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Symplectic homology via generating functions. (English) Zbl 0822.58020
Nowadays there exist two different symplectic homology theories: via Floer homology and via generating functions. The first theory has been developed by Floer, Hofer and their collaborators, the second one has been developed by the author of this paper based on a circle of idea in generating functions by Viterbo and Eliashberg. Since the generating functions are based on symplectic reduction, the author constructs her version of symplectic homology only for open subsets of \(\mathbb{R}^{2n}\) with the standard symplectic structure (while the symplectic homology by Floer and Hofer can be constructed on many other symplectic manifolds). In short, the idea is following: we associate each symplectomorphism \(h\) with a generating function \(S\) which is quadratic at infinity and then define the relative homology groups \(G_ k^{(a,b]} (h)\) via relative homology groups defined by levels of \(S\) restricted on \(S^{2n} \times R^ N \subset R^{2n} \times R^ N\). She defines a symplectic homology as an inverse limit of certain symplectomorphisms associated to the given set. The author also shows certain functorial properties of these symplectic homology groups. She also calculates symplectic homology groups of an open ellipsoid and certain shells.
This paper is an important step in the study of symplectic topology.

MSC:
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
55N35 Other homology theories in algebraic topology
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