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

Title Poincaré formula PoincareFormula 2013-03-22 13:40:15 2013-03-22 13:40:15 CWoo (3771) CWoo (3771) 11 CWoo (3771) Theorem msc 05C99 Euler-Poincaré formula Euler-Poincare formula EulersPolyhedronTheorem Polytope