Knot theory is many things to many people. It is fine and important mathematics with connections to a lot of very sexy extra mathematical things including biology (e.g. DNA twisting) and physics (see, e.g, http://www.nytimes.com/1989/02/21/science/mathematicians-link-knot-theory-to-physics.html, and note the date: 1989 — this has been hot stuff for several decades). It is at the heart of topology, of course, with huge numbers of topologists doing research in the area across the world. It is a remarkably accessible subject: it provides the topics of choice for a number of my colleagues leading senior undergraduates in research projects — you don’t need to have read Spanier before starting with it, although I would recommend that one starts with, for instance, Hajime Sato’s Algebraic Topology: an Intuitive Approach, one of my favorite books.
This said, what does it all have to do with the book under review, whose focus is 3-manifolds? The answer is of course all but immediate: one generally fits knots (and links) into 3-manifolds, and, indeed, the book’s central fourth chapter has that as its exclusive focus. Schultens says in her introduction that “[i]n Chapter 4 we catch a glimpse of the interaction of pairs of manifolds, specifically pairs of the form (3-manifold, 1-manifold). Of particular interest … is the consideration of knots from the point of view of the complement (‘Not Knot’)…” Excellent: I am proud to be able to report that with my having attended many of my topologist-colleagues’ seminars over the years, as fellow traveler at best I am able to recognize just about every one of the ten themes in this chapter. Again, this is indeed a wonderfully accessible subject — even if that is a misleading phrase: accessibility certainly does not equate to triviality!
If Chapter 4 is physically central but is not the ultimate thrust of the book, what else is Schultens up to? Well, she starts with a solid discussion of manifolds per se, goes on to surfaces, and then hits 3-manifolds proper, including a brief discussion of fibre bundles and the Schoenflies theorem (which Schultens characterizes as generalizing the Jordan curve theorem). This third chapter includes some marvelous stuff, like an application of the Poincaré-Hopf index theorem, Alexander’s theorem stating that the complement of a torus in S3 must have a component whose closure is homeomorphic to a solid torus, and a discussion of Dehn’s lemma — says Schultens:
Max Dehn was among the first to investigate knots with a view toward 3-manifolds. One of his aims was to answer a fundamental question in knot theory that we will encounter again in Chapter 4: how to determine whether or not a given simple closed curve in S3 bounds a disk. In 1910 … he believed he had … established that this is the case if and only if the fundamental group of the complement of such a submanifold is abelian. But his proof hinged on a lemma he referred to simply as ‘the lemma.’ This … has since become known as “Dehn’s Lemma” … [However] Dehn’s original argument for his lemma was found to be incomplete, as pointed out by Kneser in 1929 … Kneser believed he had extended Dehn’s results. Then as his article went to press, he realized that Dehn’s argument concerning the removal of double curves of an immersed disk was incomplete. He added a brief note to this effect at the end of his article. Dehn never fixed the proof of his lemma … A proof … was finally given by Christos Papakyriakopoulos in 1957.
Schultens goes on to discuss a variant of the proof given by Jaco and Rubinstein. Serious stuff and all the more piquant in light of the historical framework Schultens introduces.
Finally, the last three chapters (following Chapter 4) deal with triangulated manifolds, Heegard splitting, and “further topics” which include, for example, hyperbolic manifolds, Dehn surgery, and foliations. Heegard splitting is, to my mind, particularly interesting. Schultens says pithily: “A Heegard splitting is, roughly speaking, a splitting of a 3-manifold into two simple pieces called handlebodies,” and the algebraic topological game is afoot, seeing that a handlebody is nothing else than a torus. Just think of the benefits one can reap by applying, say, Mayer-Vietoris (the first thing that comes to mind, I guess).
I think Schultens has written an excellent book that richly illustrates the scope of her chosen subject. It is very well written, clear and explicit in its presentation. There are plenty of exercises, and the book has been laboratory tested, so to speak, since it grew out of Schulten’s graduate course on 3-manifolds at Emory University, with rather minimal requirements in the way of algebraic topology proper and differential geometry, which indicates the level at which the book is pitched. I claim that it will also serve very well indeed as a source for self-study.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.