Poincaré formula

Let K be finite oriented simplicial complexMathworldPlanetmath of dimensionMathworldPlanetmath n. Then


where χ(K) is the Euler characteristicMathworldPlanetmath of K, and Rp(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 formulaPlanetmathPlanetmath for polyhedra.

If K is a compactPlanetmathPlanetmath connectedPlanetmathPlanetmath orientable surface with no boundary and with genus h, then χ(K)=2-2h. If K is non-orientable instead, then χ(K)=2-h.

Title Poincaré formula
Canonical name PoincareFormula
Date of creation 2013-03-22 13:40:15
Last modified on 2013-03-22 13:40:15
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 11
Author CWoo (3771)
Entry type Theorem
Classification msc 05C99
Synonym Euler-Poincaré formula
Synonym Euler-Poincare formula
Related topic EulersPolyhedronTheorem
Related topic Polytope