Topological Euler Number Explorer

The Euler characteristic χ = 2 − 2g − b for an orientable surface with genus g (handles) and b (boundary components). Add and remove handles and holes — χ never lies about your surface's topology.

χ = 2 − 2g − b

g=0 handles, b=0 holes

Sphere

Surface base:

Handles (genus):

Holes (boundary circles):

Connected sum with:

Gauss-Bonnet: ∮M K dA = 2πχ — curvature integrates to exactly 2πχ for any metric on the surface. Adding a handle inserts a torus-like tube: χ decreases by 2. Adding a disk-hole removes a contractible region: χ decreases by 1. The Klein bottle has χ=0 like the torus, but is non-orientable.