proposition 0.0.1
Let \(X\) be a Liouville domain. There is a map \[H^\bullet(X)\to \SH(X).\]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.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.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. |