5 - Topological Models for Oriented Matroids

Summary

It was described already in Chapter 1 how realizable oriented matroids correspond to arrangements of hyperplanes in R^d. The content of the Topological Representation Theorem is that general oriented matroids similarly correspond to arrangements of generalized hyperplanes, each obtained from a flat hyperplane by tame topological deformation. This important result, the proof of which was begun already in the preceding chapter, is the main concern of this chapter. In the last section it is discussed to what extent points rather than hyperplanes can be used to represent the elements of an oriented matroid in a topologically deformed arrangement. It turns out that in the non-realizable case there are severe limitations to such an analogue of projective polarity.

Arrangements of pseudospheres

A linear oriented matroid can be described as an arrangement A of oriented hyperplanes through the origin in R^d+1. This information is equivalently represented by the arrangement A′ = {H ∩ S^d : H ε A} of linear (d – l)-subspheres in S^d together with a choice of orientation (positive and negative hemisphere) for each H ∩ S^d. If these subspheres are topologically deformed in a tame way that preserves their intersection pattern, the arrangement will still represent an oriented matroid. This idea leads to “arrangements of pseudospheres”, which will be discussed in this section.

We start with a discussion of codimension 1 subspheres of S^d. If S is a (d–1)-sphere embedded in S^d (i.e., S ⊆ S^d and S is homeomorphic to S^d-1), then by the Jordan-Brouwer separation theorem S^d\S consists of two connected components such that S is the boundary of each.