Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I. (English) Zbl 0795.58019

Oh, Yong-Geun

Commun. Pure Appl. Math. 46, No. 7, 949-994 (1993).

[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)

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; monotonicity

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