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Poincaré formula (Theorem)

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

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



"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 MSC05C99 (Combinatorics :: Graph theory :: Miscellaneous)

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