Hamiltonian Floer Homology

Overview

Hamiltonian Floer homology converts periodic Hamiltonian trajectories into generators of a chain complex. Its differential counts solutions of a partial differential equation joining those trajectories. This research overview restricts to closed symplectically aspherical manifolds, where sphere bubbling is excluded. It gives the organizing definitions and analytical requirements, rather than pretending that counting solutions alone completes the construction.

Prerequisites

Use Hamiltonian flows, compatible almost complex structures, Morse homology, and basic elliptic equations. Understanding a full construction additionally requires functional analysis, Fredholm operators, transversality, compactness, and gluing. These are substantive proof obligations, not optional technical refinements.

Definition and conventions

Let $(M^{2n},\omega)$ be compact without boundary, smooth, Hausdorff, and second countable. Require $\omega$ and $c_1(TM)$ to vanish on $\pi_2(M)$. Choose a smooth one-periodic Hamiltonian $H_t$ whose contractible one-periodic orbits are nondegenerate: the return map has no eigenvalue $1$ along an orbit. Use coefficients $\mathbb F_2$.

With $\iota_{X_{H_t}}\omega=dH_t$ and compatible $J_t$, the equation in this entry is

The chain module is generated by those contractible orbits. After regularity and compactness are established, its differential counts isolated finite-energy solutions modulo translation in $s$, with specified asymptotic orbits. Grading conventions and an overall dimension shift must be fixed before a graded comparison.

Intuition

This resembles Morse homology on a loop space: periodic orbits replace critical points, and cylinders replace descending paths. The resemblance guides the construction, but infinite-dimensional analysis introduces new difficulties. The sphere-vanishing assumptions make the action and grading independent of capping choices and remove one major source of compactness failure.

Examples and non-examples

The standard symplectic torus $T^{2n}$ satisfies the sphere-vanishing assumptions because $\pi_2(T^{2n})=0$. For a sufficiently small suitable autonomous Morse Hamiltonian, the contractible generators and trajectory analysis reduce to a Morse model. The resulting ungraded Floer homology has the same total rank as ordinary homology, namely $2^{2n}$ over $\mathbb F_2$.

The standard symplectic two-sphere is outside this restricted setting: its symplectic area does not vanish on $\pi_2$. Floer homology still exists there through a broader theory, but that needs capping and bubbling machinery omitted here. A degenerate Hamiltonian also requires perturbation before this chain model applies.

Important properties

The identity $\partial^2=0$ comes from the ends of compactified one-dimensional trajectory spaces. Under these hypotheses their relevant ends are broken pairs of cylinders, whose contributions cancel modulo two. Establishing that these are all ends requires compactness and gluing.

Continuation equations compare choices of Hamiltonian and almost complex structure. The resulting homology agrees with ordinary homology, subject to the chosen grading shift. This gives fixed-point lower bounds for nondegenerate Hamiltonian diffeomorphisms. The statement does not automatically apply to arbitrary symplectomorphisms or noncompact manifolds.

Connections and further reading

Read Merry, Definition 2.24, Theorem 2.25, and §§3–4 for the inspected restricted framework. The notes explicitly warn that they were not proofread; this entry is an editorial overview, not independent analytical verification. Lagrangian, instanton, and Heegaard Floer theories use different generators and hypotheses.