The Euler characteristic of an even-dimensional graph

We write the Euler characteristic χ(G) of a four dimensional finite simple geometric graph G = (V,E) in terms of the Euler characteristic χ(G(ω)) of two-dimensional geometric subgraphs G(ω). The Euler curvature K(x) of a four dimensional graph satisfying the Gauss-Bonnet relation ∑ x∈V K(x) = χ(G) can so be rewritten as an average 1 − E[K(x, f)]/2 over a collection two dimensional “sectional graph curvatures” K(x, f) through x. Since scalar curvature, the average of all these two dimensional curvatures through a point, is the integrand of the Hilbert action, the integer 2 − 2χ(G) becomes an integral-geometrically defined relative of the Hilbert action functional. The result has an interpretation in the continuum for compact 4-manifolds M : the Euler curvature K(x), the integrand in the classical Gauss-Bonnet-Chern theorem, can be seen as an average over a probability space Ω of 1 −K(x, ω)/2 with curvatures K(x, ω) of compact 2-manifolds M(ω). Also here, the Euler characteristic has an interpretation of an exotic Hilbert action, in which sectional curvatures are replaced by surface curvatures of integral geometrically defined random twodimensional sub-manifolds M(ω) of M . This is an informal note explaining a comment which slipped into [6]. It uses the observation of [5] that the symmetric index jf (x) = (if (x) + i−f (x))/2 at a critical point x of a function has a topological interpretation as the genus 1−χ(Bf (x))/2 of a lower dimensional space Bf (x). The index if (x) = 1− χ(S− f (x)) is a discretisation of the index of a gradient vector field ∇f at a critical point which by Poincaré-Hopf add up to the Euler characteristic χ(G) of G. For four dimensional spaces, jf (x) is the genus of a two-dimensional compact surface Bf (x) obtained by intersecting a small sphere S(x) with the level surface of f at x. Since genus is additive, we can glue the local critical surfaces Bf (x) together and get for every function f a two-dimensional graph G(f) whose genus is the sum of the indices. Poincaré-Hopf assures that