Oriented Rank Three Matroids and Projective Planes

Abstract

Recently, Goodman et al. have proven two conjectures by Grünbaum right, showing that any arrangement of pseudolines in the plane can be embedded into a flat projective plane and that there exists a universal topological projective plane in which every arrangement of pseudolines is stretchable. By Folkman and Lawrence’s theorem , this plane contains every finite (simple) oriented rank three matroid. In this paper, we will also consider embeddings of oriented rank three matroids into topological projective planes, but we will take a quite different viewpoint: we shall show that there exists a projective plane Π that contains the combinatorial geometry of every finite, orientable rank three matroidMn , such that any choice of orientations χnof theMn , n∈N, extends to an orientation χ ofΠ . Furthermore, these orientations correspond to archimedean orderings of Π, hence the reorientation classes of every finite rank three matroid can be studied by the set of archimedian orderings of Π. Since, by a celebrated result of Prieß-Crampe , any archimedian projective plane can be completed and thus embedded into a flat projective plane, our results yield another proof of Grünbaum’s conjectures and a new proof of the rank three case of Folkman and Lawrence’s theorem.