“What’s it good for?” We have heard this question from disgruntled students countless times. In part to combat the attitude that mathematics is a theoretical discipline studied only for its own sake (not that there is anything wrong with that!), textbook authors now include real-world applications along with the usual abstract theory, then append the term “with applications” to the title. Even traditionally “pure” subjects such as abstract algebra have jumped on this bandwagon. With the publication of William Basener’s Topology and its Applications, topology is the latest theoretical subject to get the treatment.
First and foremost, this is a topology textbook. It covers point set topology, topological and smooth manifolds, simplicial complexes, topological groups, fixed point theorems, vector fields, the fundamental group, and homology. This may seem like an ambitious list of topics for a text just over 300 pages, but the author strategically chooses to sacrifice depth for breadth, and does not delve too deeply into any of these topics. For example, in the chapters on algebraic topology he discusses the fundamental group and simplicial and singular homology, but does not mention exact sequences, torsion, homology with arbitrary coefficients, cohomology, duality, or functors.
The applications are numerous and are spread throughout the book. They come in three varieties — topological applications (e.g., the ham sandwich theorem, the hairy ball theorem, Cantor sets), applications to other areas of mathematics (e.g., number theory, dynamical systems, geometry, differential equations, algebra), and real-world applications (e.g., population dynamics, robotics, cosmology, computer drawing software, economic game theory, computer graphics, condensed matter physics, computational algorithms). Overall the choice of applications is good — they are interesting and show a real use of the topological ideas. Most of the applications are presented as snapshots of how topology could be applied, not as an exhaustive study. Thus the text could be used a springboard to further investigation. I discovered one or two ideas that I will stash away as future research projects for talented mathematics majors. With this purpose in mind, it would have been nice to have a more extensive bibliography with suggested readings.
Perhaps most useful to the budding mathematician are the applications of topology to other areas of mathematics. Mathematicians know that topology is ubiquitous in modern mathematics, but most students have to learn this the hard way — seeing a little topology here, a little more there. This textbook does an excellent job of showing how topology pervades the other mathematical disciplines.
My main complaint about the book involves the physical presentation of the material. I was repeatedly frustrated trying to find things in the text. The bare-bones index was of little use and the internal numbering scheme made it difficult to refer back to items (for instance, Chapter 5 begins with Lemma 31, Core Intuition 13, Lemmas 32 and 33, Definition 64, Theorem 61, and Proposition 25). The artwork in the book is very inconsistent. The hand-drawn line art is often wiggly and amateurish (of course, it was the artistically-challenged Poincaré who called topology “the art of reasoning well on badly made figures.”). Color images that were taken from outside sources or generated by software were poorly reproduced in low-resolution grayscale containing distracting horizontal lines across the images. All of the images in the appendix on knot theory are missing their right half.
The material in Topology and its Applications should be accessible to any mathematics graduate student or to a strong undergraduate. Because of the emphasis on breadth over depth, it would be an excellent text for a student who wants to learn some topology but not to be a topologist. This book would not be a substitute for one of the traditional textbooks on algebraic topology such as Munkres, Hatcher, Massey, Bredon, etc., but it could be used to supplement those. On the other hand, one could argue that Topology and its Applications contains the topology we would like any non-topologist, PhD mathematician to know.
This book is a celebration of topology and its many applications. I enjoyed reading it and believe that it would be an interesting textbook from which to learn. The theory is presented well and the applications are varied, interesting, and not contrived. With a book such as this one, one can always point to topics or applications that the author could have included (differential topology, knot theory, applications to biology, etc.), but that is just personal preference. Basener does an excellent job of surveying the field of topology and arguing for its usefulness. After reading this textbook, a student will know “what topology is good for.”
Dave Richeson is an Associate Professor of Mathematics at Dickinson College in Carlisle, PA. His interests include dynamical systems, topology, and the history of mathematics.