Poincaré formula

Let $K$ be finite oriented simplicial complex of dimension $n$. Then

$$\chi (K)=\sum _{p=0}^n{(-1)}^p{R}_p(K),$$

where $\chi (K)$ is the Euler characteristic of $K$, and $R_p(K)$ is the $p$-th Betti number of $K$.

This formula also works when $K$ is any finite CW complex. The Poincaré formula is also known as the Euler-Poincaré formula, for it is a generalization of the Euler formula for polyhedra.

If $K$ is a compact connected orientable surface with no boundary and with genus h, then $\chi (K)=2-2h$. If $K$ is non-orientable instead, then $\chi (K)=2-h$.