The topological classification of compact surfaces is an immensely satisfying result. It says that a compact surface is completely determined topologically by its Euler characteristic, boundary components, and whether it is orientable. It is rare that a result so succinctly solves an interesting problem and even rarer that its proof be fairly accessible.
Understanding the classification proof on some level is a reasonable objective for an undergraduate topology course, as the proof also serves as a motivating tour of the discipline’s fundamental techniques. The basic approach is to first build the notion of a surface from elementary point-set topological machinery, arranged toward proving that any surface can be cut into triangles. Algebraic and combinatorial techniques then take over to classify how these triangles can be glued together to give surfaces.
Gallier and Xu’s A Guide to the Classification Theorem for Compact Surfaces is the book to read after completing a first pass through topology. “Guide” is exactly the right word. The purpose of the text is not to present a fully detailed proof of the classification theorem, but to outline the overall structure of the proof, compare different approaches with some historical context, and showcase how the fundamental machinery of topology can produce deep results.
The authors do an excellent job of keeping focus on the classification theorem. There is a lot of machinery to build, but it is always motivated and they do not try to do more than is necessary. The proof has lots of moving parts and the narrative carefully reminds the reader of the big picture while not allowing details to slip away. The authors cannot cover everything, but they are explicit about what is being skipped, why, and where to find more information, often including brief discussions of what the references have to offer. Most of the missing details are readily available in common topology texts.
One of the book’s greatest strengths is its awareness of the literature, offering historical context absent from most presentations. Alternate approaches are referenced and briefly discussed as they emerge and there is a short review of different proofs of classification with special attention paid to Conway’s ZIP proof. The writing is fairly lucid, although the authors do tend to pack a few more clauses into their sentences than I’d like.
Prerequisite material includes elementary point set topology, group theory, and linear algebra. The necessary algebraic topology — homotopy and homology — are introduced from scratch, but not with the detail one wants from a first presentation. (It must be said that they do a surprisingly good job of covering the essentials of simplicial and singular homology in about 15 pages!) The omission of the role of geometry in the story is sad, but understandable.
The book would make an excellent graduate or advanced undergraduate project. I suggest one audience would be instructors or researchers looking to reconnect to some topology fundamentals. Practitioners in topology-adjacent fields might find it valuable as a concise reference. It could conceivably be used as a textbook or supplemental reference for a second topology course with some careful planning, but there are no exercises. Academic libraries should strongly consider this book as it offers a broad look at material common to a standard undergraduate mathematics curriculum.
I am unaware of any exposition quite like this. This is a book I wish I’d had in graduate school or before the first time I taught topology. I am certain I will consult it again.
Bill Wood is an assistant mathematics professor at the University of Northern Iowa and tweets as @MathProfBill.