A discussion of realization spaces, including an example of an oriented matroid with disconnected extension space, is provided. In the later part, a proof of the Universality Theorem and a discussion of some of its consequences follows.

The chapters addresses the various axiomatizations and the equivalences between them and presents an introduction to the Plucker relations. The chapter finishes with some discussion of nonrealizable oriented matroids and the impossibility of an excluded minor characterization.

The geometric motivation for the theory is combinatorial data associated with matrices, vector arrangements, hyperplane arrangements, and subspaces of real vector spaces. Interpretations of this data are given in terms of linear algebra, discrete geometry, and the Plucker embedding of the Grassmannian. Elementary proofs of cryptomorphisms for realizable oriented matroids are provided. The chapter finishes with an application of Gale Diagrams.

The chapters provides a survey on the topology of various posets of oriented matroids analogous to various topological spaces, including extension spaces, combinatorial Grassmannians, and combinatorial flag spaces. A general framework for interpreting maps from spaces to posets is laid down, by way of McCord’s Theorem and the Semi-algebraic Triangulation Theorem. The chapter includes a discussion of the (now-disproved) extension space conjecture and of the various results on the topology of the MacPhersonian.

Introduction to strong and weak maps, including issues around realizability and topological representations, is the topic of this chapter.

Introduction to the oriented matroid abstraction of convex polytopes is the goal of this chapter. The chapter shows that the existence of a polar of an oriented matroid polytope is closely related to the existence of an adjoint. The Lawrence construction is introduced and used to produce interesting examples.

Some basic properties and operations, including poset structure of covector sets and flats, are provided in this chapter.

The chapter begins with localizations, including a topological proof of Las Vergnas’s characterization of localizations. Adjoints and their relationship to extensions are discussed. The final part of the chapter discusses intersection properties, particularly on the Euclidean property and non-Euclidean oriented matroids.

Several analogs to fans and triangulations of point configurations are introduced and motivated as representability issues. The equivalence of some of these analogs is established, while others remain open. Results on the topology of triangulations are proved, most notably for Euclidean oriented matroids.

A proof of the Topological Representation Theorem, including an introduction to shelling, topological interpretation of oriented matroid concepts, and an application to counting topes, are provided in this chapter.

The automorphism group is one of the most natural groups that acts on polytopes, since it captures its level of symmetry. The connection group is another group that acts naturally on the flags of a polytope; it can be interpreted as a recipe to recover the structure of the polytope from its flags. The chirality index is a measure of how far a chiral polytope is from being regular, and it is linked to a group called the chirality group. This chapter addresses these three groups and their interactions.