Oriented Matroids

Introduction to oriented matroids emphasizing connections to topology and geometry, suitable for graduate courses or independent study.

Laura Anderson (Author)

9781009494113, Cambridge University Press

Hardback, published 10 April 2025

333 pages
23.5 x 16.1 x 2.4 cm, 0.63 kg

'This book is a friendly invitation to the subject of oriented matroids, providing a geometrically inclined introduction and bridging the gap to the more demanding and encyclopaedic Red Book (Björner et al., 1999). It is not always easy to reconcile geometry with combinatorics and this textbook is certainly on the right path. For illustration, 'combinatorial Farkas Property' ('Farkas' is mentioned in the book 72 times!) offers a unified point of view throughout the book and helps the reader understand the rationale behind more technically involved parts of the subject. Both students and teachers will find a selection of relatively new developments in the book (which haven't been covered in a textbook before!), including a glimpse into the general theory of matroids over hyperfield.' Rade Živaljević, Mathematical Institute SASA, Belgrade

Oriented matroids appear throughout discrete geometry, with applications in algebra, topology, physics, and data analysis. This introduction to oriented matroids is intended for graduate students, scientists wanting to apply oriented matroids, and researchers in pure mathematics. The presentation is geometrically motivated and largely self-contained, and no knowledge of matroid theory is assumed. Beginning with geometric motivation grounded in linear algebra, the first chapters prove the major cryptomorphisms and the Topological Representation Theorem. From there the book uses basic topology to go directly from geometric intuition to rigorous discussion, avoiding the need for wider background knowledge. Topics include strong and weak maps, localizations and extensions, the Euclidean property and non-Euclidean properties, the Universality Theorem, convex polytopes, and triangulations. Themes that run throughout include the interplay between combinatorics, geometry, and topology, and the idea of oriented matroids as analogs to vector spaces over the real numbers and how this analogy plays out topologically.

1. Realizable oriented matroids
2. Oriented matroids
3. Elementary operations and properties
4. The topological representation theorem
5. Strong maps and weak maps
6. Single-element extensions
7. The universality theorem
8. Oriented matroid polytopes
9. Subdivisions and triangulations
10. Spaces of oriented matroids
11. Hints on selected exercises
References
Index.

Subject Areas: Discrete mathematics [PBD]