Book Description
From the reviews:
"The author has attempted an ambitious and most commendable project. He assumes only a modest knowledge of algebraic topology on the part of the reader to start with, and he leads the reader systematically to the point at which he can begin to tackle problems in the current areas of research centered around generalized homology theories and their applications. ... The author has sought to make his treatment complete and he has succeeded. The book contains much material that has not previously appeared in this format. The writing is clean and clear and the exposition is well motivated. ... This book is, all in all, a very admirable work and a valuable addition to the literature...
(S.Y. Husseini in Mathematical Reviews, 1976) ... Read more

Customer Reviews (1)

This might take a while...
The earlier chapters are quite good; however, some ofthe advanced topics in this book are better approached (appreciated) after one has learned about them elsewhere,at a more leisurely pace.For instance, this isn't the best place to first read about characteristic classes and topological K theory (I would recommend, without much hesitation, the books by Atiyah and Milnor & Stasheff, instead).Much to my disappointment, the chapter on spectral sequences is quite convoluted. Parts of 'user's guide' by Mclearywould certainlycome in handy here (which sets the stage rather nicely for applications).

So it turns out that supplemental reading (exluding Whitehead's massive treatise) is necessary to achieve a better understanding of algebraic topology at the level of this book.The homotopical view therein will be matched (possibly superseded) by Aguilar's book (forthcoming, to which I am very much looking forward).

Good luck! ... Read more

Book 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 (1)

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

Book Description

Book 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 (13)

One of the worst textbooks ever.
The only good thing about this textbook is that it contains possibly every topic that an instructor could conceive of covering in an introductory course in algebraic topology.However, there are several flaws which prevented me from getting much out of this book:

1. Lack of definitions:I agree with another reviewer of this text who stated that this book does not give definitions.It often does not define objects which not only have several equivalent definitions, but it is not even clear which definition is being used.I often had to search through several books to find a definition which seemed appropriate in the context of this text.

2. Useless or non-existant examples:When a definition is given, it is rarely followed by an example which illustrates it.When examples are present, they don't clearly illustrate the definition.It would be much better to have one well-written example for each definition (or at least for many definitions) rather than five examples for the same definition, none of which illustrates what each part of the definition is.

3. "It is clear":This phrase is used to such unreasonable extremes in this book that it makes me wonder why the book was written at all.If everything is so obvious, then why did you need to write a book about it?Algebraic topology is an inherently abstract and difficult subject, and the author should not have assumed that everything would be so obvious to all parties reading his text.

This book does have another redeeming quality: lots of exercises.Once you finally do understand the subject matter (whether due to a good instructor or a better reference textbook), there are lots of exercises to put your understanding to use.

Excellent book for geometers
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.

Definitely a Bible
This is certainly a modern classic that predominates algebraic topology courses like 18.905/6 at MIT and Part II and III Mathematical Tripos at Cambridge. It is also perfectly suited to personal study and reading -- savor it and its geometric beauty! I would warn the absolute beginner that the text may seem steep at first (especially if you start with Chapter 0 first, the beginning of Chapter 1 is easier) and slightly unmotivated. I would recommend Massey's "Algebraic Topology: An introduction" GTM 56 for preliminary or complementary reading. Be warned that the styles are very different. Hatcher as well as Munkres like introducing the Fundamental Group pretty much right off the bat, which I like. However, there is something to be said to getting to beef up your geometric intuition by thinking about projective space and learning some classification theorems about compact manifolds and this is the approach of Massey. Massey is also nicer if you have just finished a first undergraduate course in topology.

Hatcher is definitely every algebraic topologist's bible and this really isjust volume one in a whole series (check Hatcher's Cornell website for more info) of books that will be as monumental as Spivak's 5-volume Comprehensive Introduction to Differential Geometry (Which you should also buy as each volume is only ~$40). We should take a moment to pause and appreciate what Allen Hatcher has done by putting the book online for free. This is a tremendous statement that learning and knowledge should be free and accessible to anyone who seeks it. I know I first printed Chapter 0 out and starting reading it for free, but to be honest the quality of printing and binding done by the Cambridge University Press is worth the 30 bucks and you should pay it to keep academics warm and off the streets.

Bible of Algebraic Topology
You can not find a better book that explains and covers this beautiful subject better than Allen Hatcher's Algebraic Topology.The subject is build up very well and there are tons of examples that will help you deepen your understanding.I read this book in parallel with Sato (Algebraic Topology: An Intuitive Approach) and Munkres (Topology, 2nd Edition) for independent study.This combination is working well for me, but don't expect to get the same results as you would if you had a great teacher.

Mixed Feelings
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. ... Read more

Book Description

Customer Reviews (7)

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.

Excellent Modern Treatment of Algebraic Topology
One of the reasons that Algebraic Topology is difficult to learn is that often the more general constructions (which are algebraic) are difficult to motivate visually.In fact, I have often found that attempts at visuallizing lead to confusion.J. Peter May avoids confusing illistrations in this book.Constructions are motivated by the results they consort.Most importantly May employes a thoroughly modern point of view.For example: the language of cofibrations/fibrations is used throughout, the handy idea of fundamental groupoid is introduced early in the treatment of the fundamental groups, there are a couple of chapters dedecated to homological algebra intersperced, both homology and cohomology are developed from the axiomatic point of view.May concludes the text with introductions to several more advanced topics such as cobordism, K-theory, and characteristic classes.The list of books that May offers in the suggestions for further reading section at the end is fairily comprehensive.

[too much] for a book that will just sit on your bookself
this is not a bad book, but it isnt for real. the back of the book says: ...treatment is sophisticated, no prior knowledge of the subject is assumed.

i think not.

youbetter be armed with a few other books and be prepared to spend some hours if you want to "learn" from this book as a beginner. ... Read more

Customer Reviews (1)

Part 2, Singular Homology Theory is recommended.
This text is suitable for students of mathematics without prior knowledge of algebraic topology. The best thing with this is Part 2 which treats singular homology theory. However, you may want to resort to Maunder for aneffeective introductin to elelmentary homotopy theory, and to Dold for andintruduction to orientation and duality. ... 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
Algebraic topology is the study of the global properties of spaces by means of algebra. It is an important branch of modern mathematics with a wide degree of applicability to other fields, including geometric topology, differential geometry, functional analysis, differential equations, algebraic geometry, number theory, and theoretical physics.This book provides an introduction to the basic concepts and methods of algebraic topology for the beginner. It presents elements of both homology theory and homotopy theory, and includes various applications.The author's intention is to rely on the geometric approach by appealing to the reader's own intuition to help understanding. The numerous illustrations in the text also serve this purpose. Two features make the text different from the standard literature: first, special attention is given to providing explicit algorithms for calculating the homology groups and for manipulating the fundamental groups. Second, the book contains many exercises, all of which are supplied with hints or solutions. This makes the book suitable for both classroom use and for independent study.A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. ... Read more

Book 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)

"Intuitive" for some perhaps
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.)

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 the contents touch on many useful and up-to-date topics.

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

Book Description

A classic available again! This book traces the history of algebraic topology beginning with its creation by Henry Poincaré in 1900, and describing in detail the important ideas introduced in the theory before 1960. In its first thirty years the field seemed limited to applications in algebraic geometry, but this changed dramatically in the 1930s with the creation of differential topology by Georges De Rham and Elie Cartan and of homotopy theory by Witold Hurewicz and Heinz Hopf. The influence of topology began to spread to more and more branches as it gradually took on a central place in mathematics. Written by a world-renowned mathematician, this book will make exciting reading for anyone working with topology.

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

Book Description
Algebraic topology (also known as homotopy theory) is a flourishing branch of modern mathematics. It is very much an international subject and this is reflected in the background of the 36 leading experts who have contributed to the Handbook. Written for the reader who already has a grounding in the subject, the volume consists of 27 expository surveys covering the most active areas of research. They provide the researcher with an up-to-date overview of this exciting branch of mathematics. ... Read more

Book 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 (homotopy 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

Book Description
William S. Massey Professor Massey, born in Illinois in 1920, received his bachelor's degree from the University of Chicago and then served for four years in the U.S. Navy during World War II. After the War he received his Ph.D. from Princeton University and spent two additional years there as a post-doctoral research assistant. He then taught for ten years on the faculty of Brown University, and moved to his present position at Yale in 1960. He is the author of numerous research articles on algebraic topology and related topics. This book developed from lecture notes of courses taught to Yale undergraduate and graduate students over a period of several years. ... 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

Book 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 (2)

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

Book 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

Book Description
This text, developed from a first-year graduate course in algebraic topology, 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-and include some applications to homotopy theory. 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, "Differential Forms in Algebraic Topology" should be suitable for self-study or for 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

Book Description

Taking a direct route, Essential Topology brings the most important aspects of modern topology within reach of a second-year undergraduate student. Based on courses given at the University of Wales Swansea, it begins with a discussion of continuity and, by way of many examples, leads to the celebrated "Hairy Ball theorem" and on to homotopy and homology: the cornerstones of contemporary algebraic topology.

Customer Reviews (2)

Best undergraduate topology book
I have never seen such a beatiful explanation on continuity and its relations to series and sets. Now I understand why, when mathematics is lousily explained,everything seemms to be so hard. I recommend strongly this book for someone for self study on topology. Hope the author can write on other topics of mathematics.

A pleasure to read
I have a major in math, many years ago. I have moved into economics, but miss the elegance of math, hence I decided to revisit some old topics, and started with topology. As a student we used lecture notes and no real textbook, so my choice now was this textbook. It is a pure pleasure to read. I wish we had used it as a text book when I studied.

The topics are well motivated. Crossley does a good job in explaining why we should care about these particular lemmas and theorems. The proofs are usually elegant. I find the estetic pleasures a good math book should provide. ... 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