Algebraic Topology

In the TV series "Babylon 5" the Minbari had a saying: "Faith manages." If you are willing to take many small, some medium and a few very substantial details on faith, you will find Hatcher an agreeable fellow to hang out with in the pub and talk beer-coaster mathematics, you will be happy taking a picture as a proof, and you will have no qualms with tossing around words like "attach", "collapse", "twist", "embed", "identify", "glue" and so on as if making macaroni art. To be sure, the book bills itself as being "geometrically flavored", which over the years I have gathered is code in the mathematical community for there being a lot of cavalier hand-waving and prose that reads more like instructions for building a kite than the logical discourse of serious mathematics. Some folks really like that kind of stuff, I guess (judging from other reviews). Professors do, because they already know their stuff so the wand-waving doesn't bother them any more than it would bother the faculty at Hogwarts. When it comes to Hatcher some students do as well, I think because so often Hatcher's style of proof is similar to that of an undergrad: inconvenient details just "disappear" by the wayside if they're even brought up at all, and every other sentence features a leap in logic or an unremarked gap in reasoning that facilitates completion of an assignment by the due date. Some will say this is a book for mature math students, so any gaps should be filled in by the reader en route with pen and paper. I concede this, but only to a point. The gaps here are so numerous that, to fill them all in, a reader would be spending a couple of days on each page of prose. It is not realistic. Some have charged that this text reads like a pop science book, while others have said it is extremely difficult. Both charges are true. Never have I encountered such rigorous beer-coaster explanations of mathematical concepts. Yet this book seems to get a free ride with many reviewers, I think because it is offered for free. In the final analysis is it a good book or a bad book? Well, it depends on your background, what you hope to gain from it, how much time you have, and (if your available time is not measured in years) how willing you are to take many things on faith as you press forward through homology, cohomology and homotopy theory. First, one year of graduate algebra is not enough, you should take two. Otherwise while you may be able to fool yourself and even your professor into thinking you know what the hell is going on, you won't really. Not right away. Ignore this admonishment only if you enjoy applying chaos theory to your learning regimen. Second, you better have a well-stocked library nearby, because as others have observed Hatcher rarely descends from his cloud city of lens spaces, mind-boggling torus knots and pathological horned spheres to answer the prayers of mortals to provide clear definitions of the terms he is using. Sometimes when the definition of a term is supplied (such as for "open simplex"), it will be immediately abused and applied to other things without comment that are not really the same thing (such as happens with "open simplex") -- thus causing countless hours of needless confusion. Third: yes, the diagram is commutative. Believe it. It just is. Hatcher will not explain why, so make the best of it by turning it into a drinking game. The more shots you take, the easier things are to accept. In terms of notation, if A is a subspace of X, Hatcher just assumes in Chapter 0 that you know what X/A is supposed to mean (the cryptic mutterings in the user-hostile language of CW complexes on page 8 don't help). It flummoxed me for a long while. The books I learned my point-set topology and modern algebra from did not prepare me for this "expanded" use of the notation usually reserved for quotient groups and the like. Munkres does not use it. Massey does not use it. No other topology text I got my hands on uses it. How did I figure it out? Wikipedia. Now that's just sad. Like I said earlier: one year of algebra won't necessarily prepare you for these routine abuses by the pros; you'll need two, or else tons of free time. Now, there are usually a lot of examples in each section of the text, but only a small minority of them actually help illuminate the central concepts. Many are pathological, being either extremely convoluted or torturously long-winded -- they usually can be safely skipped. One specific gripe. The development of the Mayer-Vietoris sequence in homology is shoddy. It's then followed by Example 2.46, which is trivial and uncovers nothing new, and then Example 2.47, which is flimsy because it begins with the wisdom of the burning bush: "We can decompose the Klein bottle as the union of two Mobius bands glued together by a homeomorphism between their boundary circles." Oh really? (Cue clapping back-up chorus: "Well, ya gotta have faith...") That's the end of the "useful" examples at the Church of Hatcher on this important topic. Another gripe. The write-up for delta-complexes is absolutely abominable. There is not a SINGLE EXAMPLE illustrating a delta-complex structure. No, the pictures on p. 102 don't cut it -- I'm talking about the definition as given at the bottom of p. 103. A delta-complex is a collection of maps, but never once is this idea explicitly developed. A final gripe. The definition of the suspension of a map...? Anyone? Lip service is paid on page 9, but an explicit definition isn't actually in evidence. I have no bloody idea what "the quotient map of fx1" is supposed to mean. I can make a good guess, but it would only be a guess. Here's an idea for the 2nd edition, Allen: Sf([x,t]) := [f(x),t]. This is called an explicit definition, and if it had been included in the text it would have saved me half an hour of aggravation that, once again, only ended with Wikipedia. But still, at the end of the day, even though it's often the case that when I add the details to a one page proof by Hatcher it becomes a five page proof (such as for Theorem 2.27 -- singular and simplicial homology groups of delta-complexes are isomorphic), I have to grant that Hatcher does leave just enough breadcrumbs to enable me to figure things out on my own if given enough time. I took one course that used this text and it was hell, but now I'm studying it on my own at a more leisurely pace. It's so worn from use it's falling apart. Another good thing about the book is that it doesn't muck up the gears with pervasive category theory, which in my opinion serves no use whatsoever at this level (and I swear it seems many books cram ad hoc category crapola into their treatments just for the sake of looking cool and sophisticated). My recommendation for a 2nd edition: throw out half of the "additional topics" and for the core material increase attention to detail by 50%. Oh, and rewrite Chapter 0 entirely. Less geometry, more algebra. A final note: in actual fact I think of this book as rating around 3.4 stars, so it becomes a victim of rounding here.