Surface
Morse Theory (Marston Morse, 1930s):
Critical points where ∇f = 0 encode the topology:
● Minima (index 0) — χ contribution: +1
● Saddles (index 1) — χ contribution: −1
● Maxima (index 2) — χ contribution: +1
Euler characteristic:
χ = #min − #saddle + #max
For a sphere: χ=2. Torus: χ=0.
The level sets change topology only when the sweep passes a critical value.
Critical points where ∇f = 0 encode the topology:
● Minima (index 0) — χ contribution: +1
● Saddles (index 1) — χ contribution: −1
● Maxima (index 2) — χ contribution: +1
Euler characteristic:
χ = #min − #saddle + #max
For a sphere: χ=2. Torus: χ=0.
The level sets change topology only when the sweep passes a critical value.