The Euler characteristic of an even-dimensional graph

We write the Euler characteristic X(G) of a four dimensional finite simple geometric graph G=(V,E) in terms of the Euler characteristic X(G(w)) of two-dimensional geometric subgraphs G(w). The Euler curvature K(x) of a four dimensional graph satisfying the Gauss-Bonnet relation sum_x K(x) = X(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 2-2 X(G) becomes an integral-geometrically defined 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 W of 1-K(x,w)/2 with curvatures K(x,w) of compact 2-manifolds M(w). 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 two-dimensional sub-manifolds M(w) of M.

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