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Transactions of the American Mathematical Society

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Morse homology for generating functions
of Lagrangian submanifolds


Author: Darko Milinkovic
Journal: Trans. Amer. Math. Soc. 351 (1999), 3953-3974
MSC (1991): Primary 58E05; Secondary 57R57, 58F05
DOI: https://doi.org/10.1090/S0002-9947-99-02217-5
Published electronically: March 8, 1999
MathSciNet review: 1475690
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Abstract: The purpose of the paper is to give an alternative construction and the proof of the main properties of symplectic invariants developed by Viterbo. Our approach is based on Morse homology theory. This is a step towards relating the ``finite dimensional'' symplectic invariants constructed via generating functions to the ``infinite dimensional'' ones constructed via Floer theory in Y.-G. Oh, Symplectic topology as the geometry of action functional. I, J. Diff. Geom. 46 (1997), 499-577.


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  • 1. M. Betz and R. Cohen. Graph moduli spaces and cohomology operations. Turkish J. Math., 18:23-41, 1994. MR 95i:58037
  • 2. M. Chaperon. Une idée du type ``géodésiques brisées'' pour les systèmes hamiltoniens. C. R. Acad. Sc. Paris, 298:293-296, 1984. MR 86f:58049
  • 3. M. Chaperon. An elementary proof of the Conley-Zehnder theorem in symplectic geometry. In Braaksma, Broer, and Takens, editors, Dynamical Systems and Bifurcations, volume 1125 of Springer Lecture Notes in Mathematics, pages 1-8, 1985. MR 87a:58062
  • 4. A. Floer. Morse theory for Lagrangian intersections. J. Diff. Geom., 28:513-517, 1988. MR 90f:58058
  • 5. A. Floer. Cuplength estimates on Lagrangian intersections. Comm. Pure. Appl. Math., 42:335-356, 1989. MR 90g:58034
  • 6. A. Floer. Witten's complex and infinite dimensional Morse theory. J. Diff. Geom., 30:207-221, 1989. MR 90d:58029
  • 7. K. Fukaya. Morse theory and topological field theory. To appear in Sugaku Expositions.
  • 8. K. Fukaya and Y.-G. Oh. Zero loop open strings in the cotangent bundle and Morse homotopy. Asian J. Math., 1: 96-180, 1997. CMP 98:04
  • 9. L. Hörmander. Fourier integral operators I. Acta Math., 127:79-183, 1971. MR 52:9299
  • 10. F. Laudenbach and J.-C. Sikorav. Persistence d'intersections avec la section nulle au conours d'une isotopie Hamiltonienne dans un fibre cotangent. Invent. Math., 82:349-357, 1985. MR 82c:58042
  • 11. D. Milinkovi\'{c} and Y.-G. Oh. Generating functions versus the action functional- stable Morse theory versus Floer theory. To appear in Proceedings of Workshop on Geometry, Topology and Dynamics, Montreal, 1995.
  • 12. J. Milnor. Lectures on the h-cobordism theorem. Math. Notes. Princeton Univ. Press, 1965. MR 32:8352
  • 13. Y.-G. Oh. Symplectic topology as the geometry of action functional II. Preprint.
  • 14. Y.-G. Oh. Symplectic topology as the geometry of action functional I. J. Diff. Geom., 46:499-577, 1997. MR 99a:58032
  • 15. M. Schwarz. Morse Homology. Birkhäuser, Basel, 1993. MR 95a:58022
  • 16. J.-C. Sikorav. Problèmes d'intersections et de points fixes en géometrie Hamiltonienne. Comment. Math. Helv, 62:61-73, 1987. MR 88g:58067
  • 17. S. Smale. The generalized Poincaré conjecture in higher dimensions. Bull. Amer. Math. Soc., 66:373-375, 1960. MR 23:A2220
  • 18. D. Théret. Utilisation des fonctions generatrices en geometrie symplectique globale. PhD thesis, Université de Paris 7, 1995.
  • 19. L. Traynor. Symplectic homology via generating functions. GAFA, 4:718-784, 1994. MR 96a:58049
  • 20. C. Viterbo. Symplectic topology as the geometry of generating functions. Math. Ann., 292:685-710, 1992. MR 93b:58058
  • 21. E. Witten. Supersymmetry and Morse theory. J. Diff. Geom., 17:661-692, 1982. MR 84b:58111

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Additional Information

Darko Milinkovic
Affiliation: Department of Mathematics, University of Wisconsin–Madison, 480 Lincoln Drive, Madison, Wisconsin 53706
Address at time of publication: Department of Mathematics, University of California, Irvine, California 92697-3875
Email: dmilinko@math.uci.edu

DOI: https://doi.org/10.1090/S0002-9947-99-02217-5
Received by editor(s): August 18, 1997
Published electronically: March 8, 1999
Article copyright: © Copyright 1999 American Mathematical Society

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