Books Added: Differential Topology

Introduction

Putting out the following few books has been far harder than I expected, and has taken a lot more time. There are 6 of them: 3 texts, 1 reference, and 2 small sets of notes.

The fundamental problem is that I haven’t worked thru these books yet. Simply put, I’m effectively a grad student trying to figure out which books to read in order to introduce myself to a new field.

To put it more fancifully, I feel a bit like a wide-eyed urchin looking in a bakery window, trying to figure out what the different pastries will taste like, and I’ve picked out a few of them to try.

That simile fails, of course, because I’m not just looking at the pastries; I’ve held them in my hands and looked inside. I own these books, I’ve read each preface and table-of-contents, and I’ve read further into them. I’ve seen every one of them in other bibliographies; I’ve just read some of the reviews on Amazon….

The problem is, I haven’t gone into these books and come out the other side.

That matters, because my opinion of books has been known to change after I finished them, though usually from bad to good. Sometimes what I thought was incomprehensible and poorly written ends up seeming a nice presentation after I have a global framework. (That was true of Dugundji, and of Ahlfors’ complex analysis, and of Dean’s elements of abstract algebra: they are all very nice books, looking back on them, now that I understand more about the subjects.) OTOH, I’ve also found books that seemed inviting and appropriate, only to run into insuperable obstacles. (Hamermesh’s group theory comes to mind: even today his book is still opaque to me; I have trouble following him even on things I understand.)

So, I am encouraged by the other reviews of these books, particularly in Bloch, on Amazon, and among themselves: Guillemin & Pollack recommend all but Hirsch – which hadn’t been published yet! Bloch recommends all but Guillemin & Pollack; Milnor Topology lists all but Hirsch and Guillemin & Pollack, which didn’t exist at the time.

book reviewers on Amazon

Let me point out that there are two especially detailed Amazon reviewers all of whose reviews seemed worth reading; there’s a button for doing that. One reviewer is “Malcolm”; he appears to be an expert in differential manifolds. In fact, I’ve just ordered two books based on his reviews. If you want to see his reviews, start with any of these four: Hirsch, Guillemin & Pollack, Munkres, or Milnor Topology. He did not review either Milnor Morse Theory or Wallace.

The other reviewer seems far more trustworthy than his name, “mathwonk”, might suggest. He’s a college math professor. To find him, look under the reviews of Wallace.

What is it?

So just what is differential topology? “That sounds rather formidable,” said a physicist friend yesterday.

If nothing else, it’s a very reasonable search string on Amazon.

Seriously, it ought to be the study of differentiable manifolds (smooth manifolds). But differential manifolds are, by intention, what we can do calculus on. And the calculus is a lot of machinery.

So I took every “differentiable manifolds” book off my shelves and went thru them, just to help put the differential topology books in perspective. And that was a lot of books.

One way to think of differential topology is: the topological properties rather than the calculus-related ones. More generally, one can think of it as differential manifolds without any additional structure “… such as lie groups, Riemannian manifolds, symplectic manifolds, vector bundles, foliations….” which Lee (I need to add his Smooth Manifolds to the bibliography) says he would call differential geometry.

More importantly, I have seen the subject, whatever it may be, referred to as “differential and algebraic topology” (by Jean Dieudonne). The math arxiv (front end) lists general topology, algebraic topology, and geometric topology, and the last includes “manifolds”. Bloch says that geometric topology more narrowly defined would be topological and piecewise linear manifolds, but not differential.

I think five of these six books can be characterized as: the topological properties of differentiable manifolds without much regard to additional structure, and without using algebraic topology (that’s important for these books if not for the subject).

And I’m in trouble already. These books are going to discuss vector fields. Surely that counts as additional structure. Well, that’s why I allowed for some regard of additional structure. The fact remains that these books look different from my differentiable manifolds books, which in turn look different from my differential geometry books. I hope that as I add more books to the bibliography, I can make some sensible distinctions.

Overview

I started out with 6 books in hand. I wondered if I should add some differential manifolds books; I wondered if I should drop the Milnor “Morse theory”. In the end, I stayed with the original 6, having seen no compelling reason to add or subtract from that list.

Ah, one last point. Munkres specifically uses the phrase and title “elementary differential topology” to mean those theorems in differential topology which do not require algebraic topology for their proofs; similarly, as he says, a theorem in number theory is said to be elementary if its proof does not involve functions of a complex variable. (And that doesn’t mean the proof is easy.)

Oh, another last point. Guillemin & Pollack use an interesting rating scale, for difficulty. There is a perfectly straightforward one used by the MMA (Mathematical Association of America). Since North America uses phrases like “grade 10”, the MAA uses numerical grade levels, extending grades 1-12 up to 18 for “2nd year graduate studies”. A book suitable for seniors and first-year graduate students, then, would be marked “16-17”.

(A very little kid asked me what grade I was in in school, when I was a college freshman. Turned out, he didn’t recognize the word “freshman”, so I told him I was in the 13th grade.)

Guillemin & Pollack took another rating scale that we are familiar with: movies. G = elementary, with only analysis and linear algebra as prerequisites. PG means it requires something more, whether that’s some abstract algebra, or some topology, or whatever. R denotes graduate level mathematics, and X would be “hard going for a graduate student, meant to be read more for inspiration than for comprehension.”

I doubt that I will use either of those systems. I will comment that Guillemin & Pollack labeled Milnor Morse Theory and Wallace as PG; they suggested Milnor Topology as collateral reading with them, but did not rate it. Since Guillemin & Pollack was designed as a “leisurely first year graduate course” predominantly accessible to juniors and seniors, I would guess Guillemin & Pollack and Milnor Topology would have PG ratings.

Ok, on to the books.

Munkres is a reference book. It’s only 109 pages. Part I proves “the folklore theorems” of the subject: those things that almost everyone else says “go look somewhere else for a proof”. Skimming it, I get the impression that it would be a good place to start sharpening ones tools by working out the exercises (and Malcolm’s review concurs). Part II proves that every differentiable manifold has a unique smooth triangulation. That’s pretty major. Both Milnor Topology and Wallace cite this book.

I’ve repeatedly said that TOP, PL, and DIFF (topological, piecewise linear, and differentiable) structures are not equivalent in higher dimensions. And I also don’t understand the distinction between simplicial, piecewise linear, and triangulations.

Munkres gives me a precise definition of triangulation as he used it for the theorem. That’s a big step.

Wallace is where I want to start. I’m not sure I’ll make it all the way, because his chapter on “spherical modifications” looks rather intimidating. But Wallace seems to have been a significant researcher in the great breakthroughs of the 50’s and 60’s, and – more importantly for a first text – he is a good writer and expositor. (I own two other books by him.) More importantly for the subject, Wallace culminates in a proof of the classification theorem for compact connected smooth surfaces – but using critical points of functions on manifolds (this is the beginnings of Morse theory, I think) instead of using simplicial complexes. Since it’s a Dover paperback, you could buy it just for the classification theorem, and you shouldn’t care that it’s only 130 pages. Everyone else I’ve been reading proves the classification theorem for triangulable surfaces. (Ok, I haven’t really been reading Hirsch, but he also does it using the differentiable structure rather than simplicial; but I’ll get there sooner in Wallace.)

So I’ll be picking up Wallace first.

But he’s a special purpose text: one might say, a set of notes culminating in an important theorem.

Guillemin & Pollack is a full-blown text, aimed at graduate students but much of it accessible to undergraduates. Its four main sections are manifolds & smooth maps, transversality & intersection, oriented intersection theory, integration on manifolds. Within them, it lists immersions, submersions, and embeddings; transversality, intersection theory mod 2, and winding numbers; the poincare-hopf theorem, the Euler characteristic, and triangulations; the gauss-bonnet theorem.

I’ll be picking up this after Wallace. (Sadly, Guillemin & Pollack is out of print; the other 5 are still in print.)

Well, not quite. I’m going to do something a little strange, even for me.

Both Guillemin & Pollack and Hirsch are introductions to the subject, aimed at beginning graduate students. Guillemin & Pollack is the more elementary of the two, but Hirsch has a considerable amount of discussion.

I say, a considerable amount of discussion. Hirsch appears to be more detailed, and more advanced, so I would work thru it after Guillemin & Pollack. but I intend to read Hirsch first, and take notes on his remarks, before I start working thru Guillemin & Pollack. That’s how insightful his commentary seems.

Then I can work thru Hirsch. It looks more like an introduction to the key non-algebraic techniques of the subject: transversality (of course); vector bundles and tubular neighborhoods; degrees, intersection numbers, and the Euler characteristic; Morse theory; cobordism; isotopy; and he closes with the classification of compact surfaces using Morse theory.

This brings us to the two Milnor books. Milnor won the fields medal in 1962, and for that alone I’m very interested in what he chooses to say about a subject. Furthermore, he is famous for his lectures. The beginning of the Topology looks a lot like the beginning of Wallace, so I’ll keep it handy from the get-go. Heck, I should quote Malcolm’s review on Amazon: “… too much of the material is left out for this to be adequate as a textbook. OTOH, it does make for good bedtime reading.”

The best I can do for the Morse theory is to quote an unknown reviewer (“A Customer”, which appears to be the generic name) from Amazon. He said that his advisor gave him this book, saying, “You’re not ready for this yet, but you should have it — it’s the best piece of mathematical exposition there is.” I am pretty damned sure that any bibliography of differential topology is flawed which does not include this book.

In summary, then, 3 textbooks (which I will do in order WallaceGuillemin & PollackHirsch), 1 reference (Munkres), 1 collateral reading (Milnor Topology), and 1 for dessert (Milnor Morse Theory).

The Books Added

Wallace, Andrew H. Differential Topology: First Steps. Dover 2006 (orig 1979).
ISBN 0 486 45317 0.
[differential topology; 20 Dec 2008]
Looks like a marvelous undergraduate introduction. From the preface: “… in this field, as indeed in any branch of topology, the first steps should be geometric.” He uses differential methods to obtain the classification of 2D manifolds. Short intense guide to further reading. Structured exercises, but no solutions. Epilog.

Hirsch, Morris W. Differential Topology. Springer, 1991 (corrected 4th printing).
ISBN 0 387 90148 5.
[differential topology; 20 Dec 2008]
“Graduate Texts in Mathematics”. The discussion and remarks throughout the text are worth reading first, in my opinion. Uses Morse Theory to get the classification theorem for surfaces.

Munkres, James R. Elementary Differential Topology. Princeton, 1966 (revised ed.).
[differential topology; 20 Dec 2008]
Reference. Proves the “folklore theorems”; culminates in the proof that any smooth manifold has a smooth triangulation.

Milnor, John W. Topology from the Differentiable Viewpoint. Princeton, 1997 (orig 1965).
ISBN 0 691 04833 9.
[differential topology; 20 Dec 2008]
Notes on a few selected topics. Legendary might be more appropriate than “a classic”. Brief guide to further reading.

Milnor, John W. Morse Theory. Princeton, 1969 (orig 1963).
ISBN 0 691 08008 9.
[differential topology; 20 Dec 2008]
A compact presentation, but also legendary. I think of it as dessert after Hirsch.

Guillemin, Victor; Pollack, Alan. Differential Topology. Prentice Hall, 1976.
ISBN 0 13 212605 2.
[differential topology; 20 Dec 2008]
Graduate / advanced undergraduate. A “leisurely first year graduate course”, predominantly accessible to juniors and seniors. Guide to further reading.

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