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Mostly Surfaces

American Mathematical Society
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Mostly Surfaces is delightful reading. Schwartz gives a beautiful, careful exposition of some of the most elegant ideas, theorems, and proofs in the theory of surfaces. It’s an ideal book for casual reading in spare mathematical moments.

As Schwartz states in his introduction, the book began life as lecture notes for an undergraduate seminar on surfaces. The book certainly has a seminar-feel to it. The various sections of the book require very different background knowledge, the intent is often to motivate and inspire rather than develop technical proficiency, and there are very few exercises. Most of the expected topics are there: surfaces as topological spaces, the classification of surfaces, fundamental groups, geometrization of surfaces, Riemannian metrics, covering spaces, Riemann surfaces and uniformization, Teichmüller space. Although these ideas have applicability far exceeding the world of 2-dimensional manifolds, they are, arguably, best introduced in that context. Schwartz does an admirable job. Absent from the book are the differential geometry of surfaces in R3 and any mention of the Nielsen-Thurston classification of homeomorphisms of surfaces. Instead, there are beautiful chapters on billiards, surface dissections, and my personal favorite — the Banach-Tarski paradox.

Mostly Surfaces is published by the AMS in their Student Mathematical Library (SML) series. In that series there are two other books that are also mostly surfaces. Katok and Climenhaga’s Lectures on Surfaces is written in a similar style to Schwartz’s book (and also makes for enjoyable casual reading). The authors discuss the visualization of surfaces, homology theory, and the Riemannian geometry of surfaces much more than Schwartz does. In fact, despite both books being aimed at undergraduates and devoted to the topology and geometry of surfaces, there is surprisingly litle overlap. They overlap most in their discussion of equipping surfaces of genus at least 2 with a hyperbolic metric. I prefer Schwartz’s exposition since it is a little more relaxed and includes more pictures.

Bonahon’s Low Dimensional Geometry is the other surfaces book in the SML. It is a very different book. Two of the differences point to a couple of shortcomings in Schwartz’s book (which is, after all, the book I’m supposed to be reviewing). In describing these shortcomings, however, I think of a quotation from G. K. Chesterton:

But to call [a] man’s face ugly because it powerfully expresses [his] soul is like complaining that a cabbage has not two legs. If we did so, the only course for the cabbage would be to point out with severity, but with some show of truth, that we were not a beautiful green all over.

At the risk of failing GKC’s test, let me proceed. The main complaint I have about Schwartz’s book is its lack of coherent progression or organization. This is not as bad as it sounds — Schwartz’s book is one to dip into, not so much one to read cover-to-cover. But the lack of progression makes it difficult to use as the basis for a course with a beginning, middle, and end. Bonahon’s book, on the other hand, is ideal for such a course. It moves the reader along from euclidean planar geometry to hyperbolic 3-manifolds. Its clear goal is an exposition, with as many details as possible for an undergraduate audience, of Thurston’s geometrization conjecture (but not Perelman’s celebrated resolution).

The other shortcoming of Schwartz’s book, one not shared by Bonahon’s, is actually a by-product of the beautiful writing. The elegance of Schwartz’s exposition sometimes masks the difficulties of actually proving theorems. For example, both Bonahon and Schwartz show that the real linear fractional transformations

$$z \mapsto z+1, \qquad z \mapsto \lambda z, \qquad z \mapsto -1/z$$

(for complex numbers z and positive reals λ) are isometries of the hyperbolic plane. Bonahon takes the next step and shows that every isometry of the hyperbolic plane is the composition of these basic isometries, but Schwartz does not. This example is typical: for the material that they both share, Bonahon does not hesitate to give as complete a picture as possible, whereas Schwartz picks only the most beautiful aspects. This, of course, is part of what makes Schwartz’s book pleasurable reading, but it does diminish its usefulness as a textbook.

Recently I had the pleasure of teaching a geometry course for upper-level undergraduates. It was important to me that the course have a clear goal, that it inspire students with geometrical ideas, and that it also develop at least a certain level of technical and proof-writing proficiency. Bonahon’s and Schwartz’s books are inexpensive enough that I was able to assign them both. Bonahon’s text provided the outline of the course, a number of homework problems, and detailed proofs of important results; Schwartz’s book provided motivation and the beautiful expositions of the central ideas. Whenever possible, I had my students read both Bonahon’s and Schwartz’s accounts of the same material. (What did I, the professor, do? Well, you’ll have to ask my students.) As the semester progressed, I noticed an interesting transition in my students’ attitudes towards their texts. At the beginning of the semester, they loved Schwartz’s book and dreaded Bonahon’s, but as the semester progressed they liked Schwartz’s book less and Bonahon’s book more. By the end of the semester, Bonahon’s book was the clear favorite. The reason for this, I’m convinced, was that as they tried to actually understand and create proofs, they needed the detail that Bonahon provided. But I’m equally convinced that without Schwartz’s book to serve as their Virgilian guide to Bonahon’s Mt. Purgatorio, they would have rolled down the mountain and ended up back in the inferno of geometrical ignorance.

Scott Taylor is an assistant professor at Colby College. He constantly feels like he’s only scratching the surface.

Date Received: 
Monday, June 27, 2011
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Richard Evan Schwartz
Student Mathematical Library 60
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Scott Taylor
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  • Book overview
  • Definition of a surface
  • The gluing construction
  • The fundamental group
  • Examples of fundamental groups
  • Covering spaces and the deck group
  • Existence of universal covers
  • Euclidean geometry
  • Spherical geometry
  • Hyperbolic geometry
  • Riemann metrics on surfaces
  • Hyperbolic surfaces
  • A primer on complex analysis
  • Disk and plane rigidity
  • The Schwarz-Christoffel transformation
  • Riemann surfaces and uniformization
  • Flat cone surfaces
  • Translation surfaces and the Veech group
  • Continued fractions
  • Teichmüller space and moduli space
  • Topology of Teichmüller space
  • The Banach Tarski theorem
  • Dehn's dissection theorem
  • The Cauchy rigidity theorem
  • Bibliography
  • Index
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Monday, June 27, 2011