Floer Homology

Floer homology is an infinite-dimensional Morse theory for the action functional on the loop space of a symplectic manifold. It provides the primary tool for proving the Arnold conjecture and has become one of the most powerful invariants in symplectic topology.

Construction

Definition8.1Floer Chain Complex

For a non-degenerate Hamiltonian $H: S^1 \times M \to \mathbb{R}$ on a compact symplectic manifold $(M, \omega)$ , the Floer chain complex $CF_*(M, H)$ is the $\mathbb{Z}_2$ -vector space generated by 1-periodic orbits of $X_H$ . The differential counts Floer trajectories: solutions $u: \mathbb{R} \times S^1 \to M$ of the Floer equation $\partial_s u + J_t(\partial_t u - X_H(t, u)) = 0$ converging to periodic orbits as $s \to \pm\infty$ .

Definition8.2Floer Homology

The Hamiltonian Floer homology $HF_*(M, \omega)$ is the homology $H_*(CF_*, \partial)$ . The key properties: $\partial^2 = 0$ (from Gromov compactness for 1-dimensional moduli spaces), and $HF_*$ is independent of $H$ and $J$ (invariance via continuation maps). For a compact symplectic manifold, $HF_*(M) \cong H_*(M; \mathbb{Z}_2)$ (or $QH_*(M)$ over the Novikov ring).

ExampleFloer Homology of $T^{2n}$

For the torus $T^{2n}$ with standard symplectic form, $HF_*(T^{2n}) \cong H_*(T^{2n}; \mathbb{Z}_2)$ has total dimension $2^{2n}$ . Since $\dim CF_* \geq \dim HF_*$ , every non-degenerate Hamiltonian on $T^{2n}$ has at least $2^{2n}$ periodic orbits, proving the Arnold conjecture.

Grading and Ring Structure

RemarkConley-Zehnder Index

When $c_1(TM) = 0$ or $c_1(TM)|_{\pi_2} = 0$ , the Floer complex admits a $\mathbb{Z}$ -grading by the Conley-Zehnder index, which measures the winding of the linearized flow along a periodic orbit relative to a trivialization. The differential decreases the grading by 1.

Definition8.3Pair-of-Pants Product

The pair-of-pants product $HF_*(H_1) \otimes HF_*(H_2) \to HF_*(H_1 \# H_2)$ is defined by counting holomorphic maps from a pair-of-pants surface (sphere minus three discs) with Lagrangian/periodic boundary conditions. This gives $HF_*$ the structure of a ring isomorphic to the quantum cohomology ring $QH^*(M)$ .