Topological Representations of Matroids

One of the foundations of oriented matroid theory is the topological representation theorem of Folkman and Lawrence [8]. It says that an oriented (simple) matroid can be realized uniquely as an arrangement of pseudospheres. That there is no similar interpretation for the class of all matroids has been taken for granted. For instance, “A non-coordinatizable matroid of abstract origin may be thought of as a geometric object only in a purely formal way, whereas an oriented matroid may always be thought of as a geometric-topological configuration on the d-sphere (or in projective space)” [3, p. 19]. Our main theorem is that the class of geometric lattices, which is cryptomorphic to the category of simple matroids, is the same as the class of intersection lattices of arrangements of homotopy spheres. The interpretation of a geometric lattice as an arrangement of homotopy spheres is a natural generalization of the Folkman-Lawrence theorem. An oriented matroid realizable over R has a representation with geodesic spheres. Allowing pseudospheres, i.e., those which are homeomorphic, but possibly not isometric to the unit sphere, leads to the category of (simple) oriented matroids. If we further relax the conditions on the spheres to only homotopy equivalence to the standard sphere, then we are led to the category of all (simple) matroids. Some of the theory of oriented matroids which only depends on the underlying matroid can be extended to homotopy sphere arrangements. Zaslavsky’s enumerative theory for pseudosphere arrangements can be extended to the homotopy setting. As in the oriented matroid representation theorem, the arrangement can be forced to be antipodal, so a realization as homotopy projective spaces is also possible. The minimal cellular resolutions of orientable matroid ideals developed in [10] can be extended to arbitrary matroids by using arrangements of homotopy spheres. Our point of view is primarily through the lens of oriented matroids. Hence the homotopy spheres which represent the atoms of the geometric lattice have codimension one. In the future we hope to examine the point of view of complex hyperplane arrangements and consider spheres of even codimension. The matroid theory we require is in section 2, while section 3 presents matroid Steiner complexes. We review a very general theory of arrangements of topological