The Sphere Formula
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The sphere formula states that in an arbitrary finite abstract simplicial complex, the sum of the Euler characteristic of unit spheres centered at even-dimensional simplices is equal to the sum of the Euler characteristic of unit spheres centered at odd-dimensional simplices. It follows that if a geometry has constant unit sphere Euler characteristic, like a manifold, then all its unit spheres have zero Euler characteristic or the space itself has zero Euler characteristic. Especially, odd-dimensional manifolds have zero Euler characteristic, a fact usually verified either in algebraic topology using Poincaré duality together with Riemann-Hurwitz then deriving it from the existence of a Morse function, using that the Morse indices of the function and its negative add up to zero in odd dimensions. Gauss Bonnet also shows that odd-dimensional Dehn-Sommerville spaces have zero Euler characteristic because they have constant zero curvature. Zero curvature phenomenons can be understood integral geometrically as index expectation or as Dehn-Sommerville relations.
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