MAPS AND A-MATROIDS"

Many concepts in matroid theory are invariant by duality: connectivity, representability, base orderability. The dual of a planar matroid (i.e. the cycle matroid of a planar graph) also is a planar matroid, and this is essentially on account of the topological duality in the sphere. This cannot be extended to arbitrary imbeddings because if we consider the cycle matroids, say A4 and N, of two dual graphs, G and G* respectively, imbedded in a compact surface S with Euler characteristic x, we have