Grassmannians and pseudosphere arrangements

Michael Gene Dobbins

doi:10.5802/jep.171

Abstract

Grassmannians and pseudosphere arrangements

Highlights

If we record all the possible ways a given vector configuration or affine point set can be partitioned by a hyperplane, the resulting combinatorial data will be an oriented matroid [7]
Oriented matroids are more general objects, since not all oriented matroids arise from a vector configuration
The topological representation theorem says that every oriented matroid can be realized by a pseudosphere arrangement [13]

Summary

Introduction

If we record all the possible ways a given vector configuration or affine point set can be partitioned by a hyperplane, the resulting combinatorial data will be an oriented matroid [7]. This uses an explicit strong deformation retraction from the space of weighted pseudocircle arrangements to the space of weighted great circle arrangements. Part of this work was done at the 2019 IBS summer research program on Algorithms and Complexity in Discrete Structures, hosted by the IBS discrete mathematics group

Main definitions and theorems

Main theorems

Pseudolinear realization spaces

Deforming weighted pseudocircle arrangements

Universal vector bundles and classifying spaces