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 $\chi(K)=2-h$ .