- \(\CF(X, H_t)\) is a chain complex. The key step is to show that when \(\dim(\mathcal M(\gamma_+, \gamma_-))=1\), there exists a compactification of this moduli space by including broken cylinders. A compactification of the space comes from applying Gromov compactness, while additional requirements on \(X\) are sometimes required ensure that the only configurations which appear in the compactification are broken cylinders. In our setting, the only breaking configurations which may occur are broken cylinders, as \(X\) is exact (so \(\omega(\pi_2(X))=0)\).
- Given \(H_{t, 0}\) and \(H_{t, 1}\) two time-dependent Hamiltonians, there exists a continuation map \(\CF(X, H_{t, 0})\to \CF(X, H_{t, 1})\). Furthermore, this map is a homotopy equivalence.
- Finally, we need some way to compute \(\CF(X, H_{t, 1})\). One way to do this is to observe that for \(C^2\) small Hamiltonians the Floer cochains agree with the Morse cochains (and only consist of constant orbits). By either using the PSS-isomorphism or by analyzing Floer trajectories, the Floer cohomology can be compared to the Morse cohomology of \(X\).
example 0.0.1
The maximum modulus principle states that if \(\phi: D^2\to \CC\) is a holomorphic function from the disk to \(\CC\), that the maximum of \(|\phi|: D^2\to \RR_{\geq 0}\) is achieved on \(\partial D^2\). Let \(\hat X\) be a non-compact symplectic manifold with compatible almost complex structure \(J\), along with a \(J-\jmath\)-holomorphic projection \(W: \hat X\to \CC\). Suppose that the fibers of \(W\) are compact. Pick two loops \(\gamma_-, \gamma_+\subset \hat X\) and \(r_0\in \RR\) large enough so that \(U:=W^{-1}(\{z \st |z|\leq r\})\) contains \(\gamma_-, \gamma_+\). We will prove that every pseudoholomorphic cylinder \(u: S^1\times \RR\to \hat X\) with ends limiting to \(\gamma_-, \gamma_+\) has image contained within the compact subset \(U\). The composition \(W\circ u: S^1\times \RR\to \CC\) is a holomorphic map, with ends limiting to \(W(\gamma_\pm)\), and therefore satisfies the maximum modulus principle. Since the boundary is sent to \(W(\gamma_\pm)\), we obtain that \(|W|\) achieves a value no greater than \(r_0\) on \(u\); therefore \(\Im(u)\subset U\). It follows that the image of \(u\) is contained within a compact set.definition 0.0.2
Let \(\hat X\) be the completion of a Liouville domain. A choice of almost complex structure for \(\hat X\) is of contact type if \[d(\exp(r))\circ J = -\alpha.\]proposition 0.0.3
Let \(H: \hat X\to \RR\) be a Hamiltonian which on the symplectization takes the form of \(h(\exp(r))\). Let \(\gamma_+, \gamma_-\) be time 1 orbits of \(V_{H_{t}}\). For a contact type almost complex structure, every solution \(u: \RR\times S^1\to \hat X\) of the Floer equation with ends limiting to \(\gamma_+, \gamma_-\) has image contained in the subset \(\hat X|_{\exp(r)\leq C}\), where \(C\) is the maximum value of \(\exp(r)\) on the orbits \(\gamma_+, \gamma_-\).References
[Abo10] | Mohammed Abouzaid. A geometric criterion for generating the Fukaya category. Publications Mathématiques de l'IHÉS, 112:191--240, 2010. |
[Sei06] | Paul Seidel. A biased view of symplectic cohomology. Current developments in mathematics, 2006(1):211--254, 2006. |