At first glance, algebraic topology bears little resemblance to point-set topology. Poincaré, Hopf, Lefschetz, and others created algebraic (or combinatorial) topology in response to certain spatial puzzles that did not fit into the rigid framework of geometry — Euler’s bridges of Königsberg and his polyhedra, Riemann’s surfaces, Lord Kelvin and Tait’s knots, Lhuilier and Betti’s multiply connected spaces, Listing, Möbius, and Klein’s nonorientable surfaces, etc. Separately, Weierstrass and Cantor’s investigations of the real number line led Fréchet, Hausdorff, and others to create point-set topology. Viewed at this superficial level, the “rubber sheet geometry” of algebraic topology appears quite different from the abstract generality of point-set topology. Of course, a certain amount of point-set topology is needed to create a solid foundation for algebraic topology. The problem for those of us who teach a one semester undergraduate course in topology is that this essential layer of point-set topology is so large that it can easily consume an entire semester. Algebraic topology, if mentioned at all, gets a superficial treatment.
Topology Now! is an undergraduate textbook in algebraic topology (using the broadest definition of the term). It contains beautiful topics on 1-, 2-, and 3-manifolds such as polynomial knot invariants, Seifert surfaces, the Euler characteristic, the classification of surfaces, Heegard splittings, Sperner’s lemma, the contraction mapping theorem, the Brouwer fixed point theorem, the fundamental group, and even the Poincaré conjecture. The standard definitions of topology, open set, closed set, metric, compact, connected, interior, closure, boundary, subspace, continuity, and quotient space do not appear until the last 23 pages of the text, and most of the usual theorems of point-set topology are relegated to the exercises (as is the definition of Hausdorff).
By making a few strategic decisions the authors are able to present sophisticated topics in algebraic topology without the usual baggage from point-set topology. For instance, they consider only manifolds that are embedded in some Euclidean space. Since such spaces inherit a metric, the authors are free to use the familiar epsilon-delta definition of continuity. Since every manifold can be triangulated (they state this theorem without proof) they define a compact manifold to be one with a finite triangulation. With these and other clever choices, they are able to steer clear of inverse images of open sets and finite subcovers.
The prose in Topology Now! is very informal and conversational (Superman, Monopoly, and the Count of Monte Cristo all make an appearance). The proof of Euler’s formula is not a proof by induction, but an explanation of a magic trick. Occasionally the authors’ groan-worthy sense of humor creeps in to the text (one exercise is simply, “Homeo, homeo, wherefore art thou homeo?”). On the other hand, they tackle deep and challenging mathematics. They respect the reader and do not dumb-down the content. Initially, the mixture of formidable multidimensional arguments with chatty proofs made it seem that the authors could not decide on their audience, but the more I let the book sink in, the more it appealed to me.
The authors have collected an array of interesting and creative exercises. Some are easy, some surprising, and some thorny. Still other exercises instruct the student to consult a research article or the chapter of another book. The variety is refreshing (although the relevance of a few — such as one about covariant functors and category theory — may be lost on the readers). Each chapter concludes with a very nice annotated bibliography of suggested readings, which would be helpful to a student looking for a research project.
The blurb on the back of the book asserts that it is “accessible to undergraduate students without requiring extensive prerequisites in upper-level mathematics.” While it is true that the book is self-contained, I would hesitate to assign this text to a student who had not seen a semester of abstract algebra, and probably analysis. Moreover, I imagine that this would be an extremely challenging text for any student who does not have the ability to visualize complicated manipulations of figures in R, R2, R3, and R4. While this may be true for a student in any topology course, at least in a traditional topology course the nonvisual learner could excel in the set theoretical parts.
It is an admirable goal to take sophisticated mathematics and rewrite it in an easy-to-follow, yet rigorous, way. For the most part the authors succeed on both accounts. Although Topology Now! would not be a good text for someone teaching a course in point-set topology, it would be an excellent choice for someone wanting to share the beautiful theorems of algebraic topology with talented undergraduate students. The fascinating topics covered in this text are the kind that the student will remember long after the end of the course.
David Richeson is an Assistant Professor of Mathematics at Dickinson College in Carlisle , PA. His research explores the interplay between dynamical systems and topology (both algebraic and point-set!). Recently he has become fascinated by the Euler characteristic and its many applications.