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Algebraic Topology, by Allen Hatcher
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In most major universities one of the three or four basic first-year graduate mathematics courses is algebraic topology. This introductory text is suitable for use in a course on the subject or for self-study, featuring broad coverage and a readable exposition, with many examples and exercises. The four main chapters present the basics: fundamental group and covering spaces, homology and cohomology, higher homotopy groups, and homotopy theory generally. The author emphasizes the geometric aspects of the subject, which helps students gain intuition. A unique feature is the inclusion of many optional topics not usually part of a first course due to time constraints: Bockstein and transfer homomorphisms, direct and inverse limits, H-spaces and Hopf algebras, the Brown representability theorem, the James reduced product, the Dold-Thom theorem, and Steenrod squares and powers.
- Sales Rank: #104595 in Books
- Brand: Brand: Cambridge University Press
- Published on: 2001-12-03
- Original language: English
- Number of items: 1
- Dimensions: 9.96" h x 1.26" w x 6.97" l, 2.25 pounds
- Binding: Paperback
- 556 pages
- Used Book in Good Condition
Review
"Algebraic topoligy books that emphasize geometrical intuition usually have only a modest technical reach. Remarkably, Hatcher (Cornell Univ.) offers a highly geometrical treatment that neverheless matches the coverage of, e.g., Edwin Henry Spanier's very formidable and identically titled classic work... He promises two advanced companion volumes, one on spectral sequences, one on vector bundles. One anticipates the combined treatise doing for algebraic topology what Michael Spivak's magisterial five-volume set did for differential geometry." Choice
Most helpful customer reviews
59 of 71 people found the following review helpful.
Mixed Feelings
By Rehan Dost
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.
42 of 51 people found the following review helpful.
More Hand-Waving Than an Orchestra Conductor
By Linear Functional
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.
13 of 15 people found the following review helpful.
Excellent book for geometers
By Tom Braden
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.
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