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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.

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
55N35 Other homology theories in algebraic topology
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[1] [A]V.I. Arnold, Mathematical Methods of Classical Mechanics, 2nd Edition, Springer-Verlag, New York (1989).
[2] [C1]M. Chaperon, Une idée du type ”géodésiques brisées” pour les systèmes hamiltoniens, C.R. Acad. Sc. Paris t. 298, Série I, n0 13 (1984), 293–296.
[3] [C2]M. Chaperon, Phases génératrices en géométrie symplectique, Publication de l’institut de recherche mathématique avancée, Strasbourg, vol 41 (1989).
[4] [CiFH]K. Cieliebak, A. Floer, H. Hofer, Symplectic homology II: General symplectic manifolds, preprint (1994).
[5] [DNFo]B. Doubrovine, S. Novikov, A. Fomenko, Géométrie Contemporaine: Méthodes et Applications, 3e partie, Méthodes de la théorie de l’homologie, Editions Mir, Moscow (1987).
[6] [EH]I. Ekeland, H. Hofer, Symplectic topology and hamiltonian dynamics II, Math. Z. 203 (1990), 553–567. · Zbl 0729.53039 · doi:10.1007/BF02570756
[7] [El]Ya. Eliashberg, University of Arkansas Lecture Series on Symplectic Topology, (April 1992).
[8] [ElGr]Ya. Eliashberg, M. Gromov, Convex symplectic manifolds, Proc. Symp. Pure Math 52:2 (1991), 135–162.
[9] [FH]A. Eloer, H. Hofer, Symplectic homology I: Open sets in \(\mathbb{C}\) n , Math. Z. 215: 1 (1994), 37–88. · Zbl 0810.58013 · doi:10.1007/BF02571699
[10] [FHWy]A. Floer, H. Hofer, K. Wysocki, Applications of symplectic homology I, preprint (1992).
[11] [G]A. Givental, A symplectic fixed point theorem for toric manifolds, to appear in A. Floer Memorial Volume (1992). · Zbl 0835.55001
[12] [L]S. Lang, Algebra, Third Edition, Addison-Wesley Publishing Company, Reading, MA (1993).
[13] [LaS]F. Laudenbach, J.C. Sikorav, Persistance d’intersection avec la section nulle au cours d’une isotopie hamiltonienne dans un fibré cotangent, Invent. Math. 82 (1985), 349–357. · Zbl 0592.58023 · doi:10.1007/BF01388807
[14] [S]J. C. Sikorav, Sur les immersions lagrangiennes dans un fibré cotangent admettant une phase génétrice globale, C. R. Acad. Sc. Paris t. 302, Série I, n03 (1986), 119–122.
[15] [T]D. Theret, Thèse, Université Denis Diderot (Paris 7), to appear, 1994.
[16] [V]C. Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann. 292:4 (1992), 685–710. · Zbl 0780.58023 · doi:10.1007/BF01444643
[17] [W]A. Weinstein, Lectures on Symplectic Manifolds, CBMS Reg. Conf. Series in Math. 29, Amer. Math. Soc., Providence, R.I. (1979).
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