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Customer Reviews (9)

If U want generalization out to infinity, this is it for you, in algebraic topology basics.
This former professor, and sometime head of the math department at U of Chicago, is quite a fellow. He is so DEEP that I many times didn't have a clue about some of his books. But this one seems more down at my level of intelligence, even though it is a whirlwind romance so to speak with algebraic topology basics.

Have at it, if you like the whirlwind!

The Title Says it All
I have always believed that the "goodness" of a mathematical textbook is inversely proportional to its length. J. P. May's book "A Concise Course in Algebraic Topology" is a superb demonstration of this. While the book is indeed extremely terse, it forces the reader to thoroughly internalize the concepts before moving on. Also, it presents results in their full generality, making it a helpful reference work.

The opposite of Hatcher
This book is clear, and direct.It tells you want you want to know.

Lucid and elegant, but not for beginners
This tiny textbook is well organized with an incredible amount of information. If you manage to read this, you will have much machinery of algebraic topology at hand. But, this book is not for you if you know practically nothing about the subject (hence four stars). I believe this work should be understood to have compiled "what topologists should know about algebraic topology" in a minimum number of pages.

A Unique and Necessary Book
Ones first exposure to algebraic topology should be a concrete and pictorial approach to gain a visual and combinatorial intuition for algebraic topology. It is really necessary to draw pictures of tori, see the holes, and then write down the chain complexes that compute them. Likewise, one should bang on the Serre Spectral Sequence with some concrete examples to learn the incredible computational powers of Algebraic Topology. There are many excellent and elementary introductions to Algebraic Topology of this type (I like Bott & Tu because of its quick introduction of spectral sequences and use of differential forms to bypass much homological algebra that is not instructive to the novice).

However, as Willard points out, mathematics is learned by successive approximation to the truth. As you becomes more mathematically sophisticated, you should relearn algebraic topology to understand it the way that working mathematicians do. Peter May's book is the only text that I know of that concisely presents the core concepts algebraic topology from a sophisticated abstract point of view. To make it even better, it is beautifully written and the pedagogy is excellent, as Peter May has been teaching and refining this course for decades. Every line has obviously been thought about carefully for correctness and clarity.

As an example, ones first exposure to singular homology should be concrete approach using singular chains, but this ultimately doesn't explain why many of the artificial-looking definitions of singular homology are the natural choices. In addition, this decidedly old-fashioned approach is hard to generalize to other combinatorial constructions.

Here is how the book does it: First, deduce the cellular homology of CW-complexes as an immediate consequence of the Eilenberg-Steenrod axioms. Considering how one can extend this to general topological spaces suggests that one approximate the space by a CW-complex. Realization of the total singular complex of the space as a CW-complex is a functorial CW-approximation of the space. As the total singular complex induces an equivalence of (weak) homotopy categories and homology is homotopy-invariant, it is natural to define the singular homology of the original space to be the homology of the total singular complex. Although sophisticated, this is a deeply instructive approach, because it shows that the natural combinatorial approximation to a space is its total singular complex in the category of simplicial sets, which lets you transport of combinatorial invariants such as homology of chain complexes. This approach is essential to modern homotopy theory. ... Read more

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. ... Read more

Customer Reviews (19)

Terrible textbook
This book is horrible if regarded as a mathematics book. Like previous reviewers I feel there is a total lack of clarity and rigor. Definitions are lacking, perhaps the author feels it is better to provide a "intuitive" feel for the material, than just definingg things. He fails miserably. The fact that what we are really dealing with in this subject are functors((co)homology, homotopy ) is nearly absent from the text. Instead drawings and pictures that are meant to provide "geometric" feel are supplanted.
I would state that this book attempts to teach how to compute in and use the theory than have you understand how the theory is built. It is a book for using the oven, not understanding how it works.

More Hand-Waving Than an Orchestral Conductor
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.

Really bad as a "readable" texbookbut good reference
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 handthe "Table of contents" is excellent and is a very good book for teachers,I think this is the reason of itspopularity.

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

amazing book, but caveat emptor
I think that Allen Hatcher has given us all something very valuable in this book.If you are like me, you've had those moments when reading in a math book when you read a sentence, and your eyes shoot open and you suddenly feel like someone has been standing behind you that you never knew was there.There are lots of those kinds of sentences in this book.On the other hand, I view it as a supplement to a book like Munkres or Bredon that provides the rigor necessary to allow the learner to figure out the topologist's geometric language.I have used these three and found them to compliment one another well.

excellent modern introduction
This is an excellent introduction to the subject. It's affordable, well-written, and the topics are well chosen. The presentation is modern, but includes enough intuition that the fairly naive reader (e.g., me) can see the point of things. I needed to (re)learn topology for a research project I was part of in the intersection of math/CS/statistics and this book was a big help. I wish that he had included simplicial sets in the topics, because I like the way he writes and would like to have a more elementary exposition tied to the rest of the book (I eventually found an expository paper that did a pretty good job, but worked out examples would still help with that topic), but it can't include everything. I highly recommend this book to anyone trying to get started in this fascinating subject. It will just scratch the surface, but it does a good job of that. ... Read more

Product Description
Intended for use both as a text and a reference, this book is an exposition of the fundamental ideas of algebraic topology. The first third of the book covers the fundamental group, its definition and its application in the study of covering spaces. The focus then turns to homology theory, including cohomology, cup products, cohomology operations, and topological manifolds. The remaining third of the book is devoted to Homotropy theory, covering basic facts about homotropy groups, applications to obstruction theory, and computations of homotropy groups of spheres. In the later parts, the main emphasis is on the application to geometry of the algebraic tools developed earlier. ... Read more

Customer Reviews (6)

Pioneering text
This book was an incredible step forward when it was written (1962-1963). Lefschetz's Algebraic Topology (Colloquium Pbns. Series, Vol 27) was the main text at the time. A large number of other good to great books on the subject have appeared since then, so a review for current readers needs to address two separate issues: its suitability as a textbook and its mathematical content.
I took the course from Mr. Spanier at Berkeley a decade after the text was written.He was a fantastic teacher - one of the two best I've ever had (the other taught nonlinear circuit theory). We did NOT use this text, except as a reference and problem source. He had pretty much abandonded the extreme abstract categorical approach by then.The notes I have follow the topical pattern of the book, but are so modified as to be essentially a different book, especially after covering spaces and the first homotopy group. His statement was that his treatment had changed since the subject had changed significantly.So much more has changed since then that I would not recommend this book as a primary text these days. Bredon's Topology and Geometry (Graduate Texts in Mathematics) is much better suited to today's student.
So, why did I give it four stars?First, notice that it splits stylewise into three segments, corresponding the treatment of its material in a three quarter academic year.The first three chapters (intro, covering spaces, polyhedral) have really not been superceded in a beginning text.Topics are covered very thoroughly, aiding the student new to the subject.The next three chapters (homology) are written much with much less explanation included - indeed, some areas leave much to the reader to discover and, consequently, aren't very helpful if the instructor doesn't fill in the details (the text expects a rather rapid mathematical maturation from the first part - too much of a ramp in my opinion), but the text is comprehensive.The last section (homotopy theory, obstruction theory and spectral sequences) should just be treated as a reference - it'd be hard to find all this material in such a compact form elsewhere and the obstruction theory section has fantastic coverage of what was known as of the writing of this book.It's way too terse for a novice to learn from and there are some great books out there these days on the material.

For reference ONLY
This book is a highly advanced and very formal treatment of algebraic topology and meant for researchers who already have considerable background in the subject. A category-theoretic functorial point of view is stressed throughout the book, and the author himself states that the title of the book could have been "Functorial Topology". It serves best as a reference book, although there are problem sets at the end of each chapter.

After a brief introduction to set theory, general topology, and algebra, homotopy and the fundamental group are covered in Chapter 1. Categories and functors are defined, and some examples are given, but the reader will have to consult the literature for an in-depth discussion. Homotopy is introduced as an equivalence class of maps between topological pairs. Fixing a base point allows the author to define H-spaces, but he does not motivate the real need for using pointed spaces, namely as a way of obtaining the composition law for the loops in the fundamental group. By suitable use of the reduced join, reduced product, and reduced suspension, the author shows how to obtain H-groups and H co-groups. The fundamental group is defined in the last section of the chapter, and the author does clarify the non-uniqueness of the fundamental group based at different points of a path-connected space.

Covering spaces and fibrations are discussed in the next chapter. The author does a fairly good job of discussing these, and does a very good job of motivating the definition of a fiber bundle as a generalized covering space where the "fiber" is not discrete. The fundamental group is used to classify covering spaces.

In chapter 3 the author gets down to the task of computing the fundamental group of a space using polyhedra. Although this subject is intensely geometrical. only six diagrams are included in the discussion.

Homology is introduced via a categorical approach in the next chapter. Singular homology on the category of topological pairs and simplicial homology on the category of simplicial pairs. The author begins the chapter with a nice intuitive discussion, but then quickly runs off to an extremely abstract definition-theorem-proof treatment of homology theory. The discussion reads like one straight out of a book on homological algebra.

This approach is even more apparent in the next chapter, where homology theory is extended to general coefficient groups. The Steenrod squaring operations, which have a beautiful geometric interpretation, are instead treated in this chapter as cohomology operations. The logic used is impeccable but the real understanding gained is severely lacking.

General cohomology theory is treated in the next chapter with the duality between homology and cohomology investigated via the slant product. Characteristic classes, so important in applications, are discussed using algebraic constructions via the cup product and Steenrod squares. Characteristic classes do have a nice geometric interpretation, but this is totally masked in the discussion here.

The higher homotopy groups and CW complexesare discussed in Chapter 7, but again, the functorial approach used here totally obscures the underlying geometrical constructions.

Obstruction theory is the subject of Chapter8, with Eilenberg-Maclane spaces leading off the discussion. The author does give some motivation in the first few paragraphs on how obstructions arise as an impediment to a lifting of a map, but an explicit, concrete example is what is needed here.

The last chapter covers spectral sequences as applied to homotopy groups of spheres. More homological algebra again, and the same material could be obtained (and in more detail) in a book on that subject.

Definitely not for beginners
I gave Spanier only three stars not because I think it is a bad book: as the previous two reviewers have pointed out, Spanier is a comprehensive (and still good) account of the subject, but is by no means for beginners. Now that more user-frinedly ones like Bredon, Fomenko-Novikov, and Hatcher (forthcoming) are available,it would hardly justify giving it four or five stars.And for reference purposes, there is a small (and sometimes too terse) but attractive account by May that covers topics not touched by Spanier.

Excellent reference, poor textbook
This book is terrific as a reference for those who already know the subject, but if you teach algebraic topology it would be dangerous to use it as a graduate text (unless you're willing to supplement it extensively).The basic problem is that Spanier does not teach students how to computeeffectively because his abstract, high-powered algebraic approach obscuresthe underlying geometry, which is not developed at all. Here I'd recommendthe books by Munkres, or Greenberg; even the old-fashioned treatment ofLefschetz, with its explicit and rather cumbersome treatment of cohomology,could serve as an antidote to Spanier. Somewhere, the student has toacquire a good intuitive feeling for the geometry underlying the subject(the same can be said of algebraic geometry -- here earlier work (e.g., ofthe Italian school, Weil's old book on intersection theory, ...) should notbe neglected entirely in favor of Grothendieck et al., for somethingessential is lost)

That said, if you already know the subject Spanier'sbook is an excellent reference. Even here, though, you'll need to providesome details toward the ends of the later chapters. Each chapter starts outrelatively easily and works up to a crescendo, the treatment becomingterser and more advanced.

I give it four stars (5 for mathematicalquality, 3 for usefulness as a text). The first three chapters deal withcovering spaces and fibrations; the middle three with (co)homology andduality; the last three with general homotopy theory, obstruction theory,and spectral sequences. Some of Serre's classical results on finitenesstheorems for homotopy groups are presented.

Excellent reference, poor textbook
This book is terrific as a reference for those who already know thesubject, but if you teach algebraic topology it would be dangerous to useit as a graduate text (unless you're willing to supplement it extensively). The basic problem is that Spanier does not teach students how to computeeffectively because his abstract, high-powered algebraic approach obscuresthe underlying geometry, which is not developed at all.Here I'd recommendthe books by Munkres, or Greenberg; even the old-fashioned treatment ofLefschetz, with its explicit and rather cumbersome treatment of cohomology,could serve as an antidote to Spanier.Somewhere, the student has toacquire a good intuitive feeling for the geometry underlying the subject(the same can be said of algebraic geometry -- here earlier work (e.g., ofthe Italian school, Weil's old book on intersection theory, ...) should notbe neglected entirely in favor of Grothendieck et al., for somethingessential is lost)

That said, if you already know the subject Spanier'sbook is an excellent reference.Even here, though, you'll need to providesome details toward the ends of the later chapters.Each chapter startsout relatively easily and works up to a crescendo, the treatment becomingterser and more advanced.

I give it four stars (5 for mathematicalquality, 3 for usefulness as a text).The first three chapters deal withcovering spaces and fibrations; the middle three with (co)homology andduality; the last three with general homotopy theory, obstruction theory,and spectral sequences.Some of Serre's classical results on finitenesstheorems for homotopy groups are presented. ... Read more

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Product Description
Thorough, modern treatment, essentially from a homotopy theoretic viewpoint. Topics include homotopy and simplicial complexes, the fundamental group, homology theory, homotopy theory, homotopy groups and CW-Complexes and other topics. Each chapter contains exercises and suggestions for further reading. 1980 corrected edition.
... Read more

Customer Reviews (3)

Solid
A very interesting book which I enjoyed. I particularly found useful the "crash review" in algebra and analysis which functioned as a useful reference throughout the book.

Shouldn't be your first text in algebraic topology.
It is a decent book in algebraic topology, as a reference.At first, I found this textbook rather hard to read. Too manylemmas, theorems, etceteras. Three suggestions:

1. Needs more pictures, especially for the simplicialhomology Chapter.

2. CW complexes should be covered before duality and not after.

3. Needs more examples and exercises.

Overall, the book is very good, if you have already someexperience in Algebraic Topology. I found that the Croom'sbook "Basic concepts of Algebraic Topology" is an excellent first textbook. Too bad it is out of print, since it is very popular, every time I get it from the library, someone else recalls it. The combination of these two books probablyis the right thing to have: Maunder's book picks up whereCroom has left you.

Not bad.
Maunder's text may not be the "best" book on algebraic topology, but I still recommend this one to those who find other more advanced texts like Spanier rather inaccessible. Warning: the chapter on cohomology andduality is not very well-organaized (compared to other chapters), so youmay want to consult Bredon's book instead. ... Read more

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This book is a well-informed and detailed analysis of the problems and development of algebraic topology, from Poincaré and Brouwer to Serre, Adams, and Thom. The author has examined each significant paper along this route and describes the steps and strategy of its proofs and its relation to other work. Previously, the history of the many technical developments of 20th-century mathematics had seemed to present insuperable obstacles to scholarship. This book demonstrates in the case of topology how these obstacles can be overcome, with enlightening results.... Within its chosen boundaries the coverage of this book is superb. Read it!

—MathSciNet

Customer Reviews (1)

More than a mere "history".
This book painstakingly describes and explains algebraic topology in the chronological order of its development. I quite agree with Glen Bredon's remark in his "Geometry and Topology" that goes like "this is more than a history and should be in the bookshelf of every student of topology"(not word-for-word, as the citation is done offhand). ... Read more

Product Description
This book introduces the important ideas of algebraic topology emphasizing the relation of these ideas with other areas of mathematics. Rather than choosing one point of view of modern topology (homotropy theory, axiomatic homology, or differential topology, say) the author concentrates on concrete problems in spaces with a few dimensions, introducing only as much algebraic machinery as necessary for the problems encountered. This makes it possible to see a wider variety of important features in the subject than is common in introductory texts; it is also in harmony with the historical development of the subject. The book is aimed at students who do not necessarily intend on specializing in algebraic topology.

The first part of the book emphasizes relations with calculus and uses these ideas to prove the Jordan curve theorem. The study of fundamental groups and covering spaces emphasizes group actions. A final section gives a taste of the generalization to higher dimensions. ... Read more

Customer Reviews (3)

A book of ideas
This book is an introduction to algebraic topology that is written by a master expositor. Many books on algebraic topology are written much too formally, and this makes the subject difficult to learn for students or maybe physicists who need insight, and not just functorial constructions, in order to learn or apply the subject. Anyone learning mathematics, and especially algebraic topology, must of course be expected to put careful thought into the task of learning. However, it does help to have diagrams, pictures, and a certain degree of handwaving to more greatly appreciate this subject.

As a warm-up in Part 1, the author gives an overview of calculus in the plane, with the intent of eventually defining the local degree of a mapping from an open set in the plane to another. This is done in the second part of the book, where winding numbers are defined, and the important concept of homotopy is introduced. These concepts are shown to give the fundamental theorem of algebra and invariance of dimension for open sets in the plane. The delightful Ham-Sandwich theorem is discussed along with a proof of the Lusternik-Schnirelman-Borsuk theorem. I would like to see a constructive proof of this theorem, but I do not know of one.

Part 3 is the tour de force of algebraic topology, for it covers the concepts of cohomology and homology. The author pursues a non-traditional approach to these ideas, since he introduces cohomology first, via the De Rham cohomology groups, and these are used to proved the Jordan curve theorem. Homology is then effectively introduced via chains, which is a much better approach than to hit the reader with a HOM functor.Part 4 discusses vector fields and the discussion reads more like a textbook in differential topology with the emphasis on critical points, Hessians, and vector fields on spheres. This leads naturally to a proof of the Euler characteristic.

The Mayer-Vietoris theory follows in Part 5, for homology first and then for cohomology.

The fundamental group finally makes its appearance in Part 6 and 7, and related to the first homology group and covering spaces. The author motivates nicely the Van Kampen theorem. A most interesting discussion is in part 8, which introduces Cech cohomology. The author's treatment is the best I have seen in the literature at this level. This is followed by an elementary overview of orientation using Cech cocycles.

All of the constructions done so far in the plane are generalized to surfaces in Part 9. Compact oriented surfaces are classified and the second de Rham cohomology is defined, which allows the proof of the full Mayer-Vietoris theorem.

The most important part of the book is Part 10, which deals with Riemann surfaces. The author's treatment here is more advanced than the rest of the book, but it is still a very readable discussion. Algebraic curves are introduced as well as a short discussion of elliptic and hyperelliptic curves.

The level of abstraction increases greatly in the last part of the book, where the results are extended to higher dimensions. Homological algebra and its ubiquitous diagram chasing are finally brought in, but the treatment is still at a very understandable level.

For examples of the author's pedagogical ability, I recommend his book Toric Varieties, and his masterpiece Intersection Theory.

This is one of the great algebraic topology books!
This is a book for people who want to think about topology, not just learna lot of fancy definitions and then mechanically compute things. Fulton hasput the essence of Algebraic Topology into this book, much in the way MikeArtin has done with his "Algebra". In my opinion, he should winsome sort of expository award for it.

Probably better as a 2nd (or 3rd) course rather than 1st
Most mathematicians, I suspect, can relate to the "colloquium experience": the first minutes of a lecture go easily, followed by twenty or thirty of real edification, concluded by ten to fifteen of feeling lost.I regret to say that this was pretty much my experience with the book.Fulton writes with unusual enthusiasm and the first two- thirds of the book is a joy to read, even while it is real work.I imagine that he must be a remarkable teacher in person.He has some threads such as winding numbers and the Mayer-Vietoris Sequence that continue throughout the book, bringing unity to a wide selection of topics.There are a number of applications of the subject to other areas, such as complex analysis (Riemann surfaces) and algebraic geometry (the Riemann-Roch Theorem), to name only two.There are particularly interesting illustrations of the Brouwer Fixed Point Theorem and related results.Unfortunately, there are two rather major reservations I have about the book.The first, already alluded to, is that it seemed to me to become precipitously difficult towards the end.The second is that this book would be excellent for a second or perhaps third course in the subject rather than a first.While the topics he covers are interesting in their own right, I still favor a more "standard" approach covering simplicial complexes, homology, CW complexes, and homotopy theory with higher homotopy groups, such as in the books by Maunder, Munkres, or Rotman (the last two of which I recommend unreservedly).It is true that Fulton has some coverage these topics, and a particularly extensive discussion of group actions and G-spaces, but he presupposes a background or ability that the novice to algebraic topology is unlikely to have.I would like to recommend this book, as I found it very edifying, but it seems better suited for one with some prior acquaintance to the subject. ... Read more

Product Description
Massey's well-known and popular text is designed to introduce advanced undergraduate or beginning graduate students to algebraic topology as painlessly as possible. The principal topics treated are 2-dimensional manifolds, the fundamental group, and covering spaces, plus the group theory needed in these topics. The only prerequisites are some group theory, such as that normally contained in an undergraduate algebra course on the junior/senior level, and a one-semester undergraduate course in general topology. From the reviews: "This book is highly recommended: as a textbook for a first course in algebraic topology and as a book for selfstudy. The spirit of algebraic topology and of good mathematics is present at every page of this almost perfect book." Bulletin de la Société Mathématique de Belgique#1 ... Read more

Customer Reviews (1)

one of the best books on algebraic topology
This is a charming book on algebraic topology.It doesnt teach homology or cohomology theory,still you can find in it:about the fundamental group, the action of the fundamental group on the universal cover (and the concept of the universal cover),the classification of surfaces and a beautifull chapter on free groups and the way it is related to Van-kampen theorem .After reading this book you will have a strong intuitive picture on "what is algebraic topology all about"(well at list on part of algebraic topology)read it an enjoy it!!!. ... Read more

Product Description

Combinatorial algebraic topology is a fascinating and dynamic field at the crossroads of algebraic topology and discrete mathematics. This volume is the first comprehensive treatment of the subject in book form. The first part of the book constitutes a swift walk through the main tools of algebraic topology, including Stiefel-Whitney characteristic classes, which are needed for the later parts. Readers - graduate students and working mathematicians alike - will probably find particularly useful the second part, which contains an in-depth discussion of the major research techniques of combinatorial algebraic topology. Our presentation of standard topics is quite different from that of existing texts. In addition, several new themes, such as spectral sequences, are included. Although applications are sprinkled throughout the second part, they are principal focus of the third part, which is entirely devoted to developing the topological structure theory for graph homomorphisms. The main benefit for the reader will be the prospect of fairly quickly getting to the forefront of modern research in this active field.

Customer Reviews (1)

Help, where are the editors?
The book is poorly edited.
From the first chapter on, it is very difficult to decipher what the author is trying to say, because of linguistic and typographic errorsThere are evidently very interesting and useful things in this book, if you are interested in topology, homology and want to know about recent work on simplicial and other complexes, and especially if you are interested in applications to graph theory;but you have to be prepared to work very hard to find out what it is.With a basic introduction to simplicial homology at your side, the first few chapters should make sense. ... Read more

Product Description
Springer-Verlag began publishing books in higher mathematics in 1920, when the series Grundlehren der mathematischen Wissenschaften, initially conceived as a series of advanced textbooks, was founded by Richard Courant. A few years later, a new series Ergebnisse der Mathematik und Ihrer Grenzgebiete, survey reports of recent mathematical research, was added. Of over 400 books published in these series, many have become recognized classics and remain standard references for their subject. Springer is reissueing a selected few of these highly successful books in a new, inexpensive softcover edition to make them easily accessible to younger generations of students and researchers. ... Read more

Customer Reviews (2)

Singular homology, products and manifolds
This is a book mainly about singular (co)homology. To be able to do calculations on more complex objects, CW complexes are introduced. The book concentrates on products and manifolds. It is aimed at a graduate level audience and in that context it is self contained. Homological algebra is developed up to the level needed in the text. There is a fair amount of examples and exercises.

I am really curious about the economists, mentioned in the editorial review, using this text as a standard reference.

Elgant treatment of homology theory.
Though entitled "Algebraic Topology", this text covers only (co)homology theory. You should look for other texts if your interest is in homotopy theory. This being said, the treatment is elegant (at least forits time of publication), especially the chapter covering the mothod ofacylcic models. ... Read more

Customer Reviews (4)

Not bad..
It's worth noting that there are quite a few in number of books out there on introductory (i.e. a first course in) alg. top.
In particular, I should mention that the book by Rotman and sizeable portions of Bredon, "Geometry and Topology" can serve as good supplementary reading. I still don't think \pi_1 should have been left out; although one *could* refer to theprequel,there's still more to be desired by way of completeness, if anything, as this book is intended for beginners. For instance, the relation between the fundamental group and the first homology group would have certainly shed some light on these seemingly (at first glance, anyway) disparateinvariants (as it isheavy-going onthe (co)homological apparatus altogether).

Munkres is byno means encyclopaedic, which is good, in opposition to, say, Spanier or Whitehead, and certainly warrants attention to worked-out examples in detail and some (not-so) routine exercises which makes this book accessible to wider mathematical audiences wishing to learn a little about this fascinating subject.

A little incomplete
This well written text is one of the standard references in algebraic topology courses because of its conciseness, and I find it very useful as a reference text.

However I think it is a little incomplete because of several reasons.

(1)It pays no attention to one basic concept ofalgebraic topology: the fundamental group.

(2) It doesn't cover ^Cechhomology, important in other areas, like dimension theory forexample.

(3) It doesn't stress the most important feature of algebraictopology: its connection to other areas of mathematics (analysis,differential geometry, etc.).

(4) Its list of references is too short,and lacks almost completely HISTORICAL references which are alwaysimportant to become an expert in any field.

Conclusion: a good referenceon homology and cohomology essentials, but not "the" reference onalgebraic topology as a whole.

The book binding is horrible
The material in the book (homology and cohomology theory, universal coefficient theorems, Kunneth theorem, duality in manifolds, applications to classical theorems of point-set topology) is for the most part solid. However...

- Munkres really belabors the simplicial theory, and it getsto be quite painful (especially the*CHAPTER* on the topologicalinvariance of simplicial homology groups).

- Some very important topics(homotopy theory, fiber bundles) are not at all discussed.

- The bookbinding is horrible -- my copy is in two pieces, with several loose pages,and I don't think the hardcover edition is still in print.

Excellent text on homology and cohomology
Algebraic topology is a tough subject to teach, and this book does a very good job.Some prerequisites, however, are essential:

* point set topology (e.g. in Munkres' Topology)

* Abstract algebra

* Mathematicalmaturity to be willing to follow a definition and argument even when itseems like a weird side-track

In addition, this would not be the firstbook I would recommend to those interested in algebraic topology.Firstmight be Massey's "Algebraic Topology: and Introduction" thatintroduces the fundamental group (conceptually easier than homology andcohomology).

At some point, however, a prospective student in topologywill have to learn homological algebra and this provides the most concreteapproach I know to the subject.

Algebraic topology is a lot of fun, butmany of the previous textbooks had not given that impression.This onedoes. ... Read more

Product Description
Developed from a first-year graduate course in algebraic topology, this text is an informal introduction to some of the main ideas of contemporary homotopy and cohomology theory. The materials are structured around four core areas: de Rham theory, the Cech-de Rham complex, spectral sequences, and characteristic classes. By using the de Rham theory of differential forms as a prototype of cohomology, the machineries of algebraic topology are made easier to assimilate. With its stress on concreteness, motivation, and readability, this book is equally suitable for self-study and as a one-semester course in topology. ... Read more

Customer Reviews (7)

a masterpiece of exposition
This is a beautiful book which I have read and re-read with much profit and pleasure over the years.It presents topics in a very unusual order, which minimizes boring technicalities and develops intuition.Everything is very concrete and explicit, with lots of nice pictures and diagrams.

The book begins with a clear and concise treatment of deRham cohomology.If one hasn't seen differential forms before, then it might be a bit too brief and one might need to supplement it.But if one is comfortable with differential forms, then de Rham theory is a setting in which theorems such as Poincare duality can be proved with a minimum of pain.It is also very edifying to see the Poincare dual of a submanifold as a differential form.There is then a natural transition to Cech cohomology and double complexes.With this as a warmup, it is then a small additional step to spectral sequences (although the derived couple approach used here is perhaps not the most elementary possible).This machinery is then used to discuss an assortment of topics in homotopy theory and characteristic classes, which always sticks to the most important points without getting bogged down in technicalities.

It is highly unusual that the definition of singular homology only comes after the introduction of spectral sequences!This book might be best appreciated if one has some familiarity with singular homology and wants to better understand its geometric meaning.

Despite the avoidance of technicalities, the book is carefully written, although there is the occasional sign error.For example, the sign given for the Lefschetz fixed point theorem is wrong for odd-dimensional manifolds;try it for the circle and you will see.(Several other books make the same mistake.)

So far so good
I'm reading this book with my advisor.So far I've read through the first
five sections.My advisor is having me read this because he wanted me
to "read a really good book"So far I have no complaints. The arguments
are extremely clear and the book itself has a very smooth structure (no pun intended).

good book
It is a well written book. Useful for those whois learing algebric topology.

wonderfully clear, useful book
I agree with the other reviews, and only wanted to add to one of them that in regard to examples of chern classes, I believe they also use the whitney formula to derive the chern classes of a hypersurface from that of projective space, which really expands the realm of examples significantly.

This was all I needed in writing my notes on the Riemann Roch theorem for hypersurfaces in 3 and 4 space, for instance.I felt I knew little about concrete chern classes, but I was able to take the presentation in this book and use it for my purposes immediately.

A unique mathematics book
This book is almost unique among mathematics books in that it strives to ensure that you have the clearest picture possible of the topics under discussion.For example almost every text that discusses spectral sequences introduces them as a completely abstract machine that pumps out theorems in a mysterious way.But it turns out that all those maps actually have a clear meaning and Bott and Tu get right in there with clear diagrams showing exactly what those maps mean and where the generators of the various groups get mapped.It's clear enough that you can almost reach out and touch the things :-) And the same is true of all of the other constructions in the book - you always have a concrete example in mind with which to test out your understanding.

That makes this one of my all time favourite mathematics texts. ... Read more

Product Description
This book is intended to serve as a textbook for a course in algebraic topology at the beginning graduate level.The main topics covered are the classification of compact 2-manifolds, the fundamental group, covering spaces, singular homology theory, and singular cohomology theory.These topics are developed systematically, avoiding all unecessary definitions, terminology, and technical machinery.Wherever possible, the geometric motivation behind the various concepts is emphasized.The text consists of material from the first five chapters of the author's earlier book, ALGEBRAIC TOPOLOGY: AN INTRODUCTION (GTM 56), together with almost all of the now out-of-print SINGULAR HOMOLOGY THEORY (GTM 70).The material from the earlier books has been carefully revised, corrected, and brought up to date. ... Read more

Customer Reviews (2)

Good for newbies
Very nice way to start learning Alg topology. I am reading it for a class and it's been quite pleasant.

Excellent text on algebraic topology
The text contains material from the author's earlier two books; Algebraic Topology: An Introduction (GTM 56), and Singular Homology Theory (GTM 70). The book starts with an introductory chapter on 2-manifolds and thencontinues with the fundamental group; which is conceptually easier thanhomology, with which some books on algebraic topology start. The onlyprerequisite for this book is a basic knowledge of general topology; andthe book is easily accessible to anyone studying on his own. In short, Irecommend the book to anyone interested in algebraic topology. ... Read more

Product Description
The amount of algebraic topology a graduate student specializing in topology must learn can be intimidating. Moreover, by their second year of graduate studies, students must make the transition from understanding simple proofs line-by-line to understanding the overall structure of proofs of difficult theorems.To help students make this transition, the material in this book is presented in an increasingly sophisticated manner. It is intended to bridge the gap between algebraic and geometric topology, both by providing the algebraic tools that a geometric topologist needs and by concentrating on those areas of algebraic topology that are geometrically motivated.Prerequisites for using this book include basic set-theoretic topology, the definition of CW-complexes, some knowledge of the fundamental group/covering space theory, and the construction of singular homology. Most of this material is briefly reviewed at the beginning of the book.The topics discussed by the authors include typical material for first- and second-year graduate courses. The core of the exposition consists of chapters on homotopy groups and on spectral sequences. There is also material that would interest students of geometric topology (homology with local coefficients and obstruction theory) and algebraic topology (spectra and generalized homology), as well as preparation for more advanced topics such as algebraic $K$-theory and the s-cobordism theorem.A unique feature of the book is the inclusion, at the end of each chapter, of several projects that require students to present proofs of substantial theorems and to write notes accompanying their explanations. Working on these projects allows students to grapple with the "big picture", teaches them how to give mathematical lectures, and prepares them for participating in research seminars.The book is designed as a textbook for graduate students studying algebraic and geometric topology and homotopy theory. It will also be useful for students from other fields such as differential geometry, algebraic geometry, and homological algebra. The exposition in the text is clear; special cases are presented over complex general statements. ... Read more

Customer Reviews (1)

Godd for the second step.
This book is nicely written to explain "tools" of algebraic topology in a small number of pages. However, this is by *no* means a book for beginners, as it assumes its readers to have coverd a basic course.
For beginners I would reommend Hatcher "Algebraic Topology" or Bredon "Topology and Geometry" instead. ... Read more

Product Description
This book is a clear exposition, with exercises, of the basic ideas of algebraic topology: homology (singular, simplicial, and cellular), homotopy groups, and cohomology rings. It is suitable for a two-semester course at the beginning graduate level, requiring as a prerequisite a knowledge of point set topology and basic algebra. Although categories and functors are introduced early in the text, excessive generality is avoided, and the author explains the geometric or analytic origins of abstract concepts as they are introduced, making this book of great value to the student. ... Read more

Customer Reviews (3)

A readable alternative to Hatcher

It seems Allen Hatcher book is going to be the standard in AT
I strongly feel ¡¡ What a pity¡¡

The Rotman books is much much clearer and better, then my advice is

If you can afford the cost give up Hatcher's and get Rotman's

Rotman does it again.
Each text that I have read by Rotman is logically sound, well thought out, there are ample explanations, exercises as well as examples, and moreover, Rotman does an excellent job proving results.Sure he leaves the reader to prove certain results but, in general, all major concepts he will prove or, when it comes to familiar sticking points for students, Rotman will show that reader how to effectively prove these types of results.Now, Algebraic Topology is not an easy subject (actually it is a beautiful and far-reaching subject) and, depending upon the authors approach, the level of 'mathematical' maturity required can quickly escalate.Rotman's text is just above middle of the road with respect to this proverbial and undefined notion-'mathematical maturity'.Not as far-off as Spanier and not quite as gentle as Hatcher.For the reader who has this maturity or the necessary background, then Rotman's text is a must read provided you enjoy texts that follow the theorem-proof-theorem format.Furthermore, the logical consistecny with respect to how and when material is present to the reader places this text in a league of it's own.Without a doubt I could imagine any beginning graduate student or confident undergradute tackling this text on their own.For example, I am no math wizard but with only a background consisting of point-set topology with an introduction to the Fundamental Group, Abstract Algebra (Hungerford style) and Analysis (Rudin style) I was able to begin reading and, in particular, solving problems from Rotman's text while a senior undergraduate.For those of you who would like to learn the subject and learn it well but who are scared of this text (Springer can do that to people) I wouls strongly recommend pairing this text with Allen Hatchers or Part II of James Munkres' text depending on your level of enjoyment with respect to suffering your way through texts.In fact, I would suggest reading Munkres in its entirety since, this approach would properly prepare your for Rotman's text and the transition would be seamless.Finally, if, while reading this text you find yourself feeling lost during the initial chapters due to the use of Category Theory, I would suggest pushing forward and not becoming too hung up on acquirring a 'total' understanding.Things will make more sense as you progress through the later chapters.Enjoy and good luck!

Good textbook
Rotman's book presents all the material one would expect of an introductory text, in the language of Categories although still accessible to those who have never seen categories before. While Rotman's style andexposition is excellent, the book often gets bogged down in cumbersomenotation. Also some other textbooks(e.g. Munkres Elements of AlgebraicTopology) give more motivation to the material and explain what is actuallygoing on geometrically(as opposed to algebraically). Also, the exercisesare generally quite easy.Overall, I recommend Rotmans book to people whodon't mind being patient, and waiting to see the whole picture. ... Read more

Product Description

The authors present introductory material in algebraic topology from a novel point of view in using a homotopy-theoretic approach. This carefully written book can be read by any student who knows some topology, providing a useful method to quickly learn this novel homotopy-theoretic point of view of algebraic topology.

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Customer Reviews (1)

Not Specific and too Many Assumtions and Typos
This book claims to have no prerequisites other than general topology and algebra, and implies that even these can be taken concurrently. But in reality it assumes more advanced knowledge. For example, it talks about cell complexes without even defining them.It also has unclear definitions, confusing notation and many notational errors. ... Read more

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. ... Read more

Customer Reviews (5)

"Intuition" more a prerequisite than a result
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.

Excelent Start
In my opinion, this is a great little book to take with you to a park or on a trip to read before you start tackling a more serious book such as the one by Allen Hatcher.This book will give you a great over view of many major topics in Algebraic Topology; for a serious reader, you might want to read this book in parallel with Hatcher, Massey and Munkres (Topology, 2nd Edition).I find that these three books compliment one another very well if you are trying to learn this beautiful subject on your own.I use Sato's book to read about general ideas; once I understand the surface of the concepts I then reference the latter two books to dive deeper into the machinery.It's working well for me; however, do not be fooled, nothing replaces a great teacher!

Excellent accompaniment to Hatcher
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.

Good Supplementary Reading
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.

algeblaic topology
there are much examples. so good to understand. ... Read more