Algebraic Topology

A great book

I learned algebraic topology from this book. I've since read or at least perused a handful of other books on the subject but none of them cover as much material as well as Hatcher does. He covers much more information than any of the other introductory textbooks I have perused, and with tons of explicitly worked out examples. And I think it is indisputable that you simply will not be able to find a book that more richly and effectively conveys the intuitive geometric content of this subject. For that reason alone you are missing out in a big way if you don't buy this book. Now it is true that he doesn't assume too much background in abstract algebra (knowing the basics of groups, rings, and modules is more than sufficient). As a result he sometimes introduces the necessary algebraic structures in a more concrete, less general fashion than an algebraist would. If you want to see the full abstract nonsense type picture you will have to use a book like Spanier's. But be warned that if you're having difficulty understanding Hatcher, you have zero chance with Spanier. This brings me to my final point--the difficulty level of the text. Yes, I would rate its difficulty a notch above your typical introductory text in, say, analysis, algebra or point-set topology. If this is all you've been exposed to so far then you will feel a difference. He doesn't walk you through every tedious step, so at the beginning you will have to read more slowly until your mind gets smarter about filling in the blanks--which it will learn to do if you put in the time. What's obvious to Hatcher will usually also seem obvious to you once you get comfortable. So put on your big boy pants and stop wasting your time looking into lesser books hoping they will be easier to understand. It might (or might not) seem difficult at first, but once you're all grown and reading research papers you will look back at this book and appreciate just how unusually good Hatcher is as an expositor.

It is a comprehensive treatment.nice for a one year course at beginning graduate level or as a source of information at any point of research work. I love the paper version, at that, I am old school.

If you are like me, you live for those moments when studying a book when you read a sentence, and your eyes shoot open and you suddenly feel like the author has been watching you from somewhere just over your shoulder. There are lots of those kinds of sentences in this book, and figures too. On the other hand, the text is demanding, and not always clear. Hatcher will (generally speaking) lead you up to the point at which you can finish excavating his logic; at times his definitions are infuriatingly "pretty" and vague, and you will have to check other sources. Depending on the way you like to do things, you may get frustrated. A pity because there is so much valuable material in the book. I know of two other books, Algebraic Topology by Munkres, and Topology and Geometry by Glen Bredon, that I find helpful and not as vague as Hatcher. Each one is impressive, and each has pros and cons.

Thorough

This is a fantastic textbook, but don't take my word for it, you can download the entire book in pdf format from his website for free. He takes care to give you the intuition of whats going on as well as a complete and easy to follow rigorous development. I found it so useful that I broke down and bought the physical version of the book in the end.

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 I've learned over the years is code amongst mathematicians for there being a lot of hand-waving prose that reads more like instructions for building a kite than the logical discourse of serious mathematics. Judging from other reviews, I think it's safe to say some folks really like that kind of approach. Professors often 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 School of Witchcraft and Wizardry. And what about students? I cannot prove it, but, I think many students go ga-ga for this book because so often Hatcher's style of proof is similar to that of an undergrad's: 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 deadline. Some have said that this text reads like a pop science book, while others have said it is a supremely difficult read. Both charges are true for a simple reason: Treating hard concepts with fluff prose is bound to frustrate the more analytical reader who insists on understanding each nut and bolt in a mechanism. Hatcher's acolytes may counter that this is a book for mature students, so any gaps should be filled in by the reader along the way 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 need to spend a couple of days on each page. It is not realistic. Nevertheless this book seems to get a free ride with many reviewers, I think because it is offered for free. Whether this is a good book or a bad book 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 semester of graduate-level algebra is not enough, you should take two. Otherwise, while you may be able to fool yourself into thinking you know what the hell is going on, you won't really have anything but a superficial grasp of the basics. 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 patholgical, 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.

The more and more algebraic topology that I learn the more I continue to come back to Hatcher for motivation and examples. This book is worth its weight in gold just for all the examples both throughout the text and in the exercises. Another reviewer has said it: "You will not regret buying this book".

Classical textbook fo Algebraic Topology students

This is the printed version of the freely available pdf, and is ideal for those of us who like physical books to read. Hatcher writes clearly, and gets a lot done in a short amount of text by keeping to the point and not cluttering the narrative with unnecessary detail where it's not needed. I wish there were solutions to the exercises, though.

Only problem I have is the quality and font used for printing.. Everything else matches the online edition of the book which is free...1500 bucks for this quality of printing , the publisher could do better

Very good.

This is a must-have for the ones approaching Algebraic Topology. It is full of examples and counterexamples, and present the arguments in a geometry-flavoured way, with a very natural order. Really recommended.