Morse Theory

Critical points, index, and the topology of sublevel sets

Surface with height function

Sublevel set M_t = f⁻¹(−∞,t]

Min (index 0) Saddle (index 1) Max (index 2)

Surface Height t 0.50

Morse inequality: Σ(−1)^k cₖ = χ(M). Each critical point of index k creates/destroys a k-cell. Torus: 1 min + 2 saddles + 1 max = χ=0.