The Sphere Formula

TL;DR

The paper introduces a universal sphere formula in finite simplicial complexes linking Euler characteristics of unit spheres at different dimensions, with implications for understanding Euler characteristics of manifolds and spaces with constant curvature.

Contribution

It establishes a new general sphere formula applicable to arbitrary finite simplicial complexes, connecting local sphere Euler characteristics to global topological properties.

Findings

In odd-dimensional manifolds, the Euler characteristic is zero.

Spaces with constant unit sphere Euler characteristic have zero Euler characteristic.

The formula unifies various classical results like Poincaré duality and Gauss-Bonnet.

Abstract

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 Poincar\'e 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.