Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces. (English) Zbl 1158.53067

Let \(H\) be a Hamiltonian on a symplectic manifold \((W,\omega)\). One of the most important assumptions in the Floer homology is that the \(1\)-periodic orbits of the Hamiltonian vector field \(X_ H\) are non-degenerate, which makes \(H\) time-dependent.
In this paper, the authors generalize the notion of Floer homology by defining it for a time-independent or autonomous Hamiltonian \(H: W\to \mathbb R\) under the assumption that its \(1\)-periodic orbits are transversally non-degenerate. There is a natural class of time-independent Hamiltonians on \(W\) whose non-constant \(1\)-periodic orbits correspond precisely to closed Reeb orbits on \(M= \partial W\) and for which the Floer trajectories can be related to holomorphic cylinders in the symplectization \(M\times\mathbb R\). The authors relate the Floer trajectories of a specific time-dependent perturbation to the Floer trajectories of the unperturbed Hamiltonian in the way that Floer homology for time-independent Hamiltonians serves as a bridge between symplectic homology and linearized contact homology.


53D40 Symplectic aspects of Floer homology and cohomology
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