Manchester Geometry Seminar 1999/2000

Classical matroids were introduced in the 1930's by Whitney, and axiomatise the combinatorial properties of vector configurations in finite-dimensional vector spaces. A generalisation of the matroid concept has been introduced by Gelfand and Serganova which associates matroid-like structures with every Coxeter group, the classical matroid reappearing as an A_n-matroid.

A well-known theorem of Rado associates with every graph a vector configuration, and thus a classical matroid. Similarly, the main result of this talk associates each map on an orientable, 2-dimensional surface with a D_n-matroid. The proof is cohomological in nature.

We go on to consider the concept of orientations of vector configurations and classical matroids, and then to discuss concepts of orientation of certain (Lagrangian) B_n- and D_n-matroids.