PlanetMath: Poincaré formula

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Poincaré formula

(Theorem)

Let be finite oriented simplicial complex of dimension . Then

$\displaystyle \chi(K)= \sum_{p=0}^n (-1)^p R_p(K),$

where $\chi(K)$ is the Euler characteristic of

, and $R_{p}(K)$ is the

-th Betti number of

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

"Poincaré formula" is owned by CWoo. [ full author list (3) | owner history (2) ]

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See Also: Euler's polyhedron theorem, polytope

Other names:

Euler-Poincaré formula, Euler-Poincare formula

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Cross-references: genus, boundary, surface, orientable, connected, compact, polyhedra, Euler formula, CW complex, Betti number, Euler characteristic, dimension, simplicial complex, oriented, finite

This is version 8 of Poincaré formula, born on 2003-06-09, modified 2007-09-07.
Object id is 4336, canonical name is PoincareFormula.
Accessed 5364 times total.

Classification:

AMS MSC:

05C99 (Combinatorics :: Graph theory :: Miscellaneous)

Pending Errata and Addenda

None.[ View all 6 ]

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