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Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I. (English) Zbl 0795.58019
[This review comprises part I and part II [part II: ibid. 995-1012 (1993; see the paper below).]
A Lagrangian submanifold \(L\) in a symplectic manifold \((P^{2n},\omega)\) is an \(n\)-dimensional submanifold of \(P\) on which the symplectic form \(\omega\) vanishes. Floer defined a certain “cohomology theory”, denoted by \(I^*(L,\mathbb{Z}_ 2)\), of compact Lagrangian submanifolds on a compact symplectic manifold \((P,\omega)\) which are invariant under exact deformations of \(L\). Using this, he proved the estimate \(\# (L \cap \varphi (L)) \geq SB(L, \mathbb{Z}_ 2)\) = dimension of \(H^*(L,\mathbb{Z}_ 2))\) under the assumption \(\pi_ 2 (P,L)=\{e\}\), provided \(L\) meets \(\varphi(L)\) transversely.
In the present paper, the author refines Floer’s theory, first to generalize the construction without assuming \(\pi_ 2(P,L) = \{e\}\) but restricting to monotone \(L\)’s, and secondly to define a certain intersection invariant \(I^*(L_ 0,L_ 1:P)\) of pairs \((L_ 0,L_ 1)\) that are monotone. Again \(I^*(L_ 0,L_ 1:P)\) does not change under exact deformations of \(L_ 0\) or \(L_ 1\). In this generalization, besides Floer’s ideas, it is essential to analyze the so-called bubbling phenomena and to understand how the trajectories and pseudo-holomorphic disks (or spheres) intersect.
Reviewer: Y.-G.Oh (Madison)

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
Full Text: DOI
[1] Arnol’d, Funct. Anal. Appl. 1 pp 1– (1967)
[2] and , Casson’s Invariant for Oriented Homology 3-Spheres–An Exposition, Mathematical Notes, Princeton University Press, 1990. · Zbl 0695.57011 · doi:10.1515/9781400860623
[3] Einstein Manifolds, A series of Modern Surveys in Math. 3, Folgeband 10, Springer-Verlag, Berlin, 1987. · doi:10.1007/978-3-540-74311-8
[4] Borel, Amer. J. Math. 80 pp 458– (1958)
[5] Floer, J. Diff. Geom. 28 pp 513– (1988)
[6] Floer, Comm. Pure Appl. Math. 41 pp 393– (1988)
[7] Floer, Comm. Pure Appl. Math. 41 pp 775– (1988)
[8] Floer, J. Diff. Geom. 30 pp 207– (1989)
[9] Floer, Comm. Math. Phys. 120 pp 575– (1989)
[10] Floer, Comm. Math. Phys. 118 pp 215– (1988)
[11] , and , A note on unique continuation in the elliptic Morse theory for the action functional, preprint.
[12] Goldman, Adv. Math. 54 pp 220– (1984)
[13] Gromov, Invent. Math. 81 pp 307– (1985)
[14] Hofer, Ann. I. H. P. Analyse Non-linéaire 51 pp 465– (1988)
[15] and , A new capacity for symplectic manifolds, pp. 405–428 in: Analysis Et Cetera, and , eds., Academic Press, New York, 1990. · doi:10.1016/B978-0-12-574249-8.50023-7
[16] McDuff, Invent. Math. 89 pp 13– (1989)
[17] McDuff, Bull. AMS 23 pp 311– (1990)
[18] Oh, Invent. Math. 101 pp 501– (1990)
[19] Oh, Comm. Pure Appl. Math. 45 pp 121– (1992)
[20] Oh, Comm. Pure Appl. Math. 46 pp 995– (1993)
[21] Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks III: Arnol’d-Givental conjecture, to appear in Floer memorial volume, H. Hofer, et al, eds., Birkhäuser.
[22] Sur l’article de M. Gromov, Ecole Polytechnique, Palaiseau, 1986, preprint.
[23] Parker, J. Geom Anal. 3 pp 63– (1993) · Zbl 0759.53023 · doi:10.1007/BF02921330
[24] Viterbo, Bull. Soc. Math. Fr. 115 pp 361– (1987)
[25] Weinstein, Math. Z. 201 pp 75– (1989)
[26] Gromov’s compactness theorem for pseudo-holomorphic curves, preprint. · Zbl 0870.53002
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