Algebraic Topology (Paperback)

~ (Author) "The aim of this short preliminary chapter is to introduce a few of the most common geometric concepts and constructions in algebraic topology..." (more)
Key Phrases: cellular boundary maps, orthonormal tangent vector fields, trivial boundary maps, Basic Constructions Section, Covering Spaces Section, Formula Section (more...)

Editorial Reviews

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

Product Description

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.

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Product Details

• Paperback: 550 pages
• Publisher: Cambridge University Press; 1 edition (November 15, 2001)
• Language: English
• ISBN-10: 0521795400
• ISBN-13: 978-0521795401
• Product Dimensions: 9.9 x 7 x 1.6 inches
• Shipping Weight: 2.2 pounds (View shipping rates and policies)
• Average Customer Review: 3.8 out of 5 stars  See all reviews (17 customer reviews)
• Amazon.com Sales Rank: #48,145 in Books (See Bestsellers in Books)

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Customer Reviews

Most Helpful Customer Reviews

44 of 51 people found the following review helpful:

5.0 out of 5 stars You would not regret if you buy this., February 26, 2003

By A Customer

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.

58 of 71 people found the following review helpful:

5.0 out of 5 stars The Last Text on Introductory Algebraic Topology, January 4, 2002

By	Pisheng Ding (New York, NY United States) - See all my reviews

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.

20 of 23 people found the following review helpful:

3.0 out of 5 stars Mixed Feelings, February 4, 2007

By	Rehan Dost (Canada) - See all my reviews (REAL NAME)

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.