Winding Around: The Winding Number in Topology, Geometry, and Analysis

People who teach university-level mathematics for a living often find themselves reading lots of books on the subject. For some of us, this can often be quite pleasant — why else would anybody choose to review books for the MAA? But even for the book-lovers among us, after you’ve just read about ten linear algebra texts, all of which look like they were stamped from the same cookie cutter, the process can occasionally wear thin. It’s very pleasant, then, to stumble across a book that is genuinely unique, that addresses a topic in a way not found elsewhere, and that teaches you something that you didn’t know before. It’s even nicer when the book in question does a really good job of it, as is the case with the book under review.

Winding Around is written by John Roe, author of the excellent Elementary Geometry, a book that I have always thought is not as well known as it deserves to be. This time around, the author again discusses some geometry, but also throws in topology, complex analysis, and abstract algebra. The unifying concept that links all these ideas is that of the winding number.

Prior to looking at this book, I had at least heard of the winding number, but I must confess that it was something of a hazy memory. I first ran across the term in an undergraduate course in complex variables, where it was defined as an integral, proved to be an integer, and invoked in such things as the Cauchy Integral Formula and Cauchy Residue Theorem. Over the years I may have encountered the term again as I flipped through books on complex analysis, but that seems to have been the extent of my dealings with the concept. I certainly did not realize, until seeing this book, that the idea of the winding number could be applied profitably in other areas of mathematics.

This book starts, after some preliminary motivational material, with a chapter discussing homotopic maps. This is in preparation for a chapter that gives a topological definition of the winding number and a selection of methods for computing it. The concept of winding number is then used in the next chapter to establish some topological results in the plane (Brouwer fixed point theorem, Borsuk-Ulam theorem, “ham sandwich” theorem, Jordan curve theorem).

Chapter 5 begins with a discussion of differential 1-forms and uses these ideas to give an integration-based characterization of the winding number. Basic notions of homology are also introduced. Calculus is also used in the next chapter, which introduces some notions from differential geometry: the rotation number of a loop (i.e., the winding number of the tangent vector) is discussed and applied, culminating in a sketch of a proof of the Hopf index theorem.

The winding number is then, in chapter 7, applied to functional analysis, specifically to Fredholm and Toeplitz operators on a Hilbert space. The main result is a theorem linking the index of a Toeplitz operator to a certain winding number.

The final two chapters of the book take us back to algebraic topology. Since the winding number as discussed up to this point is about loops in the metric space of complex numbers, a natural generalization is to loops in arbitrary metric spaces, which in turn leads naturally to the fundamental group, the subject of chapter 8. This chapter also introduces covering spaces and group actions, and culminates in a statement and proof of an algebraic result due to Hopf, namely that the only finite-dimensional commutative real division algebras with identity are the fields of real and complex numbers.

Chapter 9, the final chapter of the book (described as a “coda” by the author) is essentially an explication, following a paper of Atiyah and given here mostly without proofs, of the Bott periodicity theorem, which characterizes the higher homotopy groups of the general linear group. This chapter brings together several ideas developed earlier, including Toeplitz operators.

All of this is accomplished in a surprisingly short amount of space. The text is only about 260 relatively small pages long (all the books in the AMS Student Mathematical Library series are quite compact); of these 260 pages, a full 80 are devoted to appendices, each one reviewing an area of mathematics that gets used in the book: linear algebra, metric spaces, the Stone-Weierstrass and Tietz extension theorems from topology, measure zero subspaces of the line and (briefly) the circle, calculus on normed vector spaces, Hilbert spaces, groups, and graphs. This leaves only about 160 pages to go through the material described in the preceding paragraph — material that is, obviously, sophisticated and not at all easy. I have to confess that, when I first looked at the table of contents of this book, I wondered if the author would be able to pull this off, but I do think that he has: a student who is prepared to put in some serious time and effort with these ideas will find that the author has more than met him or her halfway.

Roe’s writing style is succinct, but clear and quite elegant; I could practically hear a British accent as I read the book. This clarity of writing and the numerous appendices help make the book accessible. Moreover, the book is written so that absolute mastery of the material is neither required nor, I suspect, even expected; the author is trying to provide a sense of what’s going on, not make the readers into experts. (As he puts it in the preface, people who encounter unfamiliar concepts “can decide whether to read the appendix for a quick refresher, or to continue with the main text and hope for the best.”)

I think, therefore, that a talented, well prepared senior undergraduate mathematics major should be able to get a reasonable amount out of this text, a fact which, in conjunction with the interesting array of mathematics used in it, suggests that it would be an interesting choice for a senior seminar or capstone course. Indeed, this book has its origins, we are told, in a course given by the author as part of the Penn State MASS (Mathematics Advanced Study Seminars) program. Consistent with its use as a text, there are exercises at the end of each chapter (most of them, it seemed to me, reasonably substantial, and none with solutions).

I envy the students the experience of taking such a course. However, this book has non-textbook uses as well: as I noted in the first paragraph of this review, it would also make nice reading for more experienced people who perhaps specialize in other areas but who are interested in seeing a lot of very pretty mathematics tied together in a very adroit manner.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.