books added 9 Aug 2008: Algebraic Topology

introduction

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.

http://www.math.cornell.edu/~hatcher/

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.


I cannot resist saying that all topology is divided into three parts. It may not be true, but it was too good a line to pass up. (If you’ve never had Latin, forget it.) Within topology, we find general, algebraic, and differential. Within algebraic topology, we find homotopy and homology; within homology, we find simplicial, singular, cubical, and possibly more, and there’s cohomology, too, mixed in. (Oh, I’m not counting topological groups and lie groups among “topology”. Too specialized. Frankly, I wonder if algebraic topology and general topology shouldn’t be considered completely separate; but most of my general topology books do include an introduction to algebraic topology.)

Anyway, trying to get more comfortable with simplicial complexes, I wandered thru my algebraic topology books, and found some fun stuff. Then I went looking thru just about every book (oops, it occurs to me that I have some physics books that I didn’t check) with topology (or homology) in the title – about 30 all told – and ordered a few more, of course. If all I wanted to do was add a pile of books to the bibliography, I could do that; but I’d like to talk about the subjects, too, and I’m foundering for lack of precise knowledge of their areas and boundaries.

I’d like to get to the point where I can write that overview / introduction, but I’m going to call it a longer-term project, instead of trying to get it done now: there’s more specific math I feel a need to get to.

Nevertheless, what started out as a search for more about simplicial complexes got all wound up in counting vertices, edges, and faces. It was fun, and I have one post ready to go, and a draft that may turn into more than one post.

I used a couple of books already in the bibliography. The Firby & Gardiner “Surface Topology” was invaluable; the Biggs et al. “Graph Theory 1736-1936” provided a lot of auxiliary material. (That, in fact, was why the graph theory books were added a while ago.)

general

The following four books, by three authors, have been added to the bibliography. They are all “algebraic topology”. They are being added primarily because they are cited in the next post to come. As it happens, they are also high on my list of what to do next in topology after Bloch, but there are a couple of other books high on that list, too.

I know that both Fulton and Massey are reputed to be good expositors, and I can confirm that; I don’t know Rotman’s reputation, but I myself find him to be good at talking about what we’re doing and why. Reference books are invaluable, but textbooks should not be reference books. These four books are textbooks to learn from.

Massey

Note that both Massey books are still in print: the older was republished by Springer, which is also the publisher of the newer book.

For the older of the Massey books, let me quote his preface: “undoubtedly some experts will be shocked that a textbook purporting to be an introduction to algebraic topology does not even mention homology theory.”

His first chapter is two-dimensional manifolds; his second is the fundamental group (homotopy; e.g. that there are two distinct kinds of loops we can draw on a torus, those collapsible to a point and those which are not). His third and fourth chapters are focused on free groups and the Seifert-van Kampen theorem, respectively, but he uses these two chapters to show us “universal mapping problems” and, especially, to discuss why we use them. (A simple example, not one of his, is: “the function f is 1-1 if and only if there exists g (a left inverse) such that the following diagram commutes”).

(Before I got to Carnegie-Mellon, I had never even heard of a commutative diagram; my very first class meeting was general topology, and as I recall, I’ll bet the very first theorem was that diagram.)

His fifth chapter is covering spaces. And these five chapters are the beginning of the newer Massey book. The newer Massey also has the same appendix B as the older (but not the same appendix A!).

The older book then does the fundamental group and covering spaces of a graph (but you can find that elsewhere); the fundamental group in higher dimensions; and an appendix on the quotient space topology (but that’s general topology); and an appendix on transformation groups – which is also in the newer Massey.

The newer Massey picks up homology theory – the stuff he left out of the older book. Actually, it’s pretty much all of a 3rd book by him (“Singular Homology Theory”). About all I’m going to say is that the initial homology chapter (ch. 6) is 10 pages of background and motivation in homology theory. Worth reading even if I don’t continue in Massey.

Let me clarify that. There are actually three books involved here, two of which I own. Most of Massey’s “Algebraic Topology: An Introduction” and his “Singular Homology Theory” were combined into “A Basic Course in Algebraic Topology”.

If you are going to buy a new copy of Massey, you might as well buy the newer book, “A Basic Course….”but if, instead, you simply stumble across a cheap copy of either older book, hey, buy it, understanding that it’s about half of the combined book.

Fulton

Fulton is divided into parts, each of a few chapters, and some of the parts are divided. We find:

  • I calculus in the plane
  • II winding numbers
  • III cohomology & homology (then again parts V and VIII)
  • IV vector fields
  • VI covering spaces and fundamental groups (then again part VII)
  • IX topology of surfaces
  • X Riemann surfaces
  • XI higher dimensions

He says (p. viii), “To achieve this variety at an elementary level, we have looked at the first nontrivial instances of most of these notions: the first homology group, the first De Rham group, the first Cech group, etc…. We have tried to do this without assuming a graduate-level knowledge or sophistication.”

Finally, I should point out that the book encourages one to work along with it. For example, the section from pages 43 to 47 defines winding number, degree, and local degree; it has two lemmas and one proposition, 3 exercises, and 9 problems (requiring more ingenuity than exercises). All interspersed, suggesting things for the reader to play with.

Rotman

From the table of contents of Rotman, one might think that it was very similar to Massey or Fulton; not so. Yes, we see the usual suspects: simplexes, fundamental group, singular homology, covering spaces, etc; but we also notice chapters on “long exact sequences” and “natural transformations”. In the preface, he says, “I am an algebraist with an interest in topology.” indeed: the word “functor” first appears on page 3.

I like that. I don’t want to do it first, but I do want to do it this way eventually. Rotman’s book calls to me.

the books added

Fulton, William. Algebraic Topology: A First Course. Springer, 1995.
ISBN 0 387 94327 7.
[algebraic topology, 8 Aug 2008]
“Graduate Texts in Mathematics”. “To achieve this variety at an elementary level, we have looked at the first nontrivial instances of most of these notions: the first homology group, the first De Rham group, the first Cech group, etc…. We have tried to do this without assuming a graduate-level knowledge or sophistication.”

Massey, William S. A Basic Course in Algebraic Topology. Springer, 1991.
ISBN 0 387 97430 X.
[algebraic topology, 8 Aug 2008]
“Graduate Texts in Mathematics”. The first 5 chapters of this appear to coincide with the first 5 chapters of “Algebraic Topology: An Introduction.” In particular, it includes the excellent introduction to universal mappings. OTOH, it has a substantial introduction to the “why?” of homolgy.

Massey, William S. Algebraic Topology: An Introduction. Harcourt, Brace & World, 1967 (Springer, 1977).
[algebraic topology, 8 Aug 2008]
He is a wonderful author, and I love his style. Both versions of this text have the most readable introduction to universal mappings which I’ve seen. This book covers Surfaces and homotopy; it does not cover homology.

Rotman, Joseph J. An Introduction to Algebraic Topology. Springer, 1988.
ISBN 0 387 96678 1.
[algebraic topology, 8 Aug 2008]
“Graduate Texts in Mathematics”. In the preface, he says, “I am an algebraist with an interest in topology. The basic outline of this book corresponds to the syllabus of a first-year’s course in algebraic topology….”

Advertisements

One Response to “books added 9 Aug 2008: Algebraic Topology”

  1. AlexM Says:

    I found your site on technorati and read a few of your other posts. Keep up the good work. I just added your RSS feed to my Google News Reader. Looking forward to reading more from you down the road!


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: