Algebraic Topology: An Intuitive Approach (Translations of Mathematical Monographs) (Paperback)

~ Hajime Sato (Author), Kiki Hudson (Translator) "Throughout this book a map means a continuous map..." (more)
Key Phrases: homology axioms, integral cohomology groups, cohomology spectral sequence (more...)

Editorial Reviews

Review

A nice supplement for a topology course. -- American Mathematical Monthly

The present book is an interesting, perhaps radical, survey of current algebraic topology. -- Mathematical Reviews

Product Description

The single most difficult thing one faces when one begins to learn a new branch of mathematics is to get a feel for the mathematical sense of the subject. The purpose of this book is to help the aspiring reader acquire this essential common sense about algebraic topology in a short period of time. To this end, Sato leads the reader through simple but meaningful examples in concrete terms. Moreover, results are not discussed in their greatest possible generality, but in terms of the simplest and most essential cases.

In response to suggestions from readers of the original edition of this book, Sato has added an appendix of useful definitions and results on sets, general topology, groups and such. He has also provided references.

Topics covered include fundamental notions such as homeomorphisms, homotopy equivalence, fundamental groups and higher homotopy groups, homology and cohomology, fiber bundles, spectral sequences and characteristic classes. Objects and examples considered in the text include the torus, the Möbius strip, the Klein bottle, closed surfaces, cell complexes and vector bundles.

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Product Details

• Paperback: 118 pages
• Publisher: American Mathematical Society (February 1999)
• Language: English
• ISBN-10: 0821810464
• ISBN-13: 978-0821810460
• Product Dimensions: 7.8 x 5.5 x 0.3 inches
• Shipping Weight: 4.8 ounces (View shipping rates and policies)
• Average Customer Review: 4.2 out of 5 stars  See all reviews (5 customer reviews)
• Amazon.com Sales Rank: #1,089,811 in Books (See Bestsellers in Books)

Customer Reviews

Most Helpful Customer Reviews

13 of 13 people found the following review helpful:

4.0 out of 5 stars Excellent accompaniment to Hatcher, January 27, 2004

By A Customer

As a student just wading into the realm of Algebraic Topology, this book has been a wonderful companion. If you are looking for a book that will lay out precise proofs of theorems and get down to the nity-gritty, this book is not for you. However, if you are new to A.T. as I am, and want a book that will give you a nice easy to follow introduction to a topic before wading into your thicker text, then this book will help you tremendously. For instance, reading the chapters regarding CW-complexes and Homotopy in Sato, although thin and easy to follow (you will have to do a little bit of lifting, but not too much), helped me to more easily digest what was to come in Chapters 0 and 1 of Hatcher (which I also highly recommend, incidentally). It always helps to read material taken from a different person's perspective, and Sato has truly made Algebraic topology more transparent in this brief overview.

12 of 12 people found the following review helpful:

4.0 out of 5 stars Good Supplementary Reading, March 5, 2002

By	A. Ali "Harkonnen" (Minneapolis, MN USA) - See all my reviews (REAL NAME)

This modest 118-page book would best accompany one of the standard graduate texts -- Spanier, Dold, Switzer, Massey, Husemoller,Maunder, Munkres, Bott and Tu, Bredon, or Greenberg and Harper. It can't be used as a text.

The book presents the most basic ideas pertaining to homotopy, homology, cohomology, fibre bundles, spectral sequences, and characteristic classes. The emphasis is on simple examples and simple calculations to demonstrate what is going on. Rigorous definitions, proofs, and even frequently even the statements of theorems, are avoided.

One good aspect of the treatment is the axiomatic presentation of homology and cohomology a la Eilenberg and Steenrod. Some of the essential material is also presented, e.g. the cup product that gives a ring structure to the cohomology group, the Kunneth theorem, the Universal Coefficient theorem, and so on.

The book would afford a bird's-eye view, a conspectus, to a bright undergraduate or beginning graduate student. It goes without saying, of course, that this is for motivation, and it doesn't replace the hard technical grind required to master the subject.

The book suffers in comparison to the one by Fomenko, Fuchs and Gutenmacher (Homotopic Topology), but that, alas, can't be had for love or money.

3 of 3 people found the following review helpful:

4.0 out of 5 stars "Intuition" more a prerequisite than a result, January 24, 2008

By	A. J. Sutter (Tokyo, Japan) - See all my reviews (TOP 1000 REVIEWER)    (REAL NAME)

This was my first crack at algebraic topology, self-studying long after my university days. I thought I'd read this book as a warm-up for Bott & Tu. The book is written in the laid-back discursive style that is one of the more charming attributes of Japanese math books. It's also short, and the author has provided solutions or hints for most of the modest exercises. At first glance it looks like a pleasant way to spend a few afternoons in a cafe.

But appearances can be deceiving. The intuitions referred to are not those of a typical beginner. No less disingenuous is the occasional advice saying it's OK to skip a chapter: the concepts and definitions are inevitably used in later ones. These are what Japanese call "tatemae" -- the stuff that's said just for the sake of making a good (or at least better) impression.

The reviewers who suggested that the book supplements more advanced texts are closer to the mark. I found myself resorting to Bott & Tu and Hatcher to clear up concepts presented in this one, when I'd expected the reverse. E.g., Sato's explanation of exact sequences was ultra-concise and rather puzzling, while the two books I mentioned and even Wikipedia are quite helpful about them. B&T also uses many more diagrams when it counts, including in some clear and beautiful proofs about homotopies that Sato presents in a drier style. Nor does Sato do a good job of motivating why cohomology is more useful than homology; for all its shortcomings (including lack of coverage of De Rham cohomology), even 1970's-vintage Maunder does a better job at this. (The first few pages of Hatcher's Chapter 3 are even better on that point, but that's what one would expect from such a humongous book.) And the diagrams accompanying the description of fiber bundles don't even indicate a fiber; there are many more "intuitive" explanations of this topic elsewhere.

This may be a good tool for reinforcing material you have learnt or are learning from another source. But you might not find it as suitable for a free-standing introduction as the title and a casual inspection might suggest. I give it 3.5-4 stars instead of 3 as a handicap, considering my own amateurism, and also because of the good range of topics touched on.