Euler Characteristic

Let a closed surface have Genus $g$. Then the Polyhedral Formula becomes the Poincaré Formula

$$\chi\equiv V-E+F=2-2g,$$ (1)

where $\chi$ is the Euler characteristic, sometimes also known as the Euler-Poincaré Characteristic. In terms of the Integral Curvature of the surface $K$,

$$\int\!\!\!\int K\,da=2\pi\chi.$$ (2)

The Euler characteristic is sometimes also called the Euler Number. It can also be expressed as

$$\chi=p_0-p_1+p_2,$$ (3)

where $p_i$ is the $i$th Betti Number of the space.

See also Chromatic Number, Map Coloring