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Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry

Jiří Matoušek
Publication Date: 
Number of Pages: 
[Reviewed by
Mihaela Poplicher
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This book, which appears in Springer's Universitext series, is based on a couple of graduate courses in "topological combinatorics" taught by the author in Prague and Zurich. It is intended for readers with some mathematical knowledge beyond undergraduate studies, but it does not assume much knowledge of algebraic topology. However, the author notes that this book is "no substitute for proper foundations" in algebraic topology.

The book is concentrated on topological tools of one type: the Borsuk-Ulam theorem and similar results. The material included here is intended as a part of an extensive project: a textbook on "topological combinatorics", with Jiří Matoušek and two other mathematicians as co-authors.

Although the author claims that knowledge of algebraic topology is not required as a prerequisite for this text, I think a reader should use (at least in parallel) books/notes on algebraic topology. However, the text itself has a first chapter titled "Simplicial Complexes" where some concepts from algebraic topology are introduced: topological space, homeomorphism, compact space, geometric simplicial complex, triangulation, simplicial map, barycentric subdivision... just to name a few. Even in this chapter, that covers topics from algebraic topology courses, some less commonly treated results are included. Also, some terminology and the notations are sometimes different.

The second chapter in the book, "The Borsuk-Ulam Theorem" includes the theorem in several equivalent formulations, several proofs, as well as proofs that the different statements of the theorem are all equivalent. More emphasis is placed on a combinatorial statement, Tucker's Lemma, from which Borsuk-Ulam theorem is derived. As expected, the author introduces Borsuk-Ulam theorem very enthusiastically, giving four reasons why this is such a great theorem:

  1. several different equivalent versions,
  2. many different proofs,
  3. a host of extensions and generalizations, and
  4. numerous interesting applications.

A popular and easy to remember interpretation of Borsuk-Ulam's theorem for n = 2 states that "at any given time there are two antipodal places on Earth that have the same temperature and, at the same time, identical air pressure."

Chapter 3 includes "Direct Applications of Borsuk-Ulam", among which "the Ham Sandwich Theorem" and "On Multicolored Partitions and Necklaces" have certainly fun names. They also have informal statements that are very easy to remember. For instance, the ham sandwich theorem says that "for every sandwich made of ham, cheese, and bread, there is a planar cut that simultaneously halves the ham, the cheese, and the bread." Easy to remember... but the rigorous versions are not so easy to prove.

"The necklace's theorem" also has a memorable statement: "every (open) necklace with d kinds of stones can be divided between two thieves using no more than d cuts." Other applications are: Lovasz-Kneser's Theorem, Dol'nikov's Theorem, Gale's Lemma, and Schrijver's Theorem.

Chapter 4, titled "A Topological Interlude", includes some basic topological concepts (quotient spaces, Cartesian product), and some other concepts needed in subsequent developments: join of simplicial complexes, k-connected space, nerve theorem, CW-complex, cellular map. None of these is very easy, but all are explained very well.

Chapter 5 "Maps and Nonembeddability" and Chapter 6 "Multiple Points of Coincidence" give generalizations of Borsuk-Ulam theorem from spheres to a much wider class of spaces. The material becomes more challenging, but the exposition is very clear and a pleasure to read and study.

The book contains more than 100 exercises, many of them being compressed outlines of interesting results. As such, most of them are both challenging and interesting.

I think this book will be of interest mostly for mathematicians and graduate students. It can certainly be used in a graduate course or seminar. It has an extensive list of references and it covers many interesting and not-so-easy results, with proofs that are sometimes easier than in the original papers. I wouldn't say that this book is an "easy read", but it is very well written, very interesting, and very informative.

Mihaela Poplicher is an assistant professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is