From Geometry to Topology

The book under review, an unaltered reprint of a text first published in 1974, begins with a couple of chapters on Euclidean and projective geometry, segues into nine chapters on low-dimensional “surface-type” topology, and then ends with five chapters introducing metric and topological spaces. The exposition throughout is (by deliberate intent) intuitive and picture-based rather than rigorous, so the book is probably not suitable for upper-level mathematics courses in topology or geometry, but might find some use as supplemental reading in such courses or as a text for lower-level students in less rigorous courses. Students wanting to get a taste of what these subjects are about without getting bogged down in the details might also find this book useful.

In more detail: the first two chapters discuss geometry (Euclidean and then projective) from the standpoint of congruence, using this as a springboard for discussing transformations and their invariants. So, this is really an introduction to the ideas behind Klein’s Erlangen Programme, although that term is not used in these chapters (but is mentioned in a three-page Historical Note at the end of the book). Unfortunately, not much is actually said about the underlying geometry; it wasn’t even made clear, for example, that in projective geometry there are no parallel lines. It seems that the primary purpose of these chapters was to get the reader thinking in terms of transformations, so as to motivate the introduction to topology that begins in the next chapter, where the class of transformations now includes things like “stretching, bending and twisting” but not “cutting”.

Chapter 3 introduces these ideas, and then chapters 4 through 12 give a very informal introduction, with lots of pictures, to some of the ideas associated with low-dimensional topology, including classification of surfaces, the Euler characteristic, connections with graphs, planarity and map coloring and the Jordan curve theorem. The Brouwer fixed point theorem is also mentioned.

The chapter on coloring is, of course, now out of date, because the Four Color Theorem was proved several years after the publication of the original text and the Dover edition has not been updated to account for that. I was a little surprised, in fact, that the original text did not make more out of this then-unsolved question; there is a brief statement that “it has never been proved that the existence of a planar map requiring five colours is an impossibility”, but that’s about it; there is no really extended discussion of the history of this very important problem or the excitement it had engendered in the mathematical community.

This quibble aside, I found these chapters to be informative and nicely written; a student working his way through them would be rewarded with at least an intuitive understanding of the rudiments of basic surface theory. Since this material is not often covered in many undergraduate courses on topology that are based primarily on metric and topological spaces, these chapters could, as I indicated earlier, perhaps serve as supplemental reading for such a course.

The remaining chapters in the book take a first look at metric and topological spaces, after some prefatory work on basic set theory and functions. These chapters, I thought, were the least successful in the book. Take, for example, the final chapter, on topological spaces. It consists of a series of definitions, some examples and pictures, and the statements of some theorems, but does not provide much indication of just why these ideas are important. So, for example, after reading this chapter, the reader will know the technical definition of a compact topological space, but will not really know why anybody cares about them; surely, for example, the author could at least have referred to the familiar theorem from calculus that a continuous function defined on a closed and bounded interval is bounded, given a simple example of how this fails when the interval fails to be closed or bounded, and then explained that compactness was the real reason why this phenomenon occurred.

Another concern: The author mentions that the compact subsets of the real numbers are the closed and bounded subsets, but he does so in a single sentence, in the middle of a paragraph, without using an introductory phrase like “It can be shown that…”; students who have been told by their instructors to justify all steps in a math book as they read it might well think that they are supposed to be able to see why this is true, and that’s obviously unreasonable. Other examples like this pop up during the text as well.

There are no exercises in the main body of the text, but at the end of the book there is a collection of 40 of them, very few of which call for proofs. No solutions are provided.

Summary and conclusion: the best part of the book is the middle section on surfaces and graphs; that section might provide a beginning student with a good overview of these ideas in preparation for a more rigorous look at them.

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