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# 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.

Related:

EulersPolyhedronTheorem, Polytope

Synonym:

Euler-Poincar\'e formula, Euler-Poincare formula

Type of Math Object:

Theorem

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

05C99*no label found*

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