Customer Reviews

Algebraic Topology

5 star:		(12)
4 star:		(5)
3 star:		(2)
2 star:		(2)
1 star:		(3)

 

 

There are many really good textbooks on algebraic topology and each has its own merit: Bredon for his effort in explaining everything that can be dealt without using spectral equences, Fomenko & Novikov for their effort in unifying differential geometry and algebraic/differential topology.
Hatcher's book is intended as one of the series that cover every aspect of the subject. Separate books on vector bundles and K-theory, and spectral sequences respectively, are to appear sometime in the future. Thus this one covers ordinary homology/cohomology and homotopy theory only. His writing style is helpful and user-friendly, not demanding extensive "mathematical maturity".
I am not sure if this is "the" textbook on algebraic topology, but I bet this is among the best ones. You would not regret if you buy this, even when an electronic version is available online (for free) from the author's home page.

No serious introductory text on basic algebraic topology has ever achieved this level of clarity, readability and depth. Its richness in examples (in both the main text and the problems) exposes a beginner to the underlying mechanisms of geometry in algebraic topology; its choice and arrangement of topics strike a perfect balance between accesibility and substantiveness; its lively and motivating exposition makes a student reluctant to attend the often boring topology classes. For a novice, this should be the first reading on the subject before (s)he is ruined by the many existing daunting texts; for a veteran, this can be very nourishing, especially if (s)he is already ruined by those either unreadable or shallow 'introduction's.

This book is intended as an "introduction to alegbraic topology" and I rated the book accordingly.

I found the book refreshing at points and thorougly frustrating at other points. This was one of the first book I approached when trying to learn formal algebraic topology. Prior to reading it I had indirect exposure to algebraic topology in application to physics especially when learning about differential forms where one is usually exposed to homology cohomology and derham cohomology, etc. I found the physics texts MUCH more instructive than this text which is supposed to be from the mathematicians perspective.

The book has it's merits:

1) it is organized well and attempts to relate the main topics in algebraic topolgy - homotopy and homology
2) it has many examples to help solidify the concept presented
3) it has plenty of exercises of varying difficulty.
4) it genuinely tries to motivate the mathematical ideas of algebraic topology.

However it has many faults. I was particulary disturbed by it's lack of definitions. At some point I felt like I was having a conversation or reading a "pop" math books for the dilettante not mathematician. I found myself repeatedly going back and having to REREAD THE TEXT to get the definition of some mathematical object. In my humble opinion a math text should clearly state definitions and properties and not try to "explain" them in prose without the preceding definitions.

The author also states minimal prerequisites ( algebra and point set topology ), however, it is clear alot more is needed.

Although there are plenty of examples, the author, simply states conclusions which maybe "self-evident" to someone with previous exposure to algebraic topology but not to a novice. In the examples little effort is made to explain the assertions.

Finally, the author has a chapter 0 which goes over some geometric preliminaries with little rigour, which to his credit he admits. However, he states that you do not really need to read it thru and only refer to it as needed when going over the text. The problems is all of the notions used in chapter 0 are ASSUMED TO BE KNOWN in the text. You have to know all the constructions, definitions and properties and access them from memory at a moment's notice to follow along the proofs and examples. That is not difficult to do but he doesnt present these notion in chapter 0 in a clear and efficient way. Again it is presented in "prose" format.

Regardless, I suggest you download the electronic version and read it for yourself. Google the author and the link will pop up.

I wanted to rate this book a B- but there was no 3.5 so I gave it a 3.

Allen Hatcher has gone to great length's in order to create a text which, albeit overly verbose, can be used as a gentle introduction to modern Algebraic Topology. Why 'modern'? Compare this text with the tried and tested texts of Spanier, Munkres as well as May and, almost immediately, you will see what I mean. The obvious example is Hatcher's use of CW-complexes as opposed to the more traditional build up beginning with simplices. For the die-hard mathematician who enjoys less fluff, this book is not for you and, in particular, if this is your first venture in Algebriac Topology, you enjoy the theorem-proof-theorem style with a light sprinkling of explaination, then I would recommend J.J. Rotman's text. Whereas, if you enjoy filler, background information, and lots of side-notes or examples, then Hatcher's text would be a perfect fit. Myself, I fall into the category of those who enjoy the more terse texts but, I purchased Hatcher's (the hardcover) because of the clarity and percision found in the proofs. The majority of other texts have a tendancy to obfuscate the underlying meaning that should be unerstood by the up-and-coming mathematician. Of course this approach has it's merits since, in particular, it forces the reader to fill in the blanks but, as a matter of insight, Hatcher's approach is also beneficial. Another positive strength of Hatcher's text lies in the fact that he effectively breaks the subject into it's prime sub-categories in such a way that the reader can begin with either of the four parts of the text without having to rely too much on previous sections. This novel feature allows someone interested in, say, Cohomology to pick up an begin learning about Cohomology without having to waste time making their way through material they are not interested in. Finally, yes you can get the book for free via Hatcher's website but I highly recommend purchasing the hardback text. It is well made, it will last for years, and it becomes truely mobile as compared to burning your eyes out while reading the text on your computer. Moreover, why waste the time printing it out.

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.

I have taught graduate algebraic topology courses three times from this book. My overall feeling is that, despite a few flaws, I have not seen another book I would rather use -- and I really wish this book had been around when I was learning the subject! I appreciate its very geometric style and the way it tries to get the reader to "see" the definitions of homology, homotopy, etc, before diving into the rigorous treatment. Many algebraic topology books I have seen are nearly example-free; they build the theory but don't show the reader how to do much with it. In contrast, Hatcher spends a lot of time, appropriately, on some of first really important examples in topology, such as surfaces and projective spaces. These investigations are continued in the exercises, which I feel are the best thing about this book. Many of them contain juicy examples which really show how the geometry and algebra interact.

On the minus side, I would agree with another reviewer that sometimes the rambling style, which works quite well in the introduction to a new concept, sometimes gets in the way when it's time to get down to precise definitions and theorems.

This book is avaliable free to download from Allen Hatcher's webpage. You will also find other books he has written.

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

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.

Hatcher's 'Algebraic Topology' provides an interesting approach to the field. It serves as an introductory text for a graduate level algebraic topology course. The text treats an excellent varity of subjects, spanning most everything someone might want in a first or second course in algebraic topology. The approach that Hatcher tries to take is a geometric one, attempting to give the student a good intuitive understanding of what is happening throughout. The downside of this is that more often than not his approach will lack some measure of rigor, and all too often important (and not obvious!) arguments will be 'hand-waved' through, or omitted entirely.

While Hatcher's content is deep, and broad, the main course of the text lacks many of the nuts-and-bolts type of lemmas and propositions, these being omitted entirely, or relegated to the exercises that follow each section. The result of this is that while there is a very pleasing amount of exercises for the reader to avail themselves of, they don't always have all the tools they might in a more standard approach. This can lend more than a little extra difficulty to some of the problems, which tend to be more computational than anything else.

This text serves as an excellent reference, or an accompaniment to another text as it gives a great number of well-chosen pictures, problems, as well as the overall intuitive flow of the book. It is not, however, always suited to be the sole text for a class if the object is to enable students to obtain the tools of algebraic topology and be able to perform proofs and calculations with appropriate mathematical rigor.

I am not able to understad why people seems to love this book my feelings, beeing mixed, are perhaps closer to hate.

The book is OK if (and only if) you previously know the matter but the lack of clear definitions, the excessive reliance in reader geometrical intuition, the conversational style of demos the long paragraphs describing obscure geometric objects, etc make it very difficult to follow if it is your first approach to AT.

On the other hand has useful insigths if you already know the matter.

If the purpouse of the author has really been to write a "readable" book (as he told us repeatedly) I think the attemp is a complete failure.

On the other hand the "Table of contents" is excellent and is a very good book for teachers, I think this is the reason of its popularity.

If you can afford the cost, purchase J Rotman "An introduction to Algebraic Topology" and you really will get a "readable" book