# Proofs of Euler's characteristic formula for n-Dim polytopes

Twenty proofs of Euler's formula V - E + F - 1 = 1, which applies to convex polyhedrons, i.e., 3-dimensional polytopes, are presented at the Geometry Junkyard.

I'm interested in proofs of the more general formula for the Euler characteristic number for bounded, convex polytopes of dimension greater than three as well since the signed, refined face partition polynomials enumerating the k-dimensional faces (k=0 to n) of the n-dimensional associahedra and providing the compositional inversion of formal power series obey the extended Euler formula $$V - E + (2-D-faces) - (3-D-faces) + ...$$

$$(-1)^{n-1} ((n-1)-D-facets) + (-1)^n = 1,$$

and proofs of the formula might provide insight on derivations of the face partition polynomials. (Same applies to permutahedra and multiplicative inversion.)

I'm particularly interested in proofs related to a generalized Gauss-Bonnet theorem, proofs related to differential geometry.

1) Which of the Junkyard proofs can be extended beyond three dimensions to any n-dimensional bounded, convex polytope?

2) Do you have references to other proofs for indefinite dimensions?

• @SamHopkins, "indefinite" not "infinite", i.e., any finite dimension . (Although Jim Propp does address infinite according to the site.) – Tom Copeland Nov 14 '19 at 1:29
• sorry, I misread that! – Sam Hopkins Nov 14 '19 at 1:30
• Easy mistake, so I'll leave the comment just in case ... . – Tom Copeland Nov 14 '19 at 1:32
• Proof number 15, using "binary homology" (ie homology with coefficients in $\mathbb{Z}/2$) applies to any bounded convex polytope, and also works with any coefficient field. – HJRW Nov 14 '19 at 14:21
• See also " Intuitive reason why the Euler characteristic is an alternatibg sum" (math.stackexchange.com/questions/1828482/…) and "Five proofs that the Euler characteristic of a closed orientable surface is even" by Q. Yuan (qchu.wordpress.com/2014/10/14/…) – Tom Copeland Nov 14 '19 at 19:28