Book Description

This elegant book by distinguished mathematician John Milnor, provides a clear and succinct introduction to one of the most important subjects in modern mathematics. Beginning with basic concepts such as diffeomorphisms and smooth manifolds, he goes on to examine tangent spaces, oriented manifolds, and vector fields. Key concepts such as homotopy, the index number of a map, and the Pontryagin construction are discussed. The author presents proofs of Sard's theorem and the Hopf theorem.

Customer Reviews (8)

Exactly would it should be
I would suggest to use this book as a companion to more serious books on topology. Weighing in at a mere 51 pages, this book accomplishes what it needs to: a brief, succinct introduction to topology mostly based on the work of Brouwer. There is a nice mixture of topics, ranging from Sard's theorem to Poincare-Hopf theorem. The proofs and ideas are not fully rigorous or developed, but that would be quite a bit to expect from such a short exposition.

best math book ever written

Despite the lovely subject matter covered in this book, it more importanty gives one a taste of Mathematics as an intellectual discipline. It in outline shows how a mathematical theory - in this case Differential Topology -is constructed and consquently what mathematicians actually do and think about.
Anyone who would like to appreciate Mathematics as a field of study rather than just learn some math should open this book.

Better still, the prerequisite is only multivariate calculus!I have long thought this book should be the third year of calculus rather than differential equations or complex analysis.

Additionally, for the novice it is the only entry I know of into the mysteries of high dimensional geometry, that amazing almost unbelieveable accomplishment of the human mind.

There is a Star Trek episode in which a blind woman wears a dress of sensors which enable her to know more about her environment than a person can know from seeing. She knows exact distances and dimensions, can detect minute movements, can process the complete spectrum of light. In some sense she sees better. Modern topology and geometry are like that sensor dress for seeing higher dimensions. While we can not visualize the sphere in 5 dimensions, we know more about it from these mathematical theories than a five dimensionally sighted being ever could.

Today, mathematics is often considered to be just a practical tool - like a spread sheet - or a toaster oven. We forget its power to widen our imagination, to frame the unimaginable. This book reminds us of this and shows why Mathematics is the Queen of Sciences.

Compact and useful
This book packs a lot of interesting material into a small volume. E.g., I picked up another book recently that started talking about cobordisms right off the bat; despite my having a couple of shelves full of well-known Dover, Springer, Cambridge UP etc. books on topology, differential geometry, mathematical physics, etc., Milnor's tiny book was the only one I found that could help me understand what cobordisms are right away. The book also uses many illustrations to help understanding.

I demote this to 4 stars only because Princeton UP's price is a bit high; many years ago I was lucky enough to find a used copy of the old U. Virginia edition, and paid much less.

Yet another popular (YAP) Math text
In all practicality, for general math students this book is nice for the library, but by no means is it essential.In fact, it's not worth the $25 in most cases.There are so many outstanding Math texts / topics out there, it is doubtful and political that Milnor's Differential Viewpoint deserves its popularity.

I found this book helpful as supplementary reading for Calculus on Manifolds, so I am a minority student.(Majority student = Linear Algebra Done Right.)Spivak's book motivated the need to look carefully at the first few sections of Milnor's book.The definitions in Milnor coincide perfectly with Spivak.

I left Milnor's book with a good intuition about the inverse function theorem, manifolds, and the rank theorem.I also gave a small study to Sard's Theorem, but I had no need to venture into what apparently was the meat of the book...

It is arguable that the Inverse Function Theorem, Manifolds, and the Rank Theorem alone warrant buying this book, despite it only represents the first 2 sections of it, and is far from the total purpose of the book.Nonetheless, that's all I wanted to gain at the time I was reading it.

On the other hand, if you really are a Differential Topology student (small minority), you are the one who wouldn't need the review because you would know what you are trying to get from the book.

On the other hand, students who buy this book who don't know why will probably do nothing more than collect dust with it.

Take full advantage of the clear, encompassing exposition:
Do the exercises. Many were Ph.D. dissertation-level problems in the 1960s; today, they're aptly described as "elementary"- because Milnor MADE them elementary.

This book forms part of the toolkit you will need to fully explore the more modern work in dynamics, complexity, and applications (e.g., economics, physics).

The clarity of the exposition also forms an ideal example of how to communicate mathematics powerfully and simply. ... Read more

Book Description
This material is intended to contribute to a wider appreciation of the mathematical words "continuity and linearity". The book's purpose is to illuminate the meanings of these words and their relation to each other. ... Read more

Customer Reviews (10)

Great service!
The service overall was very good:

i) The item was as described, and
ii) It was shipped quickly

fantastic introduction to general topology
The first part of this book that deals with topology is a pedagogical masterpiece. After motivating the key concepts of compactness and continuity in the relatively concrete setting of metric spaces, the book goes on to abstract topological spaces, a beautiful section on compactness including the tychonoff theorem, and an extremely lucid development of the separation axioms and the proof of the urysohn imbedding theorem and the stone-cech compactification. I personally find the chapter on connectedness to be the weak link in this part of the book. Wherever possible, Simmons provides an exhaustive list of examples (especially when introducing the various types of spaces) that aids comprehension. Moreover, some of the central concepts (product topology) and deeper results such as the Stone-Cech compactification are easier to appreciate because the author has a section on topological properties of the relevant function spaces couple of chapters ahead and several exercises along the way. All in all, a highly recommended intro to the subject.

Didactic perfection
In the author's words in the preface, the dominant theme of this book is continuity and linearity, and its goal is to illuminate the meanings of these words and their relations to each other. The book, he says, belongs to the type of pure mathematics that is concerned with form and structure, and such a body of mathematics must be judged by its high aesthetic quality, and should exalt the mind of the reader.

The author's attitude can only be characterized as magnificent, and, if one is to judge his utterances in the preface by what is found after it, one will indeed find perfect evidence of his delight in mathematics and his high competence in elucidating very abstract concepts in topology and real analysis. Indeed, this has to be the best book ever written for mathematics at this level. It is a book that should be read by everyone that desires deep insights into modern real and functional analysis.

After a brief and informal overview of set theory, the author moves on to the theory of metric spaces in chapter 2. His emphasis is on the idea that metric spaces are easy to find, since every non-empty set has the discrete metric, and that metric spaces are good motivation for the more general idea of a topological space. The Cantor set, ubiquitous in measure theory, dynamical systems, and fractal geometry, is constructed as the most general closed set on the real line, i.e. one obtained by removing from the real line a countable disjoint class of open intervals. Continuity of mappings between metric spaces is defined, and also the concept of uniform continuity, the latter of which is motivated very nicely by the author. Then, the author takes the reader to a higher level of abstraction, wherein he asks the reader to consider all of the continuous functions on a metric space, and turn this collection into a metric space of a special type called a normed linear space, and, more specifically, a Banach space. Thus the author introduces the reader to the field of functional analysis.

A lengthy introduction to topological spaces follows in chapter 3. The author motivates well the idea of an open set, and shows that one could just as easily use closed sets as the fundamental concept in topology. And, most important for functional analysis, he introduces the weak topology, and shows how to obtain the weakest topology for a collection of mappings from a topological space to a collection of other topological spaces. The reader can see clearly that the weaker the topology on a space the harder it is for mappings to be continuous on the space.

Compactness, so essential in all areas of mathematics that make use of topology, is discussed in chapter 4. It is motivated by an abstraction of the Heine-Borel theorem from elementary real analysis, and the author shows how well-behaved things are on compact topological spaces. Some important theorems are proved in this chapter, namely Tychonoff's theorem, the Lebesgue covering lemma, and Ascoli's theorem.

Recognizing that the only functions able to be continuous on a space with the indiscrete topology are the constants, and that a space with the discrete topology has continuous functions in abundance, the author asks the reader to consider topologies that fall between these extremes, and this motivates the separation properties of topological spaces. Chapter 5 is an in-depth discussion of separation, and the reader again confronts function spaces, and their ability (or non-ability) to separate the points of a topological space. Spaces that allow such separation to occur are called completely regular, and this property has far-reaching consequences in analysis and other areas of mathematics. The Stone-Cech compactification is discussed as an imbedding theorem for completely regular spaces, analogous to one for normal spaces.

The intuitive idea of a space being connected is given rigorous treatment in chapter 6. Certain pathologies can of course arise when discussing connectedness, and the author shows this by discussing totally disconnected spaces, remarking that such spaces are very important in dimension theory and representation theory. Indeed, computational and fractal geometry is much harder to study because of the existence of these spaces.

Chapter 7 is important to all working in numerical analysis, wherein the author discusses approximation theory. The Weierstrass approximation and the Stone-Weierstrass theorems are discussed in detail.

A slight detour through algebra is given in chapter 8. Groups, rings, and fields are given a minimal treatment by the author, discussing only the basic rudiments that are needed to get through the rest of the book.

Banach spaces make their appearance in chapter 9, with the three pillars of the theory proven: the Hahn-Banach, the open mapping, and the uniform boundedness theorems. These theorems guarantee that the study of Banach spaces is worth doing, and that there are analogs of the finite dimensional theory in the (infinite)-dimensional context of Banach spaces. The theory of Banach spaces is very extensive, but this chapter gives a peek at this very interesting area of mathematics.

Banach spaces with an inner product are considered in chapter 10. These of course are the familiar Hilbert spaces, so important in physics and the subject of a huge amount of research in mathematics. The presence of the inner product allows constructions familiar from ordinary finite-dimensional vector spaces to carry over to the inifinite-dimensional setting, one example being the transpose of a matrix, which is replaced in the Hilbert space setting by a self-adjoint operator.

As a warm-up to the infinite-dimensional theory, finite-dimensional spectral theory is considered in chapter 11. The famous spectral theorem is proven. Then in chapter 12, the reader enters the world of "soft" analysis, wherein topological and algebraic constructions are used to study linear operators on spaces of infinite dimensions. Putting an algebraic structure on a Banach space gives a Banach algebra, and then the trick is deal with the spectrum of an element of this algebra. The reader can see the interplay between algebra, topology, and analysis in this chapter and the next one on commutative Banach algebras. Indeed, the Gelfand-Naimark theorem, that essentially states that elements of a commutative Banach *-algebra act like the functions on its maximal ideal space, has to rank as one of the most interesting results in the book, and indeed in all of mathematics.

Good Classical Introduction to Banach Algebras
This is a fine book, but not quite in the 5-star league. Let me elaborate. The book is divided into three parts: general topology, the theory of Banach and Hilbert spaces, and Banach algebras. The first two parts lead, by way of synthesis, to the last part, where some interesting but elementary results are proved about Banach algebras in general and C*-algebras in particular. I might mention, for example, the Spectral theorem for compact self-adjoint operators, the Stone representation theorem, and the Gelfand-Naimark theorem.

I can attest from personal experience that the book is well-written; indeed I worked through it chapter by chapter. But today there do exist a plethora of other treatments that can at least rival this text in lucidity, organisation and coverage. For example, for general topology, there is an excellent text by Willard titled 'General Topology',as well as Hocking and Young's old 'Topology'. Both of these go much further in the realm of point-set topology than Simmons. Similarly there are any number of well-written texts on functional analysis that cover the subject of Banach spaces, Hilbert spaces and self-adjoint operators very clearly. Indeed in some respects I feel the Simmons book was inadequate by itself and needed to be supplemented by a text on linear algebra; self-adjoint operators -- and by implication, the Spectral theorem -- need to be seen and manipulated in the finite-dimensional version before one examines their infinite-dimensional generalisation. The Simmons book is a bit weak here; one needs to be playing with matrices.

These are, however, minor quibbles. The book can be recommended to a junior- or senior-level undergraduate.

Topology Classic
This book was recommended for our analysis course (final year at Adelaide University). It helped me pass the course but more importantly, gave me an interest in metric spaces and topology. The book is an excellent communicator and nearly 20 years after I have read it I am looking out for a secondhand copy! ... Read more

Book Description
This classic book is a systematic exposition of general topology. It is especially intended as background for modern analysis. Based on lectures given at the University of Chicago, the University of California and Tulane University, this book is intended to be a reference and a text. As a reference work, it offers a reasonably complete coverage of the area, and this has resulted in a more extended treatment than would normally be given in a course. As a text, however, the exposition in the eariler chapters proceeds at a more pedestrian pace. A preliminary chapter covers those topics requisite to the main body of work. ... Read more

Customer Reviews (6)

a splendid technical book
I was motivated to read this book while in grad school, becasue I needed to understand the French literature in my field (probability). One particular concern is the metrizability of a general topological space. I would say Kelley's book has a spendid presentation on this subject.

Other things in this book are also practically useful. Convergence in the general sense (net or filter) is useful in mathematical finance. The part on locally compactness and paracompactness is a must for anyone working in differential geometry. And if you work in analysis, then the chapter on space of continuous functions is a good reference to look up.

The exercise problems are also good resources when you need some help. I still remember one cute problem on the neighbourhood systems. It helped me understand how a family of seminorms would yield a topology on a linear space.

Evetually, I read this book from cover to cover. And I would say this is one of the best education I've ever received.

If there has to be a complain, the proofs are somewhat hard to read. But this is more or less determined by the nature of the subjects. And when you are well-motivated and equipped with certain mathematical maturity, this problem will gradually go off.

In summary, this book is comprehensive, useful and beautifully written. It is a treasure that every mathematician's library should have.

Generally great; a few annoyances
This is a great book.The proofs are clearly presented, and generally it is easy to understand the motivation behind definitions and theorems.Exercises are relevant, interesting, and well designed, often allowing the reader to discover things that other texts describe in dull detail.Unfortunately, a few exercises (such as "Integration Theory: Junior Grade") seem to pop out of nowhere.I consider this a minor defect.A much larger annoyance is that Kelley defines partial and linear orders in an utterly non-standard and somewhat clumsy way, which ends up affecting a large number of exercises.If you already know something about orderings, you will encounter many surprises; if you know nothing about them, you may get the wrong idea.

Topology with the analyst in mind!
I don't hesitate to give this book 5 stars. It is solid! Many reviewers allow too much personal judgement to cloud their appraisal of a certain book. To me I believe it is important to be as dispassionate as possible so that a prospective buyer can make an unbiased decision. Rather than label a book as "bad" or "good" one should focus on some factors such as:
(1) Content: a summary of the main point covered by the book (this is optional). In the case of this book, this is obvious from the title.
(2) the author's approach: Kelly took what I call the "analyst's approach" to topology. This is fine for those who love analysis but don't really care for topology for it's own sake (like me!) By using this approach, those like me are much more inclined to find topology motivating because ones sees it as abstractions of what one is familiar with
(3) the presentation: Kelly gave a simple but "sophisticated" presentation. You will not describe him as very expository but the presentation is excellent. Some people seem to prefer this style and some don't. No, this has nothing to do with the so-called "mathematical maturity" (how do you define that by the way?) What the author expects you to know to understand the book - that is, the intended audience - is usually stated clearly in the preface

May have been good in its day
I cannot agree with the other reviewers on this. Back in the days when there were hardly any general texts on topology this may have been good. Nowadays there are at least a dozen such that are far better than this. The printing fonts and layout are spidery and primitive and not easy on the eye. The style is rather formal and dry for a subject as rich as this and little effort is put into illustrating the material with background, diagrams or examples. As I said before there is no shortage of better texts amongst which Hocking & Young is worth special mention.

The great classic of point set topology
John Kelley wanted the title to be "What every young analyst should know", but was convinced (by Halmos, among others) not to use it. Still, it is a very good description of the book. Barry Simon calls it"superb" and recommends that you read it by trying to do theexercises,recurring to the text as needed. But then you would perhaps notpay attention to how wonderful the text is. I believe this is thebest-written modern mathematical text. The proofs are clean and extremelyelegant. The prose itself is beautiful and frequently witty. Treatstopological and uniform spaces at depth and in detail, so as to be both atextbook and a reference. Excels in both capacities. This is mathematicsclose to poetry. ... Read more

Book Description

If mathematics is a language, then taking a topology course at the undergraduate level is cramming vocabulary and memorizing irregular verbs: a necessary, but not always exciting exercise one has to go through before one can read great works of literature in the original language.

Customer Reviews (1)

Flawless exposition, great examples, short enough to read cover to cover
This skinny little math book from the Springer Universitext series achieves excellence on many levels.First of all, anyone familiar with the old quip "A topologist is someone who cannot tell the difference between a coffee mug and a donut" will instantly smile when they see the cover.The exposition is downright beautiful, and the organization of the material could not be more perfect.The remarkable thing is that the examples not only demonstrate the concepts, but also play a large role in the development.The choice of fonts and notation is well thought-out and, although minor, contributes greatly to the excellence of the book.One of the best features of this book is its length.With less than 200 pages, one can reasonably set a goal to read it cover to cover.The well-chosen examples not only aid in understanding, but also serve to introduce the reader to concepts from other areas of mathematics.On that note, not only those seeking an introduction to topology, but also anyone new to advanced mathematics, and in addition seasoned mathematicians who are thinking about writing books themselves, will benefit greatly from reading this book.

The author divides the material into five chapters-- 1. Set Theory, 2. Metric Spaces, 3. Topological Spaces, 4. Function Spaces, and 5. Basic Algebraic Topology.There are a number of good exmples from chapters 2 and 3 that serve to compare and contrast properties of metric spaces and topological spaces, as can be expected in any topology text, however the examples used here are interesting in their own right in other areas of math.The author uses the Zariski topology on the prime ideals of a commutative ring in many places.The reader will also meet various function spaces and see how pointwise vs. uniform convergence manifest themselves through suitably chosen topologies.

A number of unique features worth noting are the proof of the Baire category theorem, which is derived from the so called Mittag-Leffler theorem (this is probably the only introductory text which proves this), and Tychonoff's theorem is proved using nets by expressing compactness as every net has a convergent subnet.Also of interest are proofs of the Stone-Weierstrass theorem and the Arzela-Ascoli theorem.On top of all this, there is still some room left at the end to introduce some basic homotopy theory.The fundamental group is defined and covering spaces are introduced.The author proves that homotopy-equivalent spaces have isomorphic fundamental groups, shows that paths and path homotopies can be lifted, and uses this to establish that the fundamental group of the circle is isomorphic to the integers.This is used to prove the Brouwer fixed-point theorem. ... Read more

Customer Reviews (9)

As a reference
This text is really great, once you know the material.It's a bit hard to learn from, but it has so much knowledge in it for people to look up.It requires a fairly sophisticated background, and I started using it after going through the first half of Hatcher, but found it friendly after getting the basics of each section down.It is one of those books that is hard to get through on the first run through, but amazing to keep and use later.It does a good job leading someone to how the Poincare duality works and what is actually going on, and starts someone on intersection theory quite welcomingly.

The star off, however, is for the outrageous cost.It is a book you'll want to keep, though.

A casual introduction
We're using this text right now for my differential topology class.Over all, I find it rather hard to learn from...The definitions at times are sloppy and the over all feel from the book is simply too casual to use as a text.If it were cheaper, I think it'd be a great way to acquaint oneself with the subject.It's just not "text book" material.I've found "Introduction to Smooth Manifolds" by John Lee to be far more useful.

A wonderful introduction to differential topology
First, I must comment about the reviewer below (who is obviously a greater mathematician than I) - I wouldn't recommend Bredon's book to anyone who wants to study differential topology. Man, I fought through a year of algebraic topology with that book, and I'm not sure I got a darn thing out of it! Being of a more analytic, geometric mindset, however, Guillemin and Pollack's book was right up my alley.
First, the authors make the wonderful assumption in the beginning that all manifolds live in R^n for some large enough n. This made study a great deal easier for me, as fighting through charts and atlases may not be the best place to start manifold theory (I don't mean to shortchange other important methods for working with differentiable manifolds, but rather I want to emphasize that many students might get lost in the machinery before learning anything of the theory). The book moves casually along (as the authors suggest, this book is nice for a smell-the-flowers two semester grad school class; we finished in Wisconsin in about a semester and a half before moving on to other pastures). The authors' reluctance to mention functors is also quite nice (I have asked many an algebraic topologist to describe these little guys, and the best answer I've heard is "A functor is an arrow"). A bit of analysis knowledge is nice, particularly in chapter four, and linear algebra (which seems to be a lost art, at least over here in the states) is absolutely critical.
For those of you out there who want to learn a little of this vast and incredibly interesting subject, I would highly recommend this book (even over Milnor's "Topology from the Differential Viewpoint", although the price of Milnor is much nicer). I must agree that this book is outrageously overpriced, but I ended up sucking it in for a month to spare the change for it. If Bredon is your cup of tea, so be it, but I think that most will find this book much more to their liking. One caveat, however: you MUST do some exercises. The authors leave important theorems entirely to exercises (some that come to mind are the "Stack of Records" theorem, the Jordan curve theorem, the Hopf degree theorem, the Cauchy integral formula, etc.).

Lightweight and overpriced
I had to study this for my degree.It was one of those books that one person bought and was passed around mainly due to it's outrageous cost.It has a lack of rigour that is not made up by being more intuitive or giving the reader insight into why differential topology is such a great subject.

Transversality is rightly given prominence, but you don't really walk away with a good feel for it's importance or power.Degrees, linking numbers etc I got for 10 GBP with Milnor's Topology from a Differential Point of View.

As an introduction to differential topology - with a little point set and alot of algebraic throw in - Bredon's Geometry and Topology sets the gold standard, with Darling's Differential Forms and Connections doing a good job on the differential geometry front and Milnor's book above providing bedtime reading beforehand.You can buy all three together for around the same price as this book.

A good start....
Differential topology has influenced many areas of mathematics, and also has many applications in physics, engineering, comptuer graphics, network engineering, and economics. The authors, well-known contributors to the field, have written a nice introduction in this book, which is suitable for readers having a background in linear algebra and advanced calculus.

The authors begin the book with a general overview of manifolds and smooth maps between them. The local behavior of smooth maps is studied first, such behavior determined by the derivative (modulo a diffeomorphism). The inverse function theorem is stated but not proved, the authors encouraging the reader to do the proof using local parametrizations. Generalizations of local diffeomorphisms, the immersions, are discussed, and the local immersion theorem proved. Immersions that are injective and proper, the embeddings, are then discussed. When the dimension of the target manifold is less than or equal to the domain manifold, the surjectivity of the derivative of the map at every point leads to the map being what is called a submersion. Submersions can be viewed as generalizations of projection maps of standard Euclidean space to one of equal or lower dimension. The authors prove this as the local submersion theorem. Regular values of smooth maps are defined and the authors show that the premiage of regular values are submanifolds. A brief discussion of Lie groups is given as an application of the preimage theorem.

The main theme of the book, a generalization of the notion of regularity, called transversality, is also introduced in this chapter. The concept of transversality is fully elaborated on in chapter 2, in the context of manifolds with boundary. Sard's theorem is proved for manifolds with and without boundary. The Transversality Theorem for families of smooth maps is proven in detail, showing that transversal maps are generic when the target manifold is Euclidean space. The authors give an excellent discussion of intersection theory modulo two, along with the famous Jordan-Brouwer separation theorem and Borsuk-Ulam theorem.

In order to make intersection numbers an invariant of homotopy, intersection theory for oriented manifolds is considered in chapter 3. It is shown that homotopic maps always have the same intersection numbers. This brings up naturally the subject of Lefschetz fixed-point theory and the authors give a very clear overview of this. Index theory and the famous Poincare-Hopf index theorem are discussed in this chapter also, along with the Hopf degree theorem. And, thankfully, the authors do not hesitate to employ a myriad of diagrams to illustrate the main points and develop intuition.

The last chapter of the book is more formal than the rest of the book, and covers integration on manifolds. A familiar subject in courses on advanced calculus, the authors do a good job of discussing exterior algebra, differential forms, how to integrate these on manifolds via suitable partitions of unity, and how to differentiate these on manifolds. A very brief introduction to de Rham cohomology is given. The famous Gauss-Bonnet theorem is shown to follow from the Poincare-Hopf theorem. ... Read more

Book Description

The Geometry and Topology of Coxeter Groups is a comprehensive and authoritative treatment of Coxeter groups from the viewpoint of geometric group theory. Groups generated by reflections are ubiquitous in mathematics, and there are classical examples of reflection groups in spherical, Euclidean, and hyperbolic geometry. Any Coxeter group can be realized as a group generated by reflection on a certain contractible cell complex, and this complex is the principal subject of this book. The book explains a theorem of Moussong that demonstrates that a polyhedral metric on this cell complex is nonpositively curved, meaning that Coxeter groups are "CAT(0) groups." The book describes the reflection group trick, one of the most potent sources of examples of aspherical manifolds. And the book discusses many important topics in geometric group theory and topology, including Hopf's theory of ends; contractible manifolds and homology spheres; the Poincaré Conjecture; and Gromov's theory of CAT(0) spaces and groups. Finally, the book examines connections between Coxeter groups and some of topology's most famous open problems concerning aspherical manifolds, such as the Euler Characteristic Conjecture and the Borel and Singer conjectures.

Book Description

From reviews of the first edition:

Customer Reviews (3)

A nice book
I liked this book because it provided me with a new perspective on metricspaces, in using them as a basis learning about fractals. I think it servesas a nice book for an undergraduate to read and get enthused about studyingfractals at a higher level.

Good starting point to study fractal geometry.
This book could be used as a bridge between traditional books on topology-analysis and the speciallized treatises on fractal geometry. More a catalog of definitions, methods, and references than a course text, itcovers the fundamental topological and measure-theoretic concepts needed tounderstand the principles of some of the different dimension theories thatexist. But warning: the book is far away of being a complete exposition onany of the subjects it includes.

Suitable for 3rd-year undergrads.Interesting examples and exercises. Extensive bibliography.

Please checkmy other reviews in my member page (just click on my name above).

A difficult but worthy book!
The programs are in LOGO: don't let the turtles fool you, this is the real stuff by a master teacher. It is hard and the examples are even harder. The problem sets are at times impossible, but in the end Dr. Edgar delivers: understanding! Your unique Associates ID is:thefractaltransl. ... Read more

Customer Reviews (2)

Written for the physicist in mind
This book, written by some of the master expositors of modern mathematics, is an introduction to modern differential geometry with emphasis on concrete examples and concepts, and it is also targeted to a physics audience. Each topic is motivated with examples that help the reader appreciate the essentials of the subject, but rigor is not sacrificed in the book.

In the first chapter the reader gets a taste of differentiable manifolds and Lie groups, the later gving rise to a discussion of Lie algebras by considering, as usual, the tangent space at the identity of the Lie group. Projective space is shown to be a manifold and the transition functions explicitly written down. The authors give a neat example of a Lie group that is not a matrix group. A rather quick introduction to complex manifolds and Riemann surfaces is given, perhaps too quick for the reader requiring more details. Homogeneous and symmetric spaces are also discussed, and the authors plunge right into the theory of vector bundles on manifolds. Thus there is a lot packed into this chapter, and the authors should have considered spreading out the discussion more, as it leaves the reader wanting for more detail.

The authors consider more fundamental questions in smooth manifolds in chapter 3, with partitions of unity used to prove the existence of Riemannian metrics and connections on manifolds. They also prove Stokes formula, and prove the existence of a smooth embedding of any compact manifold into Euclidean space of dimension 2n + 1. Properties of smooth maps, such as the ability to approximate a continuous mapping by a smooth mapping, are also discussed. A proof of Sard's theorem is given, thus enabling the study of singularities of a mapping. The reader does get a taste of Morse theory here also, along with transversality, and thus a look at some elementary notions of differential topology. An interesting discussion is given on how to obtain Morse functions on smooth manifolds by using focal points.

Notions of homotopy are introduced in chapter 3, along with more concepts from differential topology, such as the degree of a map. A very interesting discussion is given on the relation between the Whitney number of a plane closed curve and the degree of the Gauss map. This leads to a proof of the important Gauss-Bonnet theorem. Degree theory is also applied to vector fields and then to an application for differential equations, namely the Poincare-Bendixson theorem. The index theory of vector fields is also shown to lead to the Hopf result on the Euler characteristic of a closed orientable surface and to the Brouwer fixed-point theorem.

Chapter 4 considers the orientability of manifolds, with the authors showing how orientation can be transported along a path, thus giving a non-traditional characterization as to when a connected manifold is orientable, namely if this transport around any closed path preserves the orientation class. More homotopy theory, via the fundamental group, is also discussed, with a few examples being computed and the connection of the fundamental group with orientability. It is shown that the fundamental group of a non-orientable manifold is homomorphic onto the cyclic group of order 2. Fiber bundles with discrete fiber, also known as covering spaces, are also discussed, along with their connections to the theory of Riemann surfaces via branched coverings. The authors show the utility of covering maps in the calculation of the fundamental group, and use this connection to introduce homology groups. A very detailed discussion of the action of the discrete group on the Lobachevskian plane is given.

Absolute and relative homotopy groups are introduced in chapter 5,and many examples are given of their calculation. The idea of a covering homotopy leads to a discussion of fiber spaces. The most interesting discussion in this chapter is the one on Whitehead multiplication, as this is usually not covered in introductory books such as this one, and since it has become important in physics applications. The authors do take a stab at the problem of computing homotopy groups of spheres, and the discussion is a bit unorthodox since it depends on using framed normal bundles.

The theory of smooth fiber bundles is considered in the next chapter. The physicist reader should pay close attention to this chapter is it gives many insights into the homotopy theory of fiber bundles that cannot be found in the usual books on the subject. The discussion of the classification theory of fiber bundles is very dense but worth the time reading. Interestingly, the authors include a discussion of the Picard-Lefschetz formula, as an example of a class of "fiber bundles with singularities". Those interested in the geometry of gauge field theories will appreciate the discussion on the differential geometry of fiber bundles.

Dynamical systems are introduced in chapter 7, first as defined over manifolds, and then in the context of symplectic manifolds via Hamaltonian mechanics. Liouville's theorem is proven, and a few examples are given from relativistic point mechanics. The theory of foliations is also discussed, although the discussion is too brief to be of much use. The authors also consider variational problems, and given its importance in physics, they continue the treatment in the last chapter of the book, giving several examples in general relativity, and in gauge theory via a consideration of the vacuum solutions of the Yang-Mills equation. The physicist reader will appreciate this discussion of the classical theory of gauge fields, as it is good preparation for further reading on instantons and the eventual quantization of gauge fields.

A masterful sequel!
Novikov et al's first volume was the defining book on differential geometry (S-V 93). The second volume picks up on the detailed theory of manifolds and topology and other advanced theories of differentialgeometry, including homotopy groups, Lie algebras and digressing intophysical theories as well (eg.Yang-Mills) giving one of the juciest bookson the subject - an utter delight! ... Read more

Book Description
DNA as the genetic material is a topic of intense interest in the 21st century with the familiar and iconic Watson-Crick double helix having a vital importance for its function.However, there are further complexities beyond the double helix, including supercoiling, knotting and catenation, that are less widely appreciated and understood but which are critical to its function.This book explains these topological aspects of DNA structure in a clear and approachable style that will be appreciated by both students and researchers interested in DNA structure and function. ... 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

Book Description
A sweeping yet uniquely accessible introduction to a variety of central geometrical topics

Students and teachers will benefit from a uniquely unified treatment of such topics as:

• Homeomorphism
• Graph theory
• Surface topology
• Knot theory
• Differential geometry
• Riemannian geometry
• Hyperbolic geometry
• Algebraic topology
• General topology

A logical yet flexible organization makes the text useful for courses in basic geometry as well as those with a more topological focus, while exercises ranging from the routine to the challenging make the material accessible at varying levels of study. ... Read more

Customer Reviews (1)

A great introductory book
This is a great introductory book for students (or non-students) interested in the subject. The book does not delve into rigorous discussions, nor does it get too involved in proofs of theorems, however it produces a nice overall picture of the principles involved in Topology.

Anyone, looking for a relatively smooth first-time study of the subject should find this book very helpful.
... Read more

Book Description

Customer Reviews (3)

readable
proofs are very detailed almost pedagogic. the author always points out the goal first then digs in. the only flaw is its visual presentation - rather condensed type set. i think Pontryagin was a truly educational teacher. good for self study.

A cheap classic by one of the masters
This book covers, as the title says, the foundations of combinatorial topology. What that refers to, essentially, is the topology of polyhedra and the machinery of simplicial homology groups. The first part of the bookestablishes the basic facts about these topological spaces and aboutabstract complexes as well. The second part shows the topologicalinvariance of simplicial homology. The third features applications of thematerial to fixed point theory.

Throughout the book there is also arecurring digression on dimension theory, which culminates in a proof ofthe very non-trivial fact that manifolds have the appropriate topologicaldimension.

The only formal pre-requisites for this book are basiclinear algebra and point-set topology, as covered in any advanced calculusor elementary real analysis course.

The presentation is very conciseand lucid throughout. I can imagine some people not liking such anextremely concise style, but the whole book is so logical that it worksvery well.

A great little book
This is a beautiful little book on a very pretty subject. Lev here develops very comprehensively the basic properties of polyhedra, as well as many of their basic applications. The first chapter is spent on basic topological properties of these spaces, with some applications todimension, very explicitly set aside from the main development. The secondchapter focuses on proving the topological invariance of the Betti groups,with applications of the techniques used-again carefully set aside-todimension theory and Brouwer's fixed point theorem. The third chapter,titled Continuous Mappings and Fixed points, has as a goal proving a simplefixed point theorem. Some simple constructions and concepts of algebraictopology, such as homotopy and the cylinder construction, are introducedalong the way. Unfortunately, not too long after the publication ofthis book, the author began to devote his energies to applied mathematicsand preventing Jews from polluting Russian mathematics-but don't let thatinfluence your possible purchase of this book. ... Read more

Book Description

Customer Reviews (7)

Decent book with flaws
The book has its virtues, sure enough. But there are some downsides
to it as well that I feel are underrepresented in the other reviews so far.

Let me first note that, contrary to the statement of one other reviewer, there are exercises in this book, and not too few. However, I found that I did not need them, since thinking deeply about all the little flaws and omissions that are scattered through the text allowed me to mature faster than going through these exercises. Needless to say, though, that this type of exercise can be a bit frustrating. I often found myself wondering if it was my lack of maturity that made me struggle, or if the authors actually made their life too simple at various points. Luckily, I found amply evidence for the latter. For example, the reader familiar with homotopy may open the book on page 164 and inspect their proof that the curve given by f(1-x) is the inverse of that given by f(x) in the fundamental group. While this is a true statement of course, their constructed homotopy to prove this is not really continuous, and a slight modification of it could be used as a "proof" that every curve is homotopy equivalent to a constant one. A useful review of the book by a professional can be found at the following URL,

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183524657

where similar shortcomings are noted. I agree that the latter will probably not slow down an expert who chooses this book as a reference. For beginners, however, they are unnecessary obstacles. I bought this book because I got attracted by the balanced selection of topics ranging from point set topology to algebraic topology. I wanted to learn the latter, but first needed to become proficient in the former. Having now read only the first part of the book devoted to point set topology, I can say that the book did its job, and did it quite well. However, I cannot shake off the feeling that I could have learned the same material in a fraction of the time from a different book. Feeling that I do now have a solid enough background in point set topology, I am considering to not read the second half of the book, and instead learn algebraic topology from a more modern text.

Very Impressed
I am teaching myself topology with this book right now, and I must say it has an excellent balance of motivation and rigor. The very first definition in the book reveals the implications of topology to anyone who has studied limit pts (and how connectedness is defined in terms of same). After less than a week of study, I understood the big picture better than most people I know who have taken a full course. The exercises are a little sparse, perhaps, but they generally make up for their small number with increased difficulty. I have only encountered a few exercises that I could call trivial. My only gripe is that the exercises are sometimes a little tricky to find.

A good start
Very clearly written, full of examples and counterexamples, making use of pictures but never sacrificing rigor, the authors of this book have given students of topology a superb introduction to the field. Many students have been educated in topology by using this book, and it is sure to remain a classic in the field. It builds a solid understanding of the basic rudiments and intuition behind point-set, geometric, and algebraic topology. There is a lot of material covered in the book, and some very specialized subjects, such as Cech and Vietoris homology and some dimension theory, but with some preserverance and concentration, the entire book can be grasped within reasonable time constraints. Probably the only minus to the book is the lack of exercises. This is a quite serious omission, for the only way to master a subject is to work problems that require careful thought for their solution.

The beginning student of topology should probably read this book with the following mindset: try to think of ways and techniques that you would devise to study the structure of a topological space. Homotopy and homology (in various forms) are the standard techniques for doing this. These strategies have varying degrees of success, but their use in topology now seems to be reaching a saturation limit, even though the explicit calculation of homotopy groups is still a very active area. New techniques and concepts, representing sort of a "large deviation" from the standard ones discussed in this book, will be needed to make further progress in the study of complicated topological spaces. Something more is needed now, that is completely different than homology and homotopy theory, that will make more transparent the properties of these spaces. These new techniques will be somewhat radical from the standpoint of current ones, but they will be more effective from a conceptual (and computational) point of view.

A Professional Topologist loves this book.
When I was a graduate student 40 years ago there were very few texts in topology.The only two that I recall being in use were Hocking and Young and the book by Kelley.Over the years my copy of Hocking and Young has become quite worn.It is a wonderful book that gives the true flavor of topology.It is also contains a large number of topics that one can refer to later on.It becomes quite apparent very earlier that no one will be able to fully appreciate the book in the time span of one course.It is a book that must be read and reread over and over again.It is a real classic.I do not believe that it is the type of book that would be of much or any general interest but to a point set topologist it is a classic and must for his bookself.I am quite surprised over its low price.I can not help but compare it with the newer book by Munkres.I recall seeing Munkres book many years ago and disliking it.But the current edition seems much closer in flavor to HY and Munkres book is quite good.Munkres style is much clearer than HY, but both books target a very specialized group of people.Neither book is for the faint of heart and will take many years to absorb.Considering that Munkres book is 9 times as expensive as HY, HY seems to be the better buy.

Theoretical Dictionary
An excellent book, not for those persons unfamiliar with the topic of topolgy; yet, combined with simpler texts this book is a goldmine of topological theorems and their proofs. ... 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

Customer Reviews (1)

Very good product
Nicely written, in understandable language, this book should stand amongst the references of its kind.

... Read more

Book Description
Amid the cultural and political ferment of 1960s France, a group of avant-garde architects, artists, writers, theorists, and critics known as "spatial urbanists" envisioned a series of urban utopias, phantom cities of a possible future. The utopian "spatial" city most often took the form of a massive grid or mesh suspended above the ground, all of its parts (and inhabitants) circulating in a smooth, synchronous rhythm, its streets and buildings constituting a gigantic work of plastic art or interactive machine. In this new urban world, technology and automation were positive forces, providing for material needs as well as time and space for leisure.

In this first study of the French avant-garde tendency known as spatial urbanism, Larry Busbea analyzes projects by artists and architects (including the most famous spatial practitioner, Yona Friedman) and explores texts (many of which have never before been translated from the French) by Michel Ragon, the influential founder of the Groupe International d'Architecture Prospective (GIAP), Victor Vasarely, and others.

The projects of the spatial urbanists were in large part a response to the government’s planning policies, its Kafka-esque bureaucracy, and its outdated institutions, which they considered the first obstacles to the implementation of their radical urban designs. But even though the spatial city was conceived as progressive, by the end of the 1960s some critics had begun to question its ideological foundations.

Topologies maps the literal and metaphorical topologies of spatial urbanism, describing and documenting its projects and locating it within an international network of experimental architectural practice that also included the Situationist International, Archigram, the Metabolists, Architecture Principe, Superstudio, and others.

Even at its most fanciful, Busbea argues, the French urban utopia provided an image for social transformations that were only beginning to be described by cultural theorists and sociologists. The designs of spatial urbanism anticipated the ambivalence that would greet the arrival in France of capitalist modernity and globalization, marking both the apex and the end of the technological optimism of the postwar years. ... Read more