Elements of Differential Topology

As a number theorist who has recently developed a research need for both algebraic topology (pretty reasonable, given its proximity to algebraic geometry, which, in turn, has recently come to be a kissing cousin to number theory) and differential geometry (more of an idiosyncratic affair: my work has led me to themes in symplectic geometry), the subject of differential topology is a bit of an epistemological mystery to me. Of course, I know it’s one of the things Milnor is famous for, and I know about his famous introductory monograph Topology from the Differentiable Viewpoint, which first appeared in 1965. And there are serious advanced tomes, too: Guilemin and Pollack offer Differential Topology and Hirsch offers a text with the same title.

So, what’s it all about? Wikipedia says, rather pithily, that “differential topology is the field dealing with differentiable functions on differentiable manifolds.” On the other hand, Wikipedia also says that “differential geometry is [the] discipline using the techniques of differential and integral calculus, as well as linear and multilinear algebra, to study problems in geometry.” I guess this just underscores the fact that you get what you pay for, especially with different authors in the game. But there’s some hope for clarity: the Wikipedia article of differential geometry lists differential topology as a branch of differential geometry. I guess that’s the same things as saying that topology is a branch of geometry… true as far as it goes, but then again, not quite right.

Well, let’s look at what the book under review has to offer. The back cover advertises it as an exploration of “the vast and elegant theories in topology developed by Morse, Thom, Smale, Whitney, Milnor, and others. It begins with differential and integral calculus, leads you through the intricacies of manifold theory, and concludes with discussions on algebraic topology, algebraic/differential geometry, and Lie groups.” Well, now, the aforementioned Wiki article also lists Lie groups as a part of differential geometry, and now Shastri proposes to take us to differential geometry itself within the present differential topological context.

Genug! Let’s not worry about hairsplitting any more and just restrict attention to what Shastri offers in his text —- which turns out to be quite unusual.

Shastri starts off by noting that the aforementioned books by Milnor and Guillemin-Pollack constitute big influences on his presentation; in fact he used to use the latter textbook for the course from which the material in his own book is sprang forth. But Shastri’s book is unusual in the sense that it’s pitched to a rather less sophisticated audience than a graduate school affair like Guillemin and Pollack, for instance, or a Springer GTM book like Hirsch’s. Shastri says that “[his] book assumes that the reader has gone through a semester course each in real analysis, multivariable calculus, and point-set topology. So this book should be accessible to a smart second-semester junior with an analysis bent, or certainly a senior: what an opportunity, given that Shastri’s later chapters indeed explore the themes he alluded to in his preface. We meet the Whitney imbedding theorem in Chapter 5 and Morse theory in Chapter 8, for example.

By the way, I guess I am a bit more optimistic than F. Thomas Farrell, the author of the book’s Foreword, who claims that “[Shastri’s] book is accessible to a serious first year graduate student reading it either independently or as a text in a graduate course.”

But there’s more. I first studied differential forms from Spivak’s Calculus on Manifolds, as a subject unto itself. Here, in Shastri’s treatment, the subject is developed in the larger context of the author’s stated goals, which makes for very good motivation and increased accessibility. Shastri does an excellent job with this foundational material. The first five chapters take the reader from a review of vector calculus all the way to de Rham cohomology and the theory of abstract manifolds (including gluing, a discussion of both tangent spaces and tangent bundles, and, as I already noted, Whitney’s imbedding theorem).

This sets the stage for more advanced themes: isotopy, intersection theory [!], the geometry of manifolds (see my earlier waffling), and, as closer, a dense chapter titled, “Lie groups and Lie algebras: the basics” (and it couldn’t really be otherwise). It’s altogether a solid introduction to serious themes likely to persuade the reader to go deeper into the subject.

Shastri’s exposition is rigorous at the same time that it evinces a light touch, and this of course makes for a very readable book. Examples abound, proofs are done in detail and include discussion along the lines of what one might hear in a good lecture presentation, and there are exercises replete with hints or solutions. Pedagogically, Elements of Differential Topology clearly gets very high marks. It is a good and useful textbook.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.