Lagrangian Intersection Theory

The study of intersections of Lagrangian submanifolds is one of the deepest areas of symplectic topology. The Arnold conjecture predicts that Lagrangian intersections are constrained by topology, and Floer homology provides the tool to prove this.

Arnold Conjecture

Definition6.4Arnold Conjecture for Lagrangians

Let $L$ be a compact Lagrangian in $(M, \omega)$ and $\phi \in \mathrm{Ham}(M, \omega)$ a Hamiltonian diffeomorphism. The Lagrangian Arnold conjecture states: $|L \cap \phi(L)| \geq \sum_{k} \dim H_k(L; \mathbb{Z}_2),$ provided the intersection is transverse. The non-degenerate version (requiring non-degeneracy of the intersection) was proved by Floer for monotone Lagrangians.

ExampleClifford Torus in $\mathbb{CP}^n$

The Clifford torus $L = \{[z_0 : \cdots : z_n] : |z_0| = \cdots = |z_n|\} \subset \mathbb{CP}^n$ is a monotone Lagrangian with $H_*(L; \mathbb{Z}_2)$ of total dimension $2^n$ . The Arnold conjecture predicts $|L \cap \phi(L)| \geq 2^n$ for Hamiltonian perturbations $\phi$ .

Maslov Index

Definition6.5Maslov Index

The Maslov index $\mu: \pi_2(M, L) \to \mathbb{Z}$ is a homomorphism that measures how the Lagrangian boundary condition "winds" relative to a reference. For a disc $u: (D^2, \partial D^2) \to (M, L)$ , $\mu(u)$ counts (with signs) the rotations of $u^*TL$ in the symplectic frame $u^*TM$ . The minimal Maslov number $N_L = \min\{\mu(\beta) > 0 : \beta \in \pi_2(M,L)\}$ controls the grading in Floer homology.

RemarkLagrangian Floer Homology

For two Lagrangians $L_0, L_1$ intersecting transversally, the Lagrangian Floer cohomology $HF(L_0, L_1)$ is the cohomology of a chain complex generated by intersection points, with differential counting pseudo-holomorphic strips (bigons). When well-defined, $HF(L, L) \cong H^*(L)$ , proving the Arnold conjecture.