Floer Homology Visualizer

Floer homology counts pseudo-holomorphic strips connecting critical points of an action functional. For Morse theory on a manifold: trajectories flow "downhill" along gradient lines. Floer's insight: replace the finite-dimensional Morse complex with an infinite-dimensional one on loop spaces, recovering symplectic invariants.

Potential
Perturbation ε0.00
Flow Speed1.0
ℤ = 0
Potential f(x) with critical points
Gradient flow lines (Morse trajectories)

Morse / Floer Chain Complex

HF*(f) = ker ∂ / im ∂
∂: Cₖ → Cₖ₋₁
∂² = 0 (key lemma)

#(flow lines p→q) mod 2
gives ∂ matrix entries

Floer: loop space L(M)
Action A(γ)=∮H dt
∂ = hol. strips u(s,t)

Morse Numbers

IndexCrit Pointsβₖ