**Memoirs of the American Mathematical Society**

2002;
80 pp;
Softcover

MSC: Primary 52;

Print ISBN: 978-0-8218-2769-7

Product Code: MEMO/156/741

List Price: $56.00

Individual Member Price: $33.60

**Electronic ISBN: 978-1-4704-0334-8
Product Code: MEMO/156/741.E**

List Price: $56.00

Individual Member Price: $33.60

# Triangulations of Oriented Matroids

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*Francisco Santos*

We consider the concept of triangulation of an oriented
matroid. We provide a definition which generalizes the previous ones by
Billera–Munson and by Anderson and which specializes to the usual notion of
triangulation (or simplicial fan) in the realizable case.

Then we study the relation existing between triangulations of an
oriented matroid \(\mathcal{M}\) and extensions of its dual
\(\mathcal{M}^*\), via the so-called lifting triangulations. We
show that this duality behaves particularly well in the class of Lawrence
matroid polytopes. In particular, that the extension space
conjecture for realizable oriented matroids is equivalent to the
restriction to Lawrence polytopes of the Generalized Baues problem for
subdivisions of polytopes.

We finish by showing examples and a characterization
of lifting triangulations.

#### Readership

Graduate students and research mathematicians interested in convex and discrete geometry.

#### Table of Contents

# Table of Contents

## Triangulations of Oriented Matroids

- Contents vii8 free
- Introduction 110 free
- Chapter 1. Preliminaries on Oriented Matroids 716 free
- Chapter 2. Triangulations of Oriented Matroids 1524
- Chapter 3. Duality between Triangulations and Extensions 3140
- Chapter 4. Subdivisions of Lawrence Polytopes 4352
- Chapter 5. Lifting Triangulations 5968
- Bibliography 7988