Matroid Duality From Topological Duality In Surfaces Of Nonnegative Euler Characteristic

One of the most basic examples of matroid duality is the following. Let G be a graph imbedded in the plane and let G∗ be its topological dual graph. If M(G) is the cycle matroid of G, then the dual matroid M∗(G) = M(G∗). If G is a connected graph that is 2-cell imbedded in a surface of demigenus d > 0 (the demigenus is equal to 2 minus the euler characteristic of the surface), then M∗(G) 6= M(G∗) for the simple reason that |V (G)| − |E(G)|+ |F (G)| = 2− d (|V (G)| − 1) + (|V (G∗)| − 1) = |E(G)| − d rk(M(G)) + rk(M(G∗)) = |E(G)| − d 6= |E(G)|