Product Description

This book brings the most important aspects of modern topology within reach of a second-year undergraduate student. It successfully unites the most exciting aspects of modern topology with those that are most useful for research, leaving readers prepared and motivated for further study. Written from a thoroughly modern perspective, every topic is introduced with an explanation of why it is being studied, and a huge number of examples provide further motivation. The book is ideal for self-study and assumes only a familiarity with the notion of continuity and basic algebra.

Customer Reviews (4)

Best Intro to Topology
Topics are well motivated.
Theorems are proved in a rigorous yet intuitive style that one feels like it was an explanation rather than a dry proof typically found in the advanced math books.
Important key ideas are also sufficiently illustrated through examples and exercises.

If one finds it verbose, I'd recommendcroom--a bit more like the typical math books but accessible.

Surely not optimal
We used this book in an introductory topology class I took.Some of the exercises are poor (e.g. counting the number of topologies) and the exposition wasn't anything to go crazy about.After a while, I found myself reading Munkres exclusively; it's much more comprehensive.Maybe this book is well suited for folks looking to get a flavor of topology but nothing super concrete.

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

Product Description
This excellent introduction to topology eases first-year math students and general readers into the subject by surveying its concepts in a descriptive and intuitive way, attempting to build a bridge from the familiar concepts of geometry to the formalized study of topology. The first three chapters focus on congruence classes defined by transformations in real Euclidean space. As the number of permitted transformations increases, these classes become larger, and their common topological properties become intuitively clear. Chapters 4–12 give a largely intuitive presentation of selected topics. In the remaining five chapters, the author moves to a more conventional presentation of continuity, sets, functions, metric spaces, and topological spaces. Exercises and Problems. 101 black-and-white illustrations. 1974 edition.
... Read more

Customer Reviews (3)

Perfect book for the right reader
This book is not a textbook -- nor is it a rigorous buildup of topology from geometry. It is simple and conceptual bridge from the concepts of congruence classes in geometry, and basic ideas from map theory, into topology.
For any person wishing to acquire a grasp of the subject, I would recommend this book as a primer to any more conventional treatment of topology. In many texts, you are given general theorems in terms of sets, with no real idea of why these ideas have come about. From Geometry to Topology provides the necessary intuitive background... which is not to mention -- it's a darn interesting read.

Easy, but okay...
This book is quite pictorial and thus easy to read, as it's intended to be: a first introduction on undergraduate level to topology. However, it's a pity it doesn't go into topology very much. It stays very informal.

Very nice and intuitive introducation to topology
I bought this book shortly after my introductory analysis professor had once mentioned what is topology. That was the first topology book I had read, and I fell in love with the subject ever since. It's easy to understand,and very enlightening as well.
The book walks you through the transition from geometry to topology, then eleborates on several basic topological concepts.
Very interesting stuff ! ... Read more

Product Description
A fresh approach to introductory topology, this volume explains nontrivial applications of metric space topology to analysis, clearly establishing their relationship. Also, topics from elementary algebraic topology focus on concrete results with minimal algebraic formalism. The first two chapters consider metric space and point-set topology; the second two, algebraic topological material. 1983 ed. Solutions to Selected Exercises. List of Notations. Index. 51 illus.
... Read more

Customer Reviews (8)

Great book for starters
Introduction to Topology by Gamelin is a great book for starters. There are a considerable number of exercises with answer sugestions and most part of the material in the book is self contain. Thus, Introduction to Topology is a reference for those who are familiarising with this subject.

A Wonderful Book
I highly recommend this book as a supplement to another topology text since it has a lot of hints and answers. Students may not learn very from it because they may just copy the answers in the back of the text. It is a great book though.

I'm not good at math
I wanted to teach myself some topology and a friend with mutiple Math PhD's reccomended this book to me.
This is a tremendous value, and is comprehensible. But it is prety lean and direct, so be prepared to work on this in a quiet place where you can concentrate for a sustained period of time. Proofs are direct, and expect you to be familiar with notation through all of Algebra.
I re-emphasize: there is zero, no, nada, blank, null coddeling here. Every single word, every single notation is important, and if you haven't read, marked, and inwardly digested each one it is a promise you will be lost in a page or two and have to go back. There is no fat here at all and the authors don't babysit you or expalin anything five different ways. This is direct on the coal face math.
Still, I knew only basic basic basic totpology before this, and now I have a vague understanding of all the major areas of further inquiry.
A very good value.

Okay, not great. Overall, I give it a C+
The exposition, while clear and not without attention to subtleties of the theory, is a little scattered. The metric spaces chapter is very good, but after that, it goes downhill. In particular, I was pretty disappointed that the mean value theorem was not proved as an application of connectedness. Everybody sees the mean value theorem in calculus, and the proof is really quite elegant. Also, a lot of important notions in topology are relegated to the exercises, and the rest of the exercises are like applications to analysis. It's kinda nice to be challenged to see the definitions in multiple ways through the exercises, but it would be nicer to get all the perspectives in the exposition, and be given exercises that would deepen one's understanding of the material.

All in all, this book feels like "topology as a branch of analysis" and only helps the reader to develop a modest working topological intuition. For readers interested in topology as its own subject, Munkres' book is the only book. For those readers desiring a more introductory approach, I found Mendelson's book to be an excellent introduction - the chapters on connectedness and compactness are thorough and quite helpful - though that book is lacking in that it doesn't discuss separation axioms at all, and contains few exercises. But that book is unique in that it despite its brevity it touches on metric spaces, categories, and the fundamental group.

If you're going to read this book, get a copy of Mendelson's book - it will flesh out your understanding of topology.

excellent introduction to topology
I used this book to teach myself the basics of point-set topology and homotopy theory. What makes this book so great is that the author doesn't waste words in delving into the heart of a concept, while providing insight into it. A good collection of interesting problems, most with solutions in the back of the book. This makes this book very good for self study. If you liked Rudin, you'll probably like this book as well, as it is written in a similar style. If someone knows of a better introduction, do let me know. ... Read more

Product Description
This book presents the classical theorems about simply connected smooth 4-manifolds: intersection forms and homotopy type, oriented and spin bordism, the index theorem, Wall's diffeomorphisms and h-cobordism, and Rohlin's theorem. Most of the proofs are new or are returbishings of post proofs; all are geometric and make us of handlebody theory. There is a new proof of Rohlin's theorem using spin structures. There is an introduction to Casson handles and Freedman's work including a chapter of unpublished proofs on exotic R4's. The reader needs an understanding of smooth manifolds and characteristic classes in low dimensions. The book should be useful to beginning researchers in 4-manifolds. ... Read more

Customer Reviews (1)

Excellent
For those genuinely interested in understanding the proof of the 4-dimensional Poincare conjecture, and for those who need a more geometric, intuitive view of some of the main results in topological 4-manifolds, rather than one based on the heavy machinery of algebraic topology, this book is an excellent beginning. The author endeavors in this book to be as clear as possible, and he does not hesitate to use diagrams to get the point across. Rigor however, is not sacrificed. One of the main goals in the book is to get a more geometric proof of Rohlin's theorem, which states that cobordism ring in 4-dimensions over the special orthogonal group and over the spin group is the integers.

The author starts the book with an overview of handlebody theory, noting that for the case of interest, 4-dimensional toplogical manifolds must be smooth in order for them to be handlebodies. Smooth handlebody decompositions can be described by Morse theory, and one smooth handlebody decomposition can be related to another via an isotopy of attaching maps and creation or annihilation of handle pairs. The author visualizes handlebodies in four dimensions by drawing their attaching maps in the 3-sphere. This results in the use of framed links to model the attaching maps, with examples of the 3-torus, the Poincare homology 3-sphere, and a homotopy 4-sphere, the latter of which is homeomorphic to the 4-sphere and is a double cover of an exotic smooth structure on 4-d real projective space. The author also gives a brief but interesting discussion on why the methods of this chapter are difficult to do in three dimensions.

The theory of intersection forms appears in chapter two, with the author proving first that for a closed, smooth, oriented, 4-d manifold M any element of the second integer homology group is represented by a smoothly imbedded oriented surface. Any two such surfaces can be joined by smooth oriented 3-manifold imbedded in M. The isomorphism between the second homology and cohomology groups (over the integers) modulo torsion is the famous "intersection pairing". The author then proves that two simply-connected, closed, oriented 4-manifolds are homotopy equivalent if and only if their intersection forms are isometric. The proof emphasizes the geometric connection between homotopy type and intersection forms. A brief review of symmetric bilinear forms and characteristic classes is then given, as preparation for the classification results given later in the book.

The author treats classification theorems in chapter three, which he describes as deciding which forms, whether symmetric, integral, or unimodular, can be represented by simply connected closed 4-manifolds. The relation between forms and homotopy type makes this implicitly a classification for the homotopy type of the manifold. Rohlin's theorem was historically the first major result in this problem, but the author delays its proof until chapter eleven. The author briefly discusses the work of Freedman in the topological case, and Donaldson, in the smooth case.

Spin structures are discussed in chapter four and several examples are given. The author also shows how to relate spin structures on the boundary of a manifold to spin structures on the manifold itself, to set up later discussions on cobordism. Chapter five then concentrates on the Lie group spin structure of the 3-torus T3(Lie) and the surface constructed by taking the nine-fold direct sum of complex 2-d projective space and its reverse orientation. The latter is a complex analytic projection, which is a smooth fiber bundle with fiber the two-torus except for a finite number of singular fibers. The author shows in detail how to use this object to obtain a spin manifold with spin boundary T3(Lie).

Chapter six is devoted to showing how to immerse closed, smooth, oriented 4-manifolds in Euclidean 6-d space. This involves the calculation of a characteristic class in the second integral cohomology group. Then as a warm-up to showing that a spin 4-manifold with index zero spin bounds a spin 5-manifold, the author proves in chapter 7 that every orientable 3-manifold is spin, bounds an orientable 4-manifold, and if spin bounds a spin 4-manifold with only 0-handles.

In chapter eight, the author proves that a closed, smooth, connected, and orientable 4-manifold is the boundary of a smooth 5-manifold if the first Pontryagin class is 0. If the 4-manifold is spin, and the first Pontryagin class is 0, then there exists a smooth, spin 5-manifold whose boundary is the 4-manifold, where both manifolds are considered as spin manifolds. Chapter nine proves the Hirzebruch index theorem in dimension 4, and the author shows that the cobordism ring for SO and Spin is the integers. Chapter ten is devoted to a proof of Wall's theorem and the h-cobordism theorem in dimension 4. The geometric proof of Rohlin's theorem promised by the author is finally done in chapter eleven.

Casson handles, so important in the proof of the 4-d Poincare conjecture, are discussed in chapter twelve. The author shows the role of the Whitney trick in dimensions 5 or more, and how its failure in dimension 4 results in the use of Casson handles, which are constructed using the famous "finger moves". He gives an explicit handlebody description of the simplest Casson handle, and then relates it to the Whitehead continuum.

The most fascinating part of the book is chapter thirteen, which outlines briefly Freedman's proof of the 4-dimensional Poincare conjecture. The proof makes use of 4-dimensional handlebody theory and decomposition space theory. Casson handles are decomposed via an imbedding of a Cantor set of Casson handles inside them. The "Big Reimbedding theorem" of Freedman, which points to the existence of an exotic smooth structure on the 3-sphere cross the real line, is quoted but not proved. The book ends with chapter fourteen being a brief discussion of exotic structures, their existence following from the non-smoothness of Casson handles. ... 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

Customer Reviews (12)

Topology Starter Book
I chanced upon this new arrival book in my local National Library book shelf 2 weeks ago. I congratulate the librarian who put this book at the public loanable section. Truely as the author claims, this Topology is for anyone with or without advanced math backgroud.
Topology has been the 'scarest' subject in the University where my 'sadistic' Math professors used it to 'kill' (to fail) students.
This book tells you Topology is fun and intuitive, it is a 'Rubber-sheet' Geometry as opposed to rigid objects in the axiomatic Euclidean Geometry.
Starting from the Euler formula "V-E+R = 2" (replace F = Face equivalently by R=Region,'V-E+R=2" easier remember as "VERsion 2" with a hyphen '-' before E), the central topic of Topology begins from Descartes / Euler to Riemann, Poincare...
The Epilog on "The Million-Dollar Question" (Poincare Conjecture) details the legendary Russian mathematician Perelman, who refused Fields Medal and in June 2010 rejected the Clay US$ 1 million prize.
Appendix A "Build your own Polyhedra and Surfaces", useful for students to make the 5 Plato Polyhedra on paper.
All in all, this book is excellent as a starter of Topology.

This book is a real gem itself
I just finished reading this book, and it only took me a couple of days. Admittedly, I have some exposition to math, having taken linear algebra, calculus, and differential equations, all of which are very useful in understanding this little book. But even a lay person with only some basic knowledge of geometry and algebra can grasp the content fairly easily. Most of the proofs are visual, and all of them are extremely elegant and simple. If you didn't take much math before but you are interested in the subject, get this book, and don't be scared of the apparent difficulty.

It is very rare that a math book is both so simple and so insightful. The topic is quite advanced, and the concepts, especially in the later chapters, are quite complex. And yet the author explains them in great simplicity. He doesn't go into some details I would have liked to see as someone with a math background, but that makes the book much more clear. And it's very well written. The author is very involved and obviously loves the subject. He also introduces other, related branches of mathematics like graph theory and knot theory, which could have made his book too complicated. And yet he deals with them so simply you might think that all math is like that. He presents all the beauty and elegance with little of the complexity which can make math seem so ugly and incomprehensible.

I really, really recommend this book, especially to the lay audience, and high school/undergraduate students in particular. Many of my friends like math but think that they are not smart enough. This book can show you that this is not the case, you just didn't think about it in the right way before. It's important that people stop thinking of math as something out of their reach, and all that is needed for that is a good teacher. And Richeson is certainly a good teacher.

That said, even a more advanced reader can enjoy this book, both for the incredible presentation, including many illustrations, and the elegant proofs and their sketches, which one can carry out to completion during leisure hours. The historical background is fascinating and the book reads almost like a novel.

The author did a really good job on this book, if only there were more books like it.

Very Good, But Challenging
Euler's Gem is a fascinating & well written book.However, it is also a pretty challenging read, one can not really sit back & read it straight through.But this is also what mathematics & learning is all about, as you often have to stop, re-read, & think a bit about what is being said.The claim is made that someone with only high school mathematics could read the book, & while this is probably true, it would be a steep climb.Especially as one progresses further & further into the book, many references are made to calculus, differential equations, & other related ideas, which the author does a fantastic job of explaining the ideas to people that never had the courses, but in the end it really would help the reader to have that knowledge beforehand.

What makes this a five star book is that it is so rich in knowledge.The average person won't be able to read it in a week, but if you're willing to put the time into the book, you'll get a lot of out it as it really is a great introduction to topology.Even if you can't pick up all the concepts, you're sure to be able to pick up many of the neat tricks the author points out, such as the wedding ring knot, coloring map problem, etc.Overall, one of the best books I've ever read, & one day I'll probably have to re-read it again because it's just so rich & packed with knowledge.

A Gem Indeed!
This is as good an introduction to topology as any for someone who isn't a professional mathematician. Even the professional can learn the history behind very familiar material.

A gem of mathematical results produced by one of the masters of mathematics
The title of the book is derived from the formula V - E + F = 2 that holds for any polyhedron. V is the number of vertices, E the number of edges and F the number of faces. First demonstrated by Euler, the proof of this result is surprisingly simple. As is the case with most such formulas and their proofs, there is at least one near miss in the history of mathematics. Descartes was close; in retrospect it is somewhat surprising that he didn't reach the appropriate conclusion. Of course, we are considering the great master Euler here, a giant of mathematics who was able to see things in his mathematical sight that people with the physical vision that he lacked overlooked.
Topology is a relatively recent area of mathematics, one of the few that can be considered to have had a point of origin and a creator. Richison works through the historical mathematical preliminaries of the formula, the shapes it describes were well known to the ancient Greeks yet they were nowhere close to the formula. Some historical and mathematical background on Euler follows this and it includes some of his other accomplishments. The last chapters describe some of the results that follow from topology in general and Euler's gem in particular. One of the most interesting is the theorem of combing a sphere, where the conclusion is that there must always be at least one hair that stands straight up. This may seem like an absurd thing for mathematicians to be concerned about but it has a major conclusion, that at all times there must be at least one point on Earth where there is no wind. Even more significantly it means that there will always be a zero.
Richison uses a large number of diagrams and formulas when needed, which is to his credit. Mathematics is based on equations so when an author deliberately avoids them in an attempt to increase sales, it is hard to claim that they are actually writing mathematics. This is an excellent book about a great man and a timeless formula. Well within the reach of the intelligent layperson, it is also a good book to use as a resource for a course where the students are required to make presentations.

Published in Journal of Recreational Mathematics, reprinted with permission.
... Read more

Product Description
Topology, for many years, has been one of the most exciting and influential fields of research in modern mathematics.Although its origins may be traced back several hundred years, it was Poincaré who "gave topology wings" in a classic series of articles published around the turn of the century.While the earlier history, sometimes called the prehistory, is also considered, this volume is mainly concerned with the more recent history of topology, from Poincaré onwards.

As will be seen from the list of contents the articles cover a wide range of topics.Some are more technical than others, but the reader without a great deal of technical knowledge should still find most of the articles accessible.Some are written by professional historians of mathematics, others by historically-minded mathematicians, who tend to have a different viewpoint. ... Read more

Customer Reviews (2)

A lot of very good essays
This book collects 40 essays on various themes, events, and personalities in the history of topology. I have browsed half of them and carefully read 6 or 8.Plus I have used the book as a reference, and so gained isolated facts from many of the essays.

The essays assume various amounts of background, proportionate to their subjects.In generally they are very clear and if you have any knowledge at all of a topic you can read an essay on it here at least well enough to decide if you want to work harder on it.

The book concentrates more on manifolds, fixed point theorems, algebraic topology, and homological algebra, than on issues in general or point set topology. It covers Poincare, Brouwer, Weyl, and gets up to derived categories.It could not possibly cover everything in that range.For one, Lefschetz is mentioned often but he could well have deserved a whole article on himself.Or there could have been an article on the Princeton school. But incomplete we must all resign ourselves to be.

The book is a huge amount of information, very well organized and presented.

Absolutely Brilliant
A sublime account of the history of topology from it's earliest incarnation as analysis situs to contemporary research.

No home should be without one. ... Read more

Product Description
This book is intended as an elementary introduction to differential manifolds. The authors concentrate on the intuitive geometric aspects and explain not only the basic properties but also teach how to do the basic geometrical constructions. An integral part of the work are the many diagrams which illustrate the proofs. The text is liberally supplied with exercises and will be welcomed by students with some basic knowledge of analysis and topology. ... Read more

Customer Reviews (1)

Excellent graduate-level introduction, slightly marred by poor editing & translation
Broecker & Jaenich's "Introduction to Differential Topology" is the best book for (reasonably proficient) first-year graduate students to acquire the basic tools for studying the topological aspects of smooth manifolds. Originally written in German in 1973 (as Einführung in die Differentialtopologie) and then translated into English in 1982, it has a high reputation among mathematicians, being praised by, e.g., Milnor and Brieskorn. In fact, Barden & Thomas were motivated to write their own book, An Introduction to Differential Manifolds, in part by the fact that this book was out of print; but now Cambridge has reprinted it.

Although an introduction, it would probably be considered too difficult by most undergraduates, as it moves rather quickly and assumes knowledge of basic topology and analysis. In contrast to, say, Guillemin & Pollack's Differential Topology, terms are always defined precisely (e.g., manifolds are 2nd countable and not assumed to be subsets of R^n) and there is relatively little motivating discussion, but rather it immediately launches into the subject. While it thoroughly covers the basics of differential topology - immersions and submanifolds, tangent and vector bundles, partitions of unity, transversality, isotopies, tubular neighborhoods, flows, Whitney's and Sard's theorems - there is no treatment of more advanced topics, such as Morse theory, surgery, or handlebodies (as in Hirsch's Differential Topology or Kosinski's Differential Manifolds), and there is only a brief mention of (co)bordism. Moreover, Riemann metrics are barely used and other diffeo-geometric/analytic aspects of smooth manifolds - differential forms, integration, Lie groups, de Rham cohomology, the Frobenius theorem - are not even hinted at, so this is definitely not fungible with Lee's Introduction to Smooth Manifolds or even Barden & Thomas.

Where the book really distinguishes itself is its conciseness, efficiency, and rigorousness. Despite keeping verbosity to a minimum (in contrast to, say, Lee), some very clear and complete explanations of key concepts are presented, such as the comparison of 3 different definitions of tangent spaces (the "algebraist's, physicist's, and geometer's" definitions), which helps to sort out any confusion the reader may have acquired from other sources. Usually, more general versions of theorems are given, yet with short proofs, such as that of Whitney's embedding theorem, Sard's theorem, the existence of collars, and the transversality theorems, and theorems are expressed in precise modern language, such as by the use of germs in the rank and inverse function theorems. Following Lang's Differential and Riemannian Manifolds (but more accessibly), dynamical systems and sprays are introduced and used to construct isotopies of embeddings and tubular neighborhoods. There's a refreshing lack of handwaving, with, e.g., connected sums and manifolds with corners being handled properly in the differential case; in fact, at the beginning of a chapter they state, "The differential topologist sometimes 'pushes' a submanifold aside, 'dents' it somewhere, 'bends' or 'deforms' it, and the handwaving which accompanies such operations all the more undermines the confidence of the observer. He believes the assertions are plausible but that they have not been proven. We propose to make such 'bending' precise by means of isotopies and embeddings...," and then they follow through on that promise. These reasons, combined with the book's wealth of useful technical lemmas and observations and many figures, all packed into only 150 pages, make it one of my 3 favorites (along with Kosinski and Milnor's Topology from the Differentiable Viewpoint) on the subject.

Every chapter includes 10-30 exercises, which are good practice for applying the theorems, with hints for the more difficult ones (which aren't that hard anyway). None of these exercises are used in the text.

There are a few faults with the book. First of all, as noted above, it would have been better to include more material, as neither more advanced topics in differential topology nor any of the analysis is covered, necessitating that this be supplemented with another text regardless of the emphasis of the course. Then there were a few errors/omissions (e.g., on p. 71 they fail to acknowledge that a theorem about locally compact spaces that they cite only holds if the target space is Hausdorff), and near the end of the book they start skipping steps in some proofs and are not as careful as in earlier chapters; the most egregious example of this is on p. 148, where they assume that a spray with certain special properties exists without demonstrating it. Also, there are a few sentences where the meaning is a bit hard to decipher, perhaps due to a poor translation (which is odd since the book was translated by the mathematician C. B. Thomas), and this is compounded by a more serious problem, namely, the copyediting was atrocious. I don't recall when I last saw this many meaning-altering misplaced commas or adverbs used as conjunctions; other editor's errors include a theorem number being used twice and different terminology alternately being used for the same thing. But being a former copyeditor, I am probably disturbed by this more than most people.

Overall this book, combined with Hirsch for the Morse theory and surgery, would constitute the ideal 1st-year graduate course in differential topology (for topology students). It also covers the core preparatory material for Kosinski as well. However, students with no prior exposure to the subject would probably be better served by looking at Guillemin & Pollack or Lee first.
... 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 offers an introductory course in algebraic topology. Starting with general topology, it discusses differentiable manifolds, cohomology, products and duality, the fundamental group, homology theory, and homotopy theory.

From the reviews: "An interesting and original graduate text in topology and geometry...a good lecturer can use this text to create a fine course....A beginning graduate student can use this text to learn a great deal of mathematics."—-MATHEMATICAL REVIEWS

Customer Reviews (7)

never really liked it
but maybe you will.
i think it was Vick's book that i preferred.
Massey--I preferred Massey.
i think that this is the one for the grown-ups.

a different perspective
While I agree with reviewers generally that this is a good book, i should warn that bredon isnt for the faint of heart. He makes use of simple language from category theory, doesnt always completely introduce his discussions (see for example the chapter on the tangent bundle where tangent bundle is never defined), and some other things that are nuisances to the newcomer.

I do think this is a good modern readable textbook, but for the student who has a solid foundation in mathematics. I didnt find it as accessable as other topology books, say Hatcher or Lee's books (but lee's are not as complete).

excellent for first year graduate study
This was the assigned book in my first year grad topology course. It has good examples, interesting exercises. I like the emphasis on geometrical examples, constructions. It's not easy to read, but interesting.

Among the best textbooks in algebraic topology.
As the previous reviewers have commented, this book is very accessible and detailed. I should add that the authour never lets you get lost in the labyrinth of abstract nonsense; the treatment is always geometric rather than homologico-algebraic. The only complaint I have is, the book would be more useful with chapters on spectral sequences, cobordism and K-theory.

The Graduate Sudent's Topology Bible
If you want to learn topology, this book is the place.Though this text can require some maturity, the range of topics and the clarity of exposition are outstanding.My only complaint is that an additional appendix covering the basics of category theory would have been quite useful.Bredon not infrequently uses the language of category theory (though always in a non-essential way).Since this text is aimed at 1st year graduate students, I think the tacit assumption that the student has already encountered these topics is not justified.That such a minor point is my chief complaint speaks volumes of my esteem for this text. ... Read more

Product Description

From the reviews of the 1st edition:

"This book provides a comprehensive and detailed account of different topics in algorithmic 3-dimensional topology, culminating with the recognition procedure for Haken manifolds and including the up-to-date results in computer enumeration of 3-manifolds. Originating from lecture notes of various courses given by the author over a decade, the book is intended to combine the pedagogical approach of a graduate textbook (without exercises) with the completeness and reliability of a research monograph…

All the material, with few exceptions, is presented from the peculiar point of view of special polyhedra and special spines of 3-manifolds. This choice contributes to keep the level of the exposition really elementary.

In conclusion, the reviewer subscribes to the quotation from the back cover: "the book fills a gap in the existing literature and will become a standard reference for algorithmic 3-dimensional topology both for graduate students and researchers".

Zentralblatt für Mathematik 2004

For this 2^nd edition, new results, new proofs, and commentaries for a better orientation of the reader have been added. In particular, in Chapter 7 several new sections concerning applications of the computer program "3-Manifold Recognizer" have been included.

Product Description

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
The goal of this publication is to provide basic tools of differential topology to study systems of nonlinear equations, and to apply them to the analysis of general equilibrium models with complete and incomplete markets. The main content of general equilibrium analysis is to study existence, (local) uniqueness and efficiency of equilibria. To study existence Differential Topology and General Equilibrium with Complete and Incomplete Markets combines two features. As a first step, first order conditions (of agents' maximization problems) and market clearing conditions, instead of aggregate excess demand functions, are used. As a second step, a homotopy argument, stated and proved in relatively elementary manner, is applied to that "extended systemof equations. Local uniqueness and smooth dependence of the endogenous variables from the exogenous ones are studied using a version of a so called parametric transversality theorem. ... Read more

Product Description
One copy ... Read more