Contents
Introduction 1
Chapter 1. Preliminaries on Oriented Matroids 7
1.1. Convexity. 7
1.2. Extensions. Lexicographic extensions. 9
1.3. Euclideanness. 12
Chapter 2. Triangulations of Oriented Matroids 15
2.1. Definition, characterizations and remarks 15
2.2. Equivalence of the different characterizations 18
2.3. Some properties of triangulations 23
2.4. Topology of triangulations 26
Chapter 3. Duality between Triangulations and Extensions 31
3.1. Circuit, cocircuit, extension and triangulation vectors 31
3.2. The affine span of characteristic vectors of triangulations 33
3.3. Mutations versus geometric bistellar flips 37
Chapter 4. Subdivisions of Lawrence Polytopes 43
4.1. Lifting subdivisions. Subdivisions 43
4.2. Lawrence polytopes only have lifting subdivisions 49
4.3. The extension space conjecture and the Baues problem 53
4.4. A reoriented Lawrence construction 55
Chapter 5. Lifting Triangulations 59
5.1. Some properties. Lifting versus regular triangulations. 59
5.2. Three interesting non-lifting triangulations 64
5.3. Two characterizations of lifting subdivisions. 72
Bibliography 79
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