Customer Reviews (1)

Not for the Undergraduate
Professor Cullen says in the preface that this book should be read by the advanced undergraduate or first-year graduate student, but this book in my opinion should only be read by a first- or second-year graduate student. The mathematics in this book is as rigourous as math can possibly get; the proofs are often quite long and sometimes difficult, and the concepts Professor Cullen tries to convey are sometimes very difficult to follow. If you like your math rigourous (trust me, there are people out there that like rigourous mathematics) and you have some backround in topology and real analysis, then this might be for you. But remember, this book is serious mathematics, and if you try to pick this book up with no backround then you'll get eaten alive. ... Read more

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

Customer Reviews (2)

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

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

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

Product Description
This book is an introduction to elementary topology presented in an intuitive way, emphasizing the visual aspect. Examples of nontrivial and often unexpected topological phenomena acquaint the reader with the picturesque world of knots, links, vector fields, and two-dimensional surfaces. The book begins with definitions presented in a tangible and perceptible way, on an everyday level, and progressively makes them more precise and rigorous, eventually reaching the level of fairly sophisticated proofs. This allows meaningful problems to be tackled from the outset. Another unusual trait of this book is that it deals mainly with constructions and maps, rather than with proofs that certain maps and constructions do or do not exist. The numerous illustrations are an essential feature. The book is accessible not only to undergraduates but also to high school students and will interest any reader who has some feeling for the visual elegance of geometry and topology. ... Read more

Product 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. ... Read more

Customer Reviews (9)

it's ggrrrrrrrrrrreat!
I consider myself to be a pretty lousy graduate student and I still found this book to be very readable. This book is also cheap enough that you may want keep an extra copy around, as it makes a great gift item/stocking stuffer.

a must-read supplement for topology students
Milnor's "Topology from the Differentiable Viewpoint" is a brief sketch of differential topology, well written, as are all books by Milnor, with clear, concise explanations. For students who wish to learn the subject, it should be read as a companion to a more substantive text, such as Guillemin & Pollack's Differential Topology or Hirsch's Differential Topology, as too much of the material is left out for this to be adequate as a textbook. OTOH, it does make for good bedtime reading.

While this book is highly regarded among mathematicians, it is not without its faults, namely,
- it fails to cover many topics of importance, such as transversality (only mentioned in an exercise), embeddings, differential forms, integration, Morse theory, and the intersection form;
- it only cites some theorems without proving them, or it leaves the proofs to the reader;
- it offers proofs of many theorems that are really only sketches without all the details;
- manifolds are only defined as subsets of Euclidean spaces;
- there is only 1 collection of 17 problems at the end of the book, which are used to introduce important concepts; and
- it probably moves too quickly for true beginners, packing a lot into only 51 pages.

So don't buy this as your only, or even first, book on differential topology. Oddly, many of the faults that I listed above are simultaneously strengths, in that it can be read very quickly, with relatively little effort and a high rate of retention. Milnor really emphasizes the topology of the subject, giving applications such as the fundamental theorem of algebra, Brouwer's fixed point theorem, the hairy ball theorem, the Poincare-Hopf theorem, and Hopf's theorem. Most of the book focuses on degree theory, but there is also a nice introduction to framed cobordism, which is rare for an elementary book. Guillemin & Pollack's book was based in large part on this one, and could be read together, with G&P giving more elementary explanations and additional topics, while Milnor's book provides a proof of the Sard theorem and the Pontrjagin-Thom construction. The exercises, though not particularly difficult, do provide a good opportunity to practice proving theorems in the subject, as there are no hints for them, as one would find in many other differential topology books, and they are not separated by chapter.

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. ... Read more

Product Description

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

Customer Reviews (1)

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

Product Description
Proceedings of the Northeast Conference on the subject at Wesleyan University, Connecticut, in June 1988. The two dozen papers, by mathematicians from the US, Canada, and the Netherlands, report on recent advances in topology for research mathematicians and graduate students. They focus on the theor ... Read more

Product Description
This book combines mathematics (geometry and topology), computer science (algorithms), and engineering (mesh generation) in order to solve the conceptual and technical problems in the combining of elements of combinatorial and numerical algorithms.The book develops methods from areas that are amenable to combination and explains recent breakthrough solutions to meshing that fit into this category. It should be an ideal graduate text for courses on mesh generation.The specific material is selected giving preference to topics that are elementary, attractive, lend themselves to teaching, are useful, and interesting. ... Read more

Customer Reviews (2)

A very nice, clear book
And much more pleasant to read than Edelsbrunner's Computational Geometry book. I have very little interest in Mesh Generation, but most of the book is not about that, but is rather a rather lucid introduction to central topics in modern discrete and computational geometry.

Excellent read for mathematically inclined reader
OK, I must admit, I know the author personally, like his work a lot. And so I can only recommend this book. But that's not what you're looking for in a review, nor am I talking to freinds or collegues who already know hime too.This book will introduce you to simplicial complexes, and deep mathematical constructs, along with some topology and geometry, while at the same time remaining hands-on and simpler.Mostly, the notation is clean, simple, and yet rigorous enough that you'll really be in terrific shape one you integrate it. It's actually amazing that it comes so clean given how powerful it can be. Meshes are the basis for many of the computer graphics and CAD/CAM modern methods, and are an indispensible tool. That's the real value of reading this book: you'll get some real good tools for manipulating meshes (whether what you want to do is the same or different than the author), and especially you'll get a mathematically correct and rigorous treatment.

At the same time this book is quite manageable even (and foremost) if you don't have a PhD in algebraic topology (all you need is a good bachelor in some computational science with good mathematics foundations). Although it might challenge you at times, it is basically self-contained and does not rely on any other book. (Additional knowledge is always useful, but this is a good starting point.)
You'll earn about simplicial complexes not from an abstract algebraic topology point of view (although the author is well-acquainted with them) but as a tool for representing surfaces and solids.The topics revolve around reconstructing a surface from a point cloud (using so-called Delaunay triangulations, which is one of the prevalent methods for that problem). It starts simply with 2D and moves on to 3D.It builds on the research of the author for more than a decade.

I should mention in fine that the author is the creator of Alpha-Shapes(a well known Delaunay-based method for reconstruction) and of the succesful startup Raindrop Geomagic which uses these methods in an essential way. His current interests are linked to biogeometry. Most of the research in this book somehow made it in one way or the other in their products. So this is excellent reading and a must for anyone interested in meshes, whether from computer graphics, CAD/CAM, or scientific computation / finite elements. ... Read more

Customer Reviews (1)

few topics, of very limited value
Munkres' "Elementary Differential Topology" was intended as a supplement to Milnor's Differential topology notes (which were similar to his Topology from the Differentiable Viewpoint but at a higher level), so it doesn't cover most of the material that standard introductory differential topology books do. Rather, the author's purpose was to (1) give the student a feel for the techniques of differential topology and practice in using them, and (2) prove a couple of basic and important results that at the time (1961) had not appeared in book form. Thus this book could not serve as a textbook for a course in the subject, but could be useful perhaps as a workbook for a student who wanted to practice solving problems. The word "elementary" in the title merely indicates that no algebraic topology is used in the proofs (with one minor exception, to show that a disk cannot be mapped homeomorphically onto an annulus) - its use was not intended as an indication of the level of the book, although it is pretty elementary anyway.

That this is not suitable as a text for learning differential topology is apparent from what material has been omitted: Sard's theorem, Whitney's imbedding theorem, Morse theory, transversality (except for a brief mention in the last couple of pages), the degree of a map, intersection theory, differential forms, vector bundles (except for the tangent and normal bundles), etc., to say nothing of more advanced topics such as cobordism or surgery.

So what is covered? Aside from basic definitions of C^r manifolds (i.e., manifolds with charts that have transition functions that are r times continuously differentiable), submanifolds, immersions, diffeomorphisms, bump functions, partitions of unity, and the inverse and implicit function theorems (proved only for Euclidean spaces), the results are divided into 2 sets: Those having to do with approximating a map with certain features by other maps (generally, showing that the set of maps with certain properties, such as imbeddings, immersions, diffeomorphisms, etc., is open in a certain function space). From this follows the well-known result that all C^r (r>=1) manifolds are smooth, the highlight of the first 2/3 of the book. Along the way, a few results are demonstrated that are needed in the proof, such as the existence of tubular neighborhoods and an imbedding theorem that is much weaker than Whitney's, but not much time is spent on them. This part ends with a proof of the uniqueness of the double of a manifold. Virtually all of these results can be found in Hirsch's Differential Topology in the first 2 chapters, proved much simpler and with modern notation. However, by keeping his presentation more geometric and with a minimum of formalism, it may be easier to follow Munkres' proofs (not that Hirsch is hard). As an example, Munkres uses for the topology of his function spaces the strong C^1 topology, rather than the compact-open topology that Hirsch uses.

The second part of the book, the final 40 pages or so, is devoted to proving that smooth manifolds are actually PL manifolds, and that the triangulation of a smooth manifold with a given smooth structure is essentially unique (a kind of smooth Hauptvermutung - this is not true for PL manifolds in general). This classic result is not usually included in differential topology (or PL topology) books - in fact, I can't think of another book which does contain this proof, making this the best (only?) reason to own this book. The proof itself is not that interesting, consisting of the standard manipulations of simplices that one usually sees in PL topology or older homology theory.

There are many "exercises" through the book, which generally ask the reader to fill in the details of proofs or extend the results of them. These tend to be pretty easy, whereas the many "problems" are harder. For these, hints are often given, so they usually aren't that difficult either (although one problem is labeled as "unsolved"). Aside from the proof that smooth => PL, the only other benefit of reading this book is to practice doing these exercises. But overall, this is far inferior to the aforementioned works of Milnor, Hirsch, Wallace (Differential Topology: First Steps), or Guillemin and Pollack (Differential Topology). ... Read more

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

Customer Reviews (1)

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

Product Description
Contents: Introduction. - Fundamental Concepts. -Topological Vector Spaces.- The Quotient Topology. -Completion of Metric Spaces. - Homotopy. - The TwoCountability Axioms. - CW-Complexes. - Construction ofContinuous Functions on Topological Spaces. - CoveringSpaces. - The Theorem ofTychonoff. - Set Theory (by T.Br|cker). - References. - Table of Symbols. -Index. ... Read more

Customer Reviews (5)

great as motivation but not a textbook
While I agree with the other reviewers here that Jaenich's "Topology" is very well written, goes to great lengths to explain the "hows and whys" of topology, and includes many, many figures (about 1 per page on average), it is probably more popular with people who already know topology than with beginning students, even though it is an introductory text intended for undergraduates. This is due to both a frequent lack of precision or formality in proofs and definitions coupled with a tendency to discuss much more advanced material with which a student at this level wouldn't be familiar. I believe that experienced mathematicians, who perhaps learned point-set topology from books such as those of Munkres, Kelley, or Bredon (or even an analysis book such as Royden), appreciate how this book focuses on motivating the concepts, explaining how the various objects are used elsewhere in mathematics - for that purpose this is one of the finest books I have seen. However, too much material is mentioned that is certainly over the heads of most students new to topology, such as the Pontrjagin-Thom construction, the spectrum of commutative Banach algebras, or Lie groups, often in a very cursory manner that would serve only to confuse beginners. Concepts are often used before they are defined, or are not defined precisely, which is liable to frustrate these students as well. Many topics are given such short attention it makes you wonder why the author even bothered - such as a page devoted to Frechet spaces followed by a section consisting of a single paragraph on locally convex topological vector spaces. Much of the material is not covered very deeply - only a definition and maybe a theorem, which half the time isn't even proved but just cited.

Certainly this book couldn't be used as a textbook for an undergraduate course - for the reasons mentioned above and also because not enough material is actually covered, as well as the obvious deficiency in that it lacks exercises for the reader. Most of the proofs until the last chapters are of the 1- or 2-paragraph variety, with some pictures added, although as the book progresses the level becomes increasingly more sophisticated. The book also covers both point-set topology (topological spaces, compactness, connectedness, separation axioms, completeness, metric topology, TVS, quotient topology, countability, metrization, etc.) and elements of algebraic topology (homotopy, fundamental group, simplices, CW-complexes, covering spaces, but not really homology), but the presentation of the algebraic topology in particular is not liable to be helpful for the novice, except for the treatment of covering spaces, which is perhaps the highlight of the book. Half of the chapter on homotopy is actually concerned with categories and functors, probably not the best way to introduce the subject. In fact, here is direct quote from the index:
"We talk about homology (and a number of other objects beyond the realm of point-set topology) several times in this book, but the definition is not given."
That, in a nutshell, explains the difficulty with this book.

So why am I rating this 5 stars? For the wealth of examples (e.g., 4 sections on examples of quotient spaces) and explanations of how these concepts are used and why they are important. Just by looking at the contents one can see this, as there are sections titled:
"What is point-set topology about?," "What is algebraic topology?," "Homotopy - what for?," "The role of the countability axioms," "Why CW-complexes are more flexible," "Yes, but...?," "The role of covering spaces in mathematics," "What is it [Tychonoff's theorem] good for?"
The chapter on covering spaces, coming near the end of the book, is particularly good, with a proof of their classification given. This is definitely the most fleshed-out part; if only the rest of the book could go into this depth.

This book would make an excellent supplement to a more formal textbook such as Munkres, but is not a substitute for it. But I would still consider this as a must-read for all those students who plan on studying mathematics in graduate school.

A simple introduction to advanced mathematical concepts
This text gives the reason behind many advanced topological concepts and tantalizes the reader with it's varied applications.

Basic topological concepts of open, closed, continuous, product topology, connectedness,compactness and intro to separation axioms is presented in a logical concise and easy to understand way.

The author then delves into topological groups and vector spaces introducting Hilbert Banach and Frechet spaces ( albeit briefly ).

Quotient spaces,homotopy, complexes and urysohn and tietze lemma along with partitions of unity are tackled next.

I especially enjoyed the section on covering spaces with which it concludes.

Perhaps the single best accolade I can give the book is that it gives one inspiration and motivation to explore in greater detail mathematical objects discussed.

The text is useful to all students of mathematics and physics alike.

Full of motivations
This book is fun to read. In a weekly homework meeting for an Algebraic Geometry class, I complained to one grad student "Geometry textbooks should have many pictures", and he asked "Define 'many'?" I said "One on each page". Now this topology book is certainly close to that. (It has more than 180 illustrations.) Though its written style is a bit informal, 'handwaving' arguments can serve as outlines of rigorous proofs.

Since it does not have any problem sections, I can see why Munkres' book is more popular in college. It still gives some inspiring questions from time to time. Besides the basic pot-set topology, it also covers some algebraic topology and differentialtopology. The author does not hesitate to use examples from those advanced areas without formal definitions, and this was a bit annoying when I read it the first time. In this sense, the book is not really selfcontained. However, when finally a notion is formally defined, I can see it from various aspects in those examples. This really helps me understand topology better, and makes me want to explore them. After reading the existence thm of covering spaces in chapter 9, I realized that mathematics is really an art.

The index in the back of the book is in the format of short definitions, which can be used as a quick reference.

Students: BUY THIS BOOK!!!
It is not too often that a book about topology is written with the goal of actually explaining in detail what is going on behind the formalism. The author does a brilliant job of teaching the reader the essential concepts of point set topology, and the book is very fun to read. The reader will walk away with an appreciation of the idea that topology is just not abstract formalism, but has an underlying intuition that is rich in imagery. The author has a knack for allowing readers to "see into the future" of what kind of mathematics is waiting for them and how topology is indispensable in its study.

At the end of chapter three, which deals with the quotient topology, the author writes the following paragraph: "If is often said against intuitive, spatial argumentation that it is not really argumentation but just so much gesticulation - just 'handwaving'. Shall we then abandon all intuitive arguments? Certainly not. As long as it is backed by the gold standard of rigorous proofs, the paper money of gestures is an invaluable aid for quick communication and fast circulation of ideas. Long live handwaving!". This has to rank as one of the best paragraphs that has every appeared in a mathematics book, for it nicely summarizes the need for developing a feel for the concepts behind mathematics before moving on to the rigorous proofs. Physicists in particular, who must assimilate mathematics very quickly in order to apply it to real problems must have a pictorial, "playful" understanding of the mathematical constructions.

Thus the language that the author employs is informal, and a listing of the best discussions in the book would really entail a listing of every one in the book. There is not one part of the book that is not helpful or interesting, and the author delves into many different areas that involve the use of topology.

If you are a beginning student in mathematics, BUY AND STUDY THIS BOOK...BUY AND STUDY THIS BOOK. You will take away so much for the price paid.

Excellent
Excelent, clear, well-motivated introduction ... Read more

Product 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.

The present book grew out of notes for an introductory topology course at the University of Alberta. It provides a concise introduction to set theoretic topology (and to a tiny little bit of algebraic topology). It is accessible to undergraduates from the second year on, but even beginning graduate students can benefit from some parts.

Great care has been devoted to the selection of examples that are not self-serving, but already accessible for students who have a background in calculus and elementary algebra, but not necessarily in real or complex analysis.

In some points, the book treats its material differently than other texts on the subject:

* Baire's theorem is derived from Bourbaki's Mittag-Leffler theorem;

* nets are used extensively, in particular for an intuitive proof of Tychonoff's theorem;

* a short and elegant, but little known proof for the Stone-Weierstrass theorem is given.

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

Product Description

Customer Reviews (2)

Should've been better, but a bit too unorthodox
Gauld's "Differential Topology" is primarily a more advanced version of Wallace's Differential Topology: First Steps. The topics covered are almost identical, including an introduction to topology and the classification of smooth surfaces via surgery, and a few of the pictures and some of the terminology ("disconnecting surgery," "twisting surgery") are the same, too. But overall, Gauld is written at a higher level (even though it is also an introduction to the subject, for undergrads) and is much more rigorous. Also as with Wallace, the presentation focuses on topology, with no coverage of such analytic and geometric concepts such as Riemann metrics, differential forms, integration, Lie groups, etc., so this is not fungible with Lee's Introduction to Smooth Manifolds, Lang's Differential and Riemannian Manifolds, or Barden & Thomas's An Introduction to Differential Manifolds, in addition to being more elementary than these books. I probably would've rated this 5 stars if not for the unusual presentation of topological spaces and the sloppiness in the proof of the aforementioned classification.

The book begins with a 3-chapter introduction to point-set topology. Like Wallace, the amount of material covered is inadequate as a substitute for a course in general topology - only basic definitions (openness, neighborhoods, continuity) and Hausdorffness, connectedness, and compactness are covered. There is a decent amount of motivating discussion and examples, but this is done in by a serious flaw: The author uses his pet approach to topology via "nearness." Nearness is essentially the property of a point being an accumulation point (or limit point or point of closure - the terminology varies among authors) of a set. If one starts with a nearness relation, defined on the collection of points and subsets of a set, then one can define the concepts of neighborhood, openness, and continuity in terms of this relation. This is an idea that was pioneered in the '70s, first for the definition of continuity (which is particularly nice if one uses nearness) and then, by the author, for the definition of topology, where it is not as useful. This approach never caught on, so a beginning student who learns topology for the first time using this book may have problems when reading just about any other book on the subject; conversely, if that student has already learned topology elsewhere, this book will at first seem confusing. This aspect is probably the worst feature of this book.

After the brief introduction to topology, manifolds are defined and some useful analytic tools in R^n, such as the inverse function theorem, the Morse lemma, and bump functions, are treated. Another deficiency here is that manifolds are defined with no mention of the 2nd countability (or paracompactness) of the manifold. This isn't necessary for the purposes of this book since partitions of unity (and their applications) are never discussed and most of the theorems concern compact manifolds only, but the full definition should've been given for completeness (cf. Wallace, who does give the proper definition even though he, too, has no need for the 2nd countability). Morse theory is introduced fairly early in the book and then developed over the course of several chapters - the presentation is not as unified or complete as in Milnor's books, but is certainly superior to that of Wallace.

The book includes the standard proof of the easier Whitney embedding theorem (oddly, divided between chapters 7 and 15), as well as all 3 definitions of tangent spaces (similar to Broecker & Jaenich's Introduction to Differential Topology, but less concise) and some coverage of vector fields, albeit without precisely defining the tangent bundle (or mentioning vector bundles at all).

What really sets this book apart is the emphasis on differential structures and orientability via charts and bases of them. In other words, throughout the book, showing that a manifold (or map) is smooth or orientable, and comparing different smooth structure or orientations, is done by working with explicit choices of charts. It is this focus that is utterly lacking in Wallace. As an example, it is shown how to create infinitely many "different" differential structures on the real line by using different charts (but the author fails to note that these are all diffeomorphic). The many pictures of coordinate patches on manifolds and computations involving Jacobians really show the reader how one can deduce these properties in practice.

The treatment of surgery, from multiple perspectives, is also outstanding. First it is defined using embedded spheres and balls, and then the notion of differentiable gluing is used to improve the construction, to show that it is smooth (with no handwaving about smoothing corners). Next, the trace of the surgery is defined and that is related to the neighborhood of a critical point, using several different explicit representations of that neighborhood, with many excellent diagrams showing the different possibilities. Applying surgical techniques to classify surfaces starts well, even first using it to classify 1-manifolds, but the exposition starts to break down, with some results concerning moving critical points being proved in an informal manner, and a very confusing explanation accompanying an equally confusing diagram on page 200. Most disappointing is that Gauld substitutes homeomorphic for diffeomorphic in his ultimate theorem, but at least he (unlike Wallace) acknowledges this deficiency and points out that it can be overcome. After being so careful for most of the book, Chapter 14 should've been written at the same level of rigor.

The 2 appendices fill in some missing details in the proofs (Appendix A) and point toward extensions of the material (Appendix B). However, even in Appendix A, most of the proofs are just sketched. Moreover, and this is another irritating feature of the presentation, throughout the text a proof will sometimes just be given as one word: "Omitted." This means that the proof, or its sketch, is in Appendix A, but the author never tells the reader this. And then there are a number of places where a fact is used (such as that the genus of a surface is finite) but not justified, making it appear as if the author is making an unsupported assumption when in fact the claim is proved in the appendix; references to Appendix A really should've been added to the text. Appendix B touches upon other topics, such as other separation properties besides Hausdorffness, the classification of nonorientable surfaces (this should've been treated in the main body of the text, in more detail), Sard's theorem (not proved here), the smooth Brouwer fixed point theorem without homotopy theory, Hausdorff dimension (really out of place here), and the barest mention of dynamical systems. Most of this appendix is too little to be of any use.

This is a reprint of a camera-ready typescript (i.e., it looks like it was typed on an old typewriter), so it is hard on the eyes. Despite this, it is relatively free from typos, as well as other errors, until near the end of the book (e.g., in the last paragraph on p. 201 the variable S should have a tilde twice, on p. 202 the variable epsilon should've been defined before it was used, on p. 153 it should read "surgery of type (m,n)" not "to type (m,n)," on p. 112 the definition of U should include the words "range of h" not "domain of h" and the word "open" should've been inserted before "neighborhood" a few lines earlier).

There are a good amount of straightforward exercises following each chapter, with the results of some being cited in the text, so this could serve well as a textbook (it was designed for an experimental course), notwithstanding the issues that I've raised concerning some of its definitions. Don't be put off by the silly limericks that open the book, either. It's a shame - with a little more work and a little less idiosyncrasy, this could've been the best book on the differential topology for undergraduates, but even as is it makes a nice companion to Guillemin & Pollack's Differential Topology in that their overlap is not that large, so the 2 books combined do a good job of covering the subject.

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

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Product Description
This volume contains the main part of the lectures contributed to the conference. They reflect the new trends of development in general topology. ... Read more

Product Description
These papers survey the developments in General Topology and the applications of it which have taken place since the mid 1980s. The book may be regarded as an update of some of the papers in the Handbook of Set-Theoretic Topology (eds. Kunen/Vaughan, North-Holland, 1984), which gives an almost complete picture of the state of the art of Set Theoretic Topology before 1984. In the present volume several important developments are surveyed that surfaced in the period 1984-1991.

This volume may also be regarded as a partial update of Open Problems in Topology (eds. van Mill/Reed, North-Holland, 1990). Solutions to some of the original 1100 open problems are discussed and new problems are posed. ... Read more