Floer Homology

Morse theory: generic f:M→ℝ has isolated critical points. Gradient flow ẋ = −∇f connects them.

Morse complex: Cₖ = free abelian group on index-k crits. Boundary ∂: Cₖ→Cₖ₋₁ counts (signed) gradient flow lines between adjacent-index crits.

Floer's insight: extend to infinite-dim (path spaces, Lagrangians). ASD equations replace gradient flow → Floer homology HF*(L₀,L₁).

Theorem: HF*(L₀,L₁) ≅ H*(L₀) when L₀⋔L₁ (and monotone etc.) → Arnold conjecture: #(L₀∩L₁) ≥ sum Betti numbers.