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Lectures on the Topology of 3-Manifolds: An Introduction to the Casson Invariant

Walter de Gruyter
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Believe it or not, I actually remember the happy moment when I stumbled on Saveliev’s Lectures on the Topology of 3-manifolds: An Introduction to the Casson Invariant in the library stacks in graduate school. It was and remains one of the few mathematics books I’ve read from cover to cover. This book is a classic, especially for its treatment of the Casson invariant and related topics, and should be included in the library of any modern-day topologist.

Lectures is an excellent source for learning “classical” topology from the last half-century, assuming a first course in algebraic topology. Based on a course Saveliev gave at the University of Michigan, it hews to the definition-theorem-proof model, while providing many worked examples throughout. The first half of the book sets the stage with standard topics in 3- and 4-manifold theory, such as the Kirby calculus for surgery, Heegaard splittings, the intersection form, Seifert surfaces and Alexander polynomial. In this, Saveliev gives competent, concise overview; the reader might do well to also look at other texts by Rolfsen, Hempel, and Lickorish for further study.

Saveliev really excels in the second half of the book. His treatment of the Arf, Rohlin, and Casson invariants makes this book the standard text in the subject. The chapters about representations of the surface group into SU(2), the examples of calculations of the Casson invariant, and the applications thereof are detailed without being overbearing. Saveliev patiently reviews the necessary algebra background and does an excellent job building from there. This second half is what makes Lectures stand out.

While providing a detailed survey of the these topics, Saveliev also takes care to mention further topics of importance and interest. For instance, when talking about the signature of manifolds, he does mention Nikolov-Wall additivity but only tangentially and in such a way as not to slow down the pace of the exposition. He is diligent in pointing the reader to other texts and research papers for more in-depth treatment. The overarching philosophy of the book seems to be to introduce the reader to as wide a range of topics from low-dimensional topology as possible, given limited time and space. In this Saveliev certainly succeeds, and a student at the end of the book should have a broad enough understanding to follow most topology talks and would be well-poised for beginning research in a more specialized topic.

The second edition provides some welcome updates from the first edition, published in 1999. The foremost topological event since then was of course Perelman’s proof of the Poincaré Conjecture, and the second edition presents a nice discussion of the significance of the Casson invariant pre- and post-Perelman. Saveliev is very adept at pointing out theorems and proofs which used techniques then necessary in light of (a lack of) the Poincaré Conjecture, and makes a case for studying the topics in the book with present-day applications, e.g. Heegaard Floer theory, in mind. Towards this end, he includes two new sections, on Heegaard diagrams and the open book decomposition.

This book is highly recommended for two purposes. Firstly, as I had first encountered it, I would put it at or near the top of a list of references for self-study for any mathematician interested in learning more about the 3-manifolds, and particularly the Casson invariant. While in no way comprehensive, it provides an excellent first look at important and standard constructions and techniques in topology. Secondly, it is very suitable as a textbook for a graduate course, though the instructor will most likely want to pick and choose amongst the topics and include more or less details as needed. In both these endeavors, Saveliev provides plenty of references and short lists of exercises at the end of chapters.

Although the Casson invariant was named after my advisor, I don’t think that he ever mentioned the term willing in the course of my studies with him. So it was with great relief that I found Saveliev’s Lectures that momentous day at the library and finally learned what it was all about. I have since gone back to it many times over the years, much like a trusted friend. I don’t think it is possible for this reviewer to give it a more enthusiastic thumbs-up.

Helen Wong is Assistant Professor of Mathematics at Carleton College in Nothfield, MN.

Date Received: 
Tuesday, January 17, 2012
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Nikolai Saveliev
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Helen Wong
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Thursday, October 25, 2012