Books Added: Differential Topology


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.
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Books Added: general topology 2

OK, I said I wouldn’t go buy more books because mine were old. I didn’t. I bought two more old books. I was looking on the internet for more about the “separation axioms” and came across these two. One was a familiar title that I probably should have gone looking for (“Counterexamples”), but I didn’t know of the other.

(Any discussion of the separation axioms must cope with the fact that there are two distinct sets of terminology. These books were cited as the epitomes of the two terminologies.)

They’re both very well reviewed and, it seems to me, excellent. Quite apart from that, they are also Dover paperbacks, which means they are quite affordable.

Willard is in the same class as Dugundji and Kelley: a textbook which is exhaustive enough to serve as a reference. Like Kelley, it has lots of problems, and many of them investigate auxiliary material. Oh, unlike the other two, Willard has a few pictures.

It is also fun to read. No, he’s not trying to be a stand-up comic, but every once in a while he phrases something nicely. “In the next (and obvious) step to normal spaces, we find ourselves confronted with the real bad boy among the separation axioms.”

Steen and Seebach is a compact presentation of topology (40 pages), beautifully organized counterexamples (120 pages), a summary of metrization theory (24 pages), and a collection of charts and tables for finding a desired example (20 pages). I would think, speaking as an onlooker, that this is an indispensable reference if you do much topology.

Need a reference text? Unless you need something specific from Dugundji or Kelley, I suggest you get Willard.

Doing topology beyond your first course? Get Steen & Seebach on general principles.

Books Added

Steen, Lynn Arthur and Seebach, J. Arthur Jr., Counterexamples in Topology, Dover, 1995 (orig. 1978),
ISBN 0 486 68735 X
[general topology; 17 Nov 2008]
Reference. Very well organized, with many charts of relationships.

Willard, Stephen. General Topology, Dover, 2004 (orig. 1970).
ISBN 0 486 43479 6.
[general topology; 17 Nov 2008]
Textbook and reference. Well-written. Copious historical references and notes.

books added: general topology (point set topology)


Let me discuss my favorite general topology, i.e. “point set topology”, books. I have already discussed “algebraic topology” here.

Like so much other pure mathematics that I do not use professionally (for modeling power plants), topology is not on the tip of my tongue. But it’s fun, so I do it once in a while. And it’s fundamental, so I often have to go back to it when I’m playing with other mathematics.

This is a discouraging review in one respect: 4 of these 10 books are out of print: Kasriel, Dugundji, Sieradski, and Seifert & Threlfall. Heck, if you want Seifert & Threlfall, you should buy it in German! And for two of the books (Naber, Chinn & Steenrod) that Amazon claims to have in stock, there are multiple listings, many of which say the books are not available.

But I’m not going to go buy more books just because the ones I have are out of print. This is what I like, of what I have.

Buying Used Books

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Books added: no, but additional information added

I have made a major edit to the bibliography. I have tried to mark 3 pieces of information for each book. To be specific, I have tried to mark affirmative answers for 3 questions. (A search should find only affirmative answers.)

Answers. Does it have answers or hints for “many” problems? If so, that is one more reason why the book may be suitable for self-study without a teacher.

Guide to further reading. Not just a bibliography; not just detailed references; but “if you want to know more about this, consider these books, and for more about that, look at those books….” This was prompted by the magnificent guide in Bloch. I wish it were always so clear-cut. O’Neill’s “Elementary Differential Geometry” (2nd ed.), for example, has a short bibliography (8 books), which leads me to say, “I could afford to own all these.” (I own all but one; maybe I should rectify that just for completeness. I like short bibliographies.) Furthermore, he has a sentence about 2 of them. Technically, he has a guide to further reading, and I said so; but I qualified it as only two books. OTOH, a few books break things down into subcategories, but unless they said something about individual books, I didn’t view that as a guide. (The same effect is achieved by end-of-chapter bibliographies, and it’s not what I’m trying to flag.)

Epilog. What might we find in the next book? Not appendices, but for example, a final chapter entitled “advanced topics” – specifically because he said he couldn’t cover them but he wanted to mention them. This was prompted by a marvelous epilog in Massey’s “Algebraic Topology: an Introduction”.

I have tried to set it so that searching the bibliographies page for “guide”, “answers”, or “epilog” will find no extraneous occurrences of these words, except for titles which contain those words.

Finally, the edit date for every book affected by any of these changes is 5 Nov 2008, so you can search the bibliography for 5 Nov to find these changes. That is, until I have some reason to make another edit to any one of the books affected today.

No new books were added today.

I tried to be careful, both looking thru books and editing the bibliography, but don’t bet the farm on my tags; and remember that some of my editions are not current.

books added 9 Aug 2008: Algebraic Topology


Let me begin by citing a site: here you will find, among other things, a free downloadable version of an algebraic topology book, offered by its author. It looks pretty good.

Someday I’d like to write an introduction to topology (a post! not a book!), but trying to do it now is taking me too far out of my comfort zone: I am reasonably familiar with general (also called point-set) topology, but I am rather ignorant of algebraic topology; I am reasonably familiar with differential geometry, but differential topology is a different and unknown beast. I have opinions about how topology hangs together, but when I try to be precise, I find that I’m not sure I can justify my opinions. I’d rather get it more right later than get it wrong now.

Remember that the path from ignorance to knowledge in any subject is not straight and true, but is almost always rather zigzagged. One seems to learn things by a method of successive approximations to the truth.

William S. Massey, Algebraic Topology: An Introduction. p xiii.

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Books Added 1 Aug 2008: Graph Theory

I have had a long-standing question: why is the usual triangulation of a torus so large? It is a square with 18 triangles. Why so many?

While looking through algebraic topology books for more information about simplicial complexes – which is supplementary material for Bloch (“A First Course in Geometric Topology….” – I found the answer. Well, I found a clue. I found an inequality, which implied that the minimum number of triangles was 14 in any possible triangulation of the torus.

What I didn’t find was a proof of that inequality, and I couldn’t work it out for myself.

Eventually, in another algebraic topology book, I found another clue, and it was enough to let me work out the first inequality.

Along the way I found some pretty interesting things, and I want to chatter about them. The result will not really be math, but more in the nature of a travel guide.

Well, no, not even that detailed. More in the nature of some really cool pictures from a foreign country.

The search took me into both algebraic topology and graph theory. Before I put out the posts (there will be at least two), I need to put out some bibliography.
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books added 21 June

The following books have been added to the bibliography.

The Ashley book is a welcome addition to control of flight vehicles (Bryson; Blakelock): it’s got a lot more detail about the underlying dynamics. I have no idea when I bought it, but I eventually remembered that it was somewhere in my library, and was delighted to find its more detailed explanation – and excellent drawing – of the various coordinate systems in use for aircraft and missles. This is material which the control theory books assume you’ve seen in more detail.

The Ideals & Varieties book is an introductory text which I am working thru with a friend. The third author, O’Shea, is the author of a recent book on the Poincare conjecture which is what got me started on the geometry of surfaces.

The 3 mechanics books (Marion, Symon, and the Berkeley) were additional references (cf. Goldstein) for acceleration in rotating coordinate systems. I have listed the Berkeley text twice, for the same reason I list Schaum’s Outlines twice. I’ve always heard it called “the Berkeley mechanics book”, and that’s how I searched to see if it – and the rest of the series – were in print (no) and available used (yes).

I bought the Basilevsky Factor Analysis book because I wanted something more about noise in factor analysis methods (cf. Malinowski). It looks like a good and interesting book (I wasn’t expecting to find the Kalman filter in it), although it is the specific text in which I found the mistaken assertion that we could always choose the eigenvector matrix orthogonal. As I said when I corrected that very same careless error on one of my own SVD pages, I am inclined to be tolerant of other people’s mistakes: I make mistakes, too.
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