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.
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)
MSC:
| 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 |
Keywords:
symplectic manifold; Lagrangian submanifold; pseudo-holomorphic curves; Floer cohomology; exact deformations; monotonicityCitations:
Zbl 0795.58020References:
| [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 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
