Let G be a connected graph that is 2-cell embedded in a surface S, and let G* be its topological dual graph. We will define and discuss several matroids whose element set is E(G), for S homeomorphic to the plane, projective plane, or torus. We will also state and prove old and new results of the type that the dual matroid of G is the matroid of the topological dual G*.
(2002). Matroid Duality from Topological Duality in Surfaces of Nonnegative Euler Characteristic. Combinatorics Probability & Computing, 11 (5), 515-528.