# zbMATH — the first resource for mathematics

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.