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Morse theory for Lagrangian intersections. (English) Zbl 0674.57027
This important paper develops “Floer cohomology” in a symplectic context, with the aim of proving the Arnold conjecture about the number of fixed points of an exact symplectic diffeomorphism. The author’s abstract:
“Let P be a compact symplectic manifold and let $$L\subset P$$ be a Lagrangian submanifold with $$\pi_ 2(P,L)=0$$. For any exact diffeomorphism $$\phi$$ of P with the property that $$\phi$$ (L) intersects L transversally, we prove a Morse inequality relating the set $$\phi$$ (L)$$\cap L$$ to the cohomology of L. As a consequence, we prove a special case of the Arnold conjecture: If $$\pi_ 2(P)=0$$ and $$\phi$$ is an exact diffeomorphism all of whose fixed points are non-degenerate, then the number of fixed points is greater than or equal to the sum over the $${\mathbb{Z}}_ 2$$-Betti numbers of P.”
The Morse inequality comes from the “Floer cohomology” of a complex $$I^*(L,\phi,J)$$ where $$I^ p(L,\phi,J)$$ is the $${\mathbb{Z}}_ 2$$-module generated by the points in $$\phi$$ (L)$$\cap L$$ which have “index p”. In order to define the boundary operator of $$I^*$$ he observes that these intersection points are the critical points of a certain action functional $${\mathfrak a}$$ on a space $$\Omega$$ of paths z(t) in P which join a point z(0)$$\in L$$ to a point z(1)$$\in \phi (L)$$. Floer chooses a metric on $$\Omega$$ using an auxiliary almost complex structure J on P. (If P were Kähler, this could just be the Kähler metric.) This has the property that the trajectories $$z_{\tau}$$ of the gradient flow of $${\mathfrak a}$$ on $$\Omega$$ with respect to this metric fit together to form a J-holomorphic map u: ($$\tau$$,t)$$\mapsto z_{\tau}(t)$$ of the strip $${\mathbb{R}}\times [0,1]$$ into P. Even though the gradient flow of $${\mathfrak a}$$ is not globally defined on $$\Omega$$, the trajectories which go from one critical point to another are well-behaved for generic J because they are solutions of an elliptic p.d.e.. Therefore, one can define the boundary in $$I^*$$ by the recipe: $$\partial x=n(x,y)y$$, where n(x,y) is the number (mod 2) of isolated trajectories from x to y if Index(y)- $$Index(x)=1$$ and is zero otherwise. Floer shows that the cohomology of $$(I^*(L,\phi,J),\partial)$$ is invariant under deformations of $$\phi$$, and equals $$H^*(P,{\mathbb{Z}}_ 2)$$ when $$\phi$$ is small. Thus the number of critical points of $${\mathfrak a}$$ is at least dim $$H^*(P,{\mathbb{Z}}_ 2)$$. The properties of $${\mathfrak a}$$ are very close to those of the Chern- Simons functional on connections which the author uses in Commun. Math. Phys. 118, No.2, 215-240 (1988).
Reviewer: D.McDuff

##### MSC:
 57R50 Differential topological aspects of diffeomorphisms 57R70 Critical points and critical submanifolds in differential topology 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 57R45 Singularities of differentiable mappings in differential topology 57R42 Immersions in differential topology 58C25 Differentiable maps on manifolds 58K99 Theory of singularities and catastrophe theory 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 58J10 Differential complexes 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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