A generalized Euler-Poincare Equation

Abstract: "The Euler-Poincaré equation, v - e + f = 2(s - g), relates the numbers of topological elements of 2-manifold surfaces. Here v, e, f, s, and g refer to the numbers of vertices, edges, faces, shells (surfaces) and handles. However, the equation does not correctly relate the elements of non-manifold surfaces, and specifically the boundaries of r-sets. We introduce a generalized form of the Euler-Poincaréequation v ́- e ́+ f = 2(s ́- g) which relates the number of vertex uses, edge uses, faces, shell uses, and handles of both 2-manifold surfaces and non-manifold surfaces of r-sets.