Differential Topology

One of the perks about being a number theorist is that the older you get mathematically, the more you know that you don’t know what you need to know, and then you have to set out and get to know that stuff. To be sure, any number theory graduate program worth its salt instills something of this awareness into its rookie charges, and presently fledgling professional Zahlentheoretiker, with brand new Ph.D. diplomas in hand, are ideally cognizant of the prospect that the problems and questions they’ll work on in the higher arithmetic might well involve just about anything under the mathematical sun. In my own case it has come to pass that over the (by now, long) years, algebraic and differential topology have been on the ascendant. I think back to my sage undergraduate professor, V. S. Varadarajan, who said to me long ago that number theory and physics are two sides of the same coin: now my work in analytic number theory critically involves the magic coming from where physics and geometry meet. It’s my lot (and a very happy one at that) that I have to learn a host of the latter, including, as I have just mentioned, algebraic and differential topology (without which the indicated differential geometry just doesn’t get off the ground).

So, my relationship to the book under review is that of an outsider who has already spent a long time visiting, and I am happy to report that the longer I visit, the more I like my trip. Together with classics like Eilenberg-Steenrod and Cartan-Eilenberg, my favorite get-off-the-ground-fast book on algebraic topology, Sato’s Algebraic Topology: An Intuitive Approach, and the fantastic Concise Course in Algebraic Topology by May, in my opinion the most evocative and down-right seductive book in the game is Bott and Tu’s Differential Forms in Algebraic Topology. I particularly mention the latter because it’s particularly concerned with what geometers would ask of algebraic topology, meaning that smooth manifolds are everywhere, and this is of course also the case for the physicists’ geometry (and add Riemannian structure to all that).

Against this backdrop, what C. T. C. Wall gives us here is entirely of a piece, but with an interesting wrinkle (and a deep theme in its own right). Here is what he says himself along these lines:

Although the foundations have much in common with differential geometry, we approached the subject from a background in algebraic topology, and the book is written from that viewpoint. The study of differential topology stands between algebraic geometry and combinatorial topology. Like algebraic geometry, it allows the use of algebra in making local calculations, but it lacks rigidity: we can make a perturbation near a point without affecting what happens far away…

Qua structure, the book “falls roughly in two halves: introductory chapters with general techniques, the four chapters, each including a major result.” Wall’s foci include, in sequence, manifold theory (including fibre bundles, material on Riemannian metrics, and tubular neighborhoods), differentiable group actions, handlebodies and surgery, and cobordism. The major results just alluded to include h-cobordism, material on the homotopy theory of Poincaré complexes and on the homotopy types of smooth manifolds, and coverage of “groups of knots and homotopy spheres.” The latter is to be found in the last chapter of the book, on cobordism, and I must say that in this chapter I was particularly struck by Wall’s §§ 8.4 and 8.5, “Bordism as a homology theory” and “Equivariant cobordism.” The opening passage of the latter section states that “the object of this section is to give a method for reducing the calculation of equivariant cobordism groups to that of the bordism groups of certain classifying spaces.” This certainly reminds one of general strategies and tactics from algebraic topology, and suggests deep organic connections — it really couldn’t be otherwise. Also, now that we’ve gotten to cobordism theory it’s worth noting that, for example, this is the stuff that topological quantum field theory is at least in part made of, to name but one beautiful application. Of course, surgery and cobordism theory stand on their own entirely and require no support from even the coolest applications, but, still, TQFT is very, very cool.

The book is of the highest quality as far as scholarship and exposition are concerned, which fits with the fact that Wall is a very big player in this game. I have had occasion over the years to do a good deal of work from books in the Cambridge Studies in Advanced Mathematics Series, always top drawer productions, and the present volume is no exception. I very much look forward to making good use of this fine book.

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