H^\bullet(X)\to \SH(X).\] Let \(m_0\) be the minimal period of a Reeb orbit in \((\partial X, \alpha)\). Pick a slope \(0< m< m_0\), and consider a linear Hamiltonian \(H^m\) which is \(C^2\) small on \(X\). Then the only orbits for \(H^m\) will be the constant orbits, and there is a quasi-isomorphism of chain complexes between \(\CF(\hat X, H^m_t)\) and the Morse complex \(CM^\bullet(\hat X, H^m_t)\) sending each constant orbit to its associated critical point of \(H^m_t\). Since \(H^m\) has gradient which points outward along the boundary of \(X\), this is a valid Morse function for computing the Morse cohomology of \(X\).

application 0.0.2

We look at an application of proposition 0.0.1. Let \(X\) be a Liouville domain, and suppose that \((\partial X, \alpha)\) has no Reeb orbits. Then the map from proposition 0.0.1 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 0.0.3

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 \(\SH(X)=0\).

definition 0.0.4

Suppose that \((X, \lambda)\) is a Liouville domain. A Liouville subdomain is a compact submanifold with boundary \(X_0\subset X\setminus \partial X\) such that the Liouville vector \(Z\) points outwards along \(\partial X_0\).

theorem 0.0.5 [Vit99]

Let \(X_0\subset X\) be a Liouville subdomain. Then there is a restriction map \(\SH(X)\to \SH(X_0)\), which is a unital ring homomorphism. Furthermore, we have a commutative diagram where the horizontal maps are given by proposition 0.0.1. In the setting of subcritical Stein domains ( application 0.0.2 ), the sublevel sets \(X|_{\phi< t}\) form a nested sequence of Liouville subdomains. One way to prove the vanishing of \(\SH(X)\) is to show that the map \(\SH(X|_{\phi< t_{i+1}})\to \SH(X|_{\phi< t_{i}})\) is a an isomorphism for all \(t_{i} < t_{i+1}\). When the only critical points of \(\phi\) contained in \(\SH(X|_{\phi < t_0})\) are minima, then \(X|_{t_0}\) is a ball, which has vanishing symplectic cohomology.

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.
[Vit99]Claude Viterbo. Functors and computations in Floer homology with applications, i. Geometric & Functional Analysis GAFA, 9(5):985--1033, 1999.