On Arrangements of Pseudohyperplanes

TL;DR

This paper extends the theory of hyperplane arrangements to pseudohyperplanes, constructing a space analogous to the complexified complement and demonstrating its homotopy equivalence to a cell complex based on oriented matroids.

Contribution

It generalizes the complexification process and the Salvetti theorem to non-realizable oriented matroids via pseudohyperplane arrangements.

Findings

Constructed a space homeomorphic to the complexified complement for pseudohyperplane arrangements.

Proved the space has the homotopy type of a cell complex derived from the oriented matroid.

Extended classical results to non-realizable cases using topological and combinatorial methods.

Abstract

To every realizable oriented matroid there corresponds an arrangement of real hyperplanes. The homeomorphism type of the complexified complement of such an arrangement is completely determined by the oriented matroid. In this paper we study arrangements of pseudohyperplanes; they correspond to non-realizable oriented matroids. These arrangements arise as a consequence of the Folkman-Lawrence topological representation theorem. We propose a generalization of the complexification process in this context. In particular we construct a space naturally associated with these pseudo-arrangements which is homeomorphic to the complexified complement in the realizable case. Further we generalize the classical theorem of Salvetti and show that this space has the homotopy type of a cell complex defined in terms of the oriented matroid.