Morse theory for Lagrangian intersections.

*(English)*Zbl 0674.57027This 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).

“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 |