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Explorations in Topology: Map Coloring, Surfaces and Knots

David A. Gay
Publication Date: 
Number of Pages: 
Elsevier Insights
[Reviewed by
Mark Hunacek
, on

Opinions differ as to what the content of an introductory undergraduate course in topology should be. Some people believe that a student’s first look at topology should follow the standard “point set” path of metric spaces, topological spaces, connectedness, compactness and separation properties; see, e.g., A Course in Point Set Topology by Conway. Other people think that a course like this should expose the students to the basic structures of low-dimensional topology such as surfaces and knots. This book is intended for people of this persuasion, and who like the “discovery learning” approach to mathematics.

This is the second edition of a book first published in 2007, but unfortunately I couldn’t find in this text any specification of what changes were made from one edition to the next. This edition is actually a little shorter than the first, which is unusual, but in a quick comparison with the first edition I didn’t notice any particularly striking deletions from the second. I did notice that the table of contents has been changed a bit, though not in any major way, and some of the end-of-chapter material has been revised.

The basic topics covered in both editions, however, seem to be the same. There are 14 chapters. The first five talk about map-coloring and graphs: the question of coloring a map is presented, and it is soon seen that this problem can be converted to one involving graphs, which are themselves introduced via the famous Bridges of Königsberg problem. Subsequent chapters address map invariants (the number of countries, vertices and edges) and relations among them; this is of course reminiscent of Euler’s formula for polyhedra, which is also discussed. Chapter five looks at the first “proof” of the four-color problem, given by Kempe in 1879 and found by Heawood to contain a flaw in 1890.

From here it is natural to talk about maps and graphs on surfaces other than the plane, so chapters 6–8 introduce the torus, Moebius strip, Klein bottle, and projective plane (called here a “cross-cap”). It is shown in these chapters that surfaces can be created by “patterns” (i.e., a more intuitive name for quotient spaces).

The next topic covered, over the course of chapters 9–11, is the classification of surfaces, including a discussion of the Euler number. Chapter 12 then ties things up by looking at the coloring problem for maps on other surfaces. Finally, chapter 13 introduces the concept of a knot and begins to discuss the idea of knot invariants.

There is also a 14th chapter, but this is of a different nature than the 13 preceding it; it is devoted entirely to a discussion of proposed projects for further study. About fifty of them are listed, along with suggested references.

Although the choice of topics presented here is relatively standard, the method of presentation is not. As briefly noted above, the author uses the “discovery approach” throughout, asking the students to work on problems and try and discover solutions on their own. (He used the same approach in a previous book, Geometry by Discovery.) Each chapter poses a question, offers some approaches (some of which may not pan out), and then, through a series of subsections with names like “Your Turn”, “Investigation”, ”Question” and “Puzzle”, attempts to guide the students to do further investigation and problem-solving on their own.

The Preface references an Instructor’s Manual, offering not only solutions and hints to some of the problems but also a description of how the author uses the book in a class. Unfortunately, I was advised by a publisher’s representative that, at least as of this writing, no such Manual exists. Another omission that struck me as quite striking was the total lack of an Index.

Because of the discovery approach adopted by this book, any instructor using it must be one who is enthusiastic about this approach, and who has the kind of specialized skill to make it work in a classroom; being able to deliver a clear, interesting lecture, for example, does not necessarily translate into the ability to tease mathematics out of students in the way this book is structured to do.

I have never taught any discovery-based courses, partly because the traditional lecture approach seems more natural to me and partly because I have some reservations about the method in general. I think it is entirely reasonable to believe that information is retained in a better, deeper, way if the student discovers it on his or her own, so to that extent I think the method has something to recommend it. But at the same time I think that there is a trade off in the amount of information that can be presented in a given time period, and so on balance the benefits of this approach may be outweighed by the detriments. This is, of course, a matter of individual taste, and there are certainly quite a few people who think very highly of this method.

The book does not consist entirely of problems and projects for the student, however. Each chapter ends with a section marked “Notes”, typically about two pages long, which gives a somewhat broader perspective of the material covered in that chapter, typically placing each topic in historical context, and sometimes giving precise definitions and statements of theorems. In the chapter on Moebius strips, for example, the author discusses their history, gives some indication of how they appear in art and literature, and briefly looks at their applications in chemistry. It is in these sections that the book comes closest to being a “typical” text.

Another unusual feature of the book is the way in which these mathematical problems are posed. The author does so by means of conversations among the employees of an imaginary company called Acme Maps. These conversations are sometimes quite meandering and are written in a deliberately colloquial manner, using phrases like “whaddaya mean”, “maybe I kin hep ya” and “I sure daggum am”. They are written as though they were scripts of plays, complete with stage direction:

Tired high fives all around. A little bit of ‘allemand-left’ and ‘do-si-do-ing’ music comes up as the gang slowly dances about. Lights fade.

Cutesy stuff like this, it seems to me, can get old pretty quickly in an upper-level mathematics textbook; I occasionally found the conversations to be somewhat forced and tedious, and as I read them I frequently found myself thinking “yes, yes, get on with it”. And reading phrases like the ones quoted above was, to me, the textual equivalent of squeaky chalk on a blackboard. I have no idea what particular benefit the author sees in having characters talk like this, and I also think that some students might find these conversations to be almost insultingly condescending.

Perhaps some people think that this is the best way to reach students, but this is not a viewpoint that I share, or care to indulge as a teacher. My view is that students should be taught that there is a difference between high school and college, and that in the latter, they are expected to rise to the level of the mathematics rather than the other way around. I have found that students, particularly in upper-division courses (like topology!) are, for the most part, willing and able to rise to the occasion without the necessity of my having to stage imaginary plays or talk in deliberately poor English.

Based on the preceding, this is not a book that I would use in a course or recommend to a student, particularly since this material is already nicely addressed elsewhere, such as the first five chapters of Introduction to Topology and Geometry by Stahl and Stenson. However, tastes vary. An instructor of this material who likes the discovery approach and is not as bothered as I am by mathematics texts with phrases like “maybe I kin hep ya” in it, may want to give this book a look.

Mark Hunacek ( teaches mathematics at Iowa State University. 

CHAPTER 1: ACME makes maps and considers coloring them
CHAPTER 2: ACME adds tours to its services
CHAPTER 3: ACME collects data from maps
CHAPTER 4: ACME gathers more data, proves a theorem, and returns to coloring maps
CHAPTER 5: ACME’s lawyer proves the four color conjecture
CHAPTER 6: ACME adds doughnuts to its repertoire
CHAPTER 7: ACME considers the Möbius strip
CHAPTER 8: ACME creates new worlds --- Klein bottle and other surfaces
CHAPTER 9: ACME makes order out of chaos --- surface sum and Euler numbers
CHAPTER 10: ACME classifies surfaces
CHAPTER 11: ACME encounters the fourth dimension
CHAPTER 12: ACME colors maps on surfaces --- Heawood’s estimate
CHAPTER 13: ACME gets all tied up with knots
CHAPTER 14: Where to go from here --- Projects