Product Description
The uniqueness of this text in combining geometric topology and differential geometry lies in its unifying thread: the notion of a surface. With numerous illustrations, exercises and examples, the student comes to understand the relationship between modern axiomatic approach and geometric intuition. The text is kept at a concrete level, 'motivational' in nature, avoiding abstractions. A number of intuitively appealing definitions and theorems concerning surfaces in the topological, polyhedral, and smooth cases are presented from the geometric view, and point set topology is restricted to subsets of Euclidean spaces. The treatment of differential geometry is classical, dealing with surfaces in R3 . The material here is accessible to math majors at the junior/senior level. ... Read more

Customer Reviews (5)

Great Book
It is a very intiutive book in both areas. Also at the end of the book there is a good material for further study, author explains the research fields in Geometry/Topology and related books. If you are an undergraduate and want to get an overall idea about the gradute study in topology and geometry that is a nice introduction.

a remark on omissions
I have not read the book, only the reviews.In one excellent review here it is remarked that it is "unfortunate" that the author does not prove the Schoenflies theorem and the triangulability of surfaces.

later this same reviewer observes that the proof of the smooth Gauss Bonnet theorem in the book seems relatively hard.I merely wish to point out that the author has made choices in the reader's interest both by what he includes and what he omits.

The two theorems named above which are not proved, could well take another entire book to prove.They are far harder than the smooth Gauss Bonnet theorem.

I have seen entire books devoted to proving triangulability, and Schoenflies theorem was the subject of weeks of tedious work in a topology course I took as a student.I still dislike even hearing of this result.So if these omissions are the reviewer's only criticisms of the book, they should rightly be considered pluses.

Hence I also give the book at least 4 stars, by logical deduction.

Good introduction
This book is suitable for reading at an advanced undergraduate or beginning graduate level. The author is careful to present the subject from both a rigorous point of view and one that emphasizes the geometric intuition behind the subject. These two approaches to teaching topology are not mutually exclusive, with this book giving a good example of this.

After a brief overview of the elementary topology of subsets of Euclidean space in chapter 1, topological surfaces are discussed in chapter 2. Surfaces are built up from arcs, disks, and one-spheres. Unfortunately, the proofs of the theorem of invariance of domain and the Schonflies Theorem are not included, but references are given. Gluing techniques though are effectively discussed, and the author does not hesitate to use diagrams to explain the relevant concepts. The more popular constructions in surface topology, namely the Mobius strip and the Klein bottle are given as examples of the cutting and pasting techniques. The amusing fact that the Klein bottle can be obtained from gluing two Mobius strips along their boundaries is proven.

The theory of simplicial surfaces is discussed in the next chapter. Simplicial surfaces are much easier to deal with for beginning students of topology. Simplicial complexes are introduced first, and the author then studies which simplicial complexes have underlying spaces that are topological surfaces. He proves that this is the case when each one-dimensional simplex of the complex is the face of precisely two two-dimensional simplices, and the underlying space of each link of each zero-dimensional simplex of the complex is a one-dimensional sphere. Unfortunately, the author does not prove that any compact topological surface in n-dimensional Euclidean space can be triangulated. The Euler characteristic is defined first for 2-complexes and it is shown that it is the same for two simplicial surfaces that triangulate a compact topological surface. The author does prove in detail the classification of compact connected surfaces. Interestingly, the author also proves a simplicial analogue of the Gauss-Bonnet theorem, and gives a proof of the Brouwer fixed point theorem.

The author turns to smooth surfaces in the next few chapters, wherein curves are defined along with the relevant differential-geometric notions such as curvature and torsion. The fundamental theorem of curves is proven. The reader is first introduced to the concept of what in more advanced treatments is called a differentiable manifold, and several concrete examples are given of smooth surfaces. The differential geometry of smooth surfaces is outlined, with the first fundamental form and directional derivatives discussed in great detail. The reader should be familiar with the inverse function theorem to appreciate the discussion of regular values.

Even more interesting differential geometry is discussed in chapter 6, which covers the curvature of smooth surfaces. The important Gauss map is defined, along with the Weingarten map and the second fundamental form. This allows an intrinsic notion of curvature, but the author does perform explicit computations of curvature using various choices of coordinates. The proof that Gaussian curvature is intrinsic (Theorema Egregium) is proven, along with the fundamental theorem of surfaces. Geodesics, so important in physical applications, are discussed in the next chapter. The reader gets a first look at the "Christoffel symbols", even though they are not designated as such in the book.

The book ends with a thorough treatment of the Gauss-Bonnet theorem for smooth surfaces. The smooth case is much more difficult to prove than the simplicial case, as the reader will find out when studying this chapter. The author also gives a very brief introduction to non-Euclidean geometry.

Presentation of The Spirit...
Lots of times the mathematicians stuck in proofs,in that fool symbols, forgetting the ideas, the picture.One can never find the right way withclosed eyes.This book teaches to think, getting beyond the symbols. Ithas also useful advises about the research areas. The author made his Phdat Cornell with D.Henderson.A beautiful undergraduate text.

Just a mediocre book of lesser extent
First, the title reads fine. But there's a catch. This kind of title sounds like it covers all. The truth is. It ain't true. Second. The author's attitude. I'd rather say the author is talking to himself. ... Read more

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

Customer Reviews (1)

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

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

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

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

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

Overall this book, combined with Hirsch for the Morse theory and surgery, would constitute the ideal 1st-year graduate course in differential topology (for topology students). It also covers the core preparatory material for Kosinski as well. However, students with no prior exposure to the subject would probably be better served by looking at Guillemin & Pollack or Lee first.
... Read more

Product Description
Accessible, concise, and self-contained, this book offers an outstanding introduction to three related subjects: differential geometry, differential topology, and dynamical systems. Topics of special interest addressed in the book include Brouwer's fixed point theorem, Morse Theory, and the geodesic flow.

Smooth manifolds, Riemannian metrics, affine connections, the curvature tensor, differential forms, and integration on manifolds provide the foundation for many applications in dynamical systems and mechanics. The authors also discuss the Gauss-Bonnet theorem and its implications in non-Euclidean geometry models.

The differential topology aspect of the book centers on classical, transversality theory, Sard's theorem, intersection theory, and fixed-point theorems. The construction of the de Rham cohomology builds further arguments for the strong connection between the differential structure and the topological structure. It also furnishes some of the tools necessary for a complete understanding of the Morse theory. These discussions are followed by an introduction to the theory of hyperbolic systems, with emphasis on the quintessential role of the geodesic flow.

The integration of geometric theory, topological theory, and concrete applications to dynamical systems set this book apart. With clean, clear prose and effective examples, the authors' intuitive approach creates a treatment that is comprehensible to relative beginners, yet rigorous enough for those with more background and experience in the field. ... Read more

Customer Reviews (2)

Excellent book
It was a great pleasure to read the book “Differential Geometry and Topology With a View to Dynamical Systems” by Keith Burns and Marian Gidea. The topic of manifolds and its development, typically considered as “very abstract and difficult”, becomes for the reader of this outstanding book tangible and familiar.This joyful aspect of the book was achieved by the authors by setting the advanced material of differential geometry and topology as if on a “mobile bridge” or a “crossroad” that associates a(n) (primarily) unfamiliar abstract part of the text with elementary math theories. The latter pedagogical approach was mostly carried out through carefully prepared examples, in which, for essentially abstract structures and mathematical topics, well known familiar elementary settings serve as obvious motivations, which make the transition to a higher level of an abstraction smooth. Nevertheless, the scope of the main topic in this book, differential geometry and topology, is pretty far advanced.Besides the basic theory, centered around analytical properties of manifolds (mostly endowed with additional, in particular Riemannian, structures and vector or tensor fields defined on them) and their applications, it also provides a good introductory approach to some deeper topics of differential topology such as Fixed Points theory, Morse theory, and hyperbolic systems throughout the rest of the book.
The main stream of the applications that always follow or motivate the theoretical context is dynamical systems. Excellent examples reveal the close ties ofthisbeautiful mathematical theory with common problems intheoreticalphysics, classical and fluid mechanics, field theory, and, most importantly, the theory of general relativity.
The book by Burns and Gidea is also be strongly recommended for those readers who wish to enhance their mathematical tools to make possible a deeper insight into these fascinating physical theories.

Jerzy K.Filus

A very good book
A very clear and very entertaining book for a course on differential geometry and topology (with a view to dynamical systems).

First let me remark that talking about content, the book is very good. Each of the 9 chapters of the book offers intuitive insight while developing the main text and it does so without lacking in rigor. The first 6 chapters (which deal with manifolds, vector fields and dynamical systems, Riemannian metrics, Riemannian connections and geodesics, curvature and tensors and differential forms) make up an introduction to dynamical systems and Morse theory (the subject of chapter 8). Chapter 7 is devoted to fixed points and intersection numbers. The last chapter is an introduction to hyperbolic systems.

This enjoyable and highly instructive book contains a large number of examples and exercises. It is an incredible help to those trying to learn dynamical systems (and not only). It teaches all the differential geometry and topology notions that somebody needs in the study of dynamical systems.
The authors, without making use of a pedantic formalism, emphasize the connection of important ideas via examples. It completely enhanced my knowledge on the subject and took me to a higher level of understanding.

... Read more

Product Description
This first edition of this book quickly became an established text in this fast-developing branch of mathematics. This second edition has been significantly revised and expanded. It includes a section on new developments and an expanded discussion of Taubes' and Donaldson's recent results. ... Read more

Customer Reviews (2)

Perfect
That book is in perfect condition. I got it in just 1 week (free shipping). I just hope the price could be a little cheaper.

A must for researchers new to the field
An authoritative and comprehensive reference...McDuff and Salamon havedone an enormous service to the symplectic community: their book greatlyenhances the accessibility of the subject to students and researchersalike.

The discussion begins with classic topology and cover a variety offinal year undergraduate topics such as complex manifolds and inversedifferential techniques before moving into the vastly complex world ofSymplectic Topology.

A must for researchers new to the field ... Read more

Product Description
The book is intended as a text for a two-semester course in topology and algebraic topology at the advanced undergraduate orbeginning graduate level. There are over 500 exercises, 114 figures, numerous diagrams. The general direction of the book is towardhomotopy theory with a geometric point of view. This book would providea more than adequate background for a standard algebraic topology coursethat begins with homology theory. For more information seewww.bangor.ac.uk/r.brown/topgpds.htmlThis version dated April 19, 2006, has a number of corrections made. ... Read more

Customer Reviews (1)

Topology and Groupoids - great value!!
This book presents an unusual but very valuable approach to topology and homotopy in terms of groupoids, in particular, in terms of the fundamental groupoid.It is revised, and this is its third edition. It is self-contained and is beautifully written.The importance of groupoids have been amply justified by their increasing importance in noncommutative geometry and differential geometry.The book, for example, gives insight into the surprising abelian character of the higher homotopy groups by pointing out that we should be looking at the higher homotopy GROUPOIDS which are non-abelian.

The book is incredible value at $23.99.The print is excellent and delivery from BookSurge was in good time.There is also an e-version of the book.This book is wonderful value! ... Read more

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

Customer Reviews (3)

A readable alternative to Hatcher

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

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

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

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

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

Product Description
Excellent text for upper-level undergraduate and graduate students shows how geometric and algebraic ideas met and grew together into an important branch of mathematics. Lucid coverage of vector fields, surfaces, homology of complexes, much more. Some knowledge of differential equations and multivariate calculus required. Many problems and exercises (some solutions) integrated into the text. 1979 edition. Bibliography.
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Customer Reviews (7)

A reader's opinion
This is the second time I have bought this book since I offered
the first one to my son. An excellent introduction to the topic!

A good start
Historically, combinatorial topology was a precursor to what is now the field of algebraic topology, and this book gives an elementary introduction to the subject, directed towards the beginning student of topology or geometry. Due to its importance in applications, the physicist reader who is intending eventually to specialize in elementary particle physics will gain much in the perusal of this book.

Combinatorial topology can be viewed first as an attempt to study the properties of polyhedra and how they fit together to form more complicated objects. Conversely, one can view it as a way of studying complicated objects by breaking them up into elementary polyhedral pieces. The author takes the former view in this book, and he restricts his attention to the study of objects that are built up from polygons, with the proviso that vertices are joined to vertices and (whole) edges are joined to (whole) edges.

He begins the book with a consideration of the Euler formula, and as one example considers the Euler number of the Platonic solids, resulting in a Diophantine equation. This equation only has five solutions, the Platonic solids. The author then motivates the concept of a homeomorphism (he calls them "topological equivalences") by considering topological transformations in the plane. Using the notion of topological equivalence he defines the notions of cell, path, and Jordan curve. Compactness and connectedness are then defined, along with the general notion of a topological space.

Elementary notions from differential topology are then considered in chapter 2, with the reader encountering for the first time the connections between analysis and topology, via the consideration of the phase portraits of differential equations. Brouwer's fixed point theorem is proved via Sperner's lemma, the latter being a combinatorial result which deals with the labeling of vertices in a triangulation of the cell. Gradient vector fields, the Poincare index theorem, and dual vector fields,which are some elementary notions in Morse theory, are treated here briefly.

An excellent introduction to some elementary notions from algebraic topology is done in chapter 3. The author treats the case of plane homology (mod 2), which is discussed via the use of polygonal chains on a grating in the plane. Beginning students will find the presentation very understandable, and the formalism that is developed is used to give a proof of the Jordan curve theorem. Then in chapter 4, the author proves the classification theorem for surfaces, using a combinatorial definition of a surface.

The author raises the level of complication in chapter 5, wherein he studies the (mod 2) homology of complexes. A complex is defined somewhat loosely as a topological space that is constructed out of vertices, edges, and polygons via topological identification. He proves the invariance theorem for triangulations of surfaces by showing that the homology groups of the triangulation are same as the homology groups of the plane model of the surface. This is an example of the invariance principle, and the author briefly details some of the history of invariance principles, such as the Hauptvermutung, its counterexample due to the mathematician John Milnor, and Heawood's conjecture, the latter of which deals with the minimum number of colors needed to color all maps on a surface with a given Euler characteristic. Integral homology is also introduced by the author, and he shows the origin of torsion in the consideration of the "twist" in a surface.

In the last part of the book, the author returns to the consideration of continuous transformations, tackling first the idea of a universal covering space. Algebraic topology again makes its appearance via the consideration of transformations of triangulated topological spaces, i.e. simplicial transformations. He shows how these transformations induce transformations in the homology groups, thus introducing the reader to some notions from category theory. The elaboration of the invariance theorem for homology leads the author to studying the properties of the group homomorphisms via matrix algebra, and then to a proof of the Lefschetz fixed point theorem. The book ends with a brief discussion of homotopy, topological dynamics, and alternative homology theories.

The beginning student of topology will thus be well prepared to move on to more rigorous and advanced treatments of differential, algebraic, and geometric topology after the reading of this book. There are still many unsolved problems in these areas, and each one of these will require a deep understanding and intuition of the underlying concepts in topology. This book is a good start.

Splendidly intuitive yet rigorous
This covers the basics of algebraic topology with simplexes, covering in essence the fundamental ideas behind of the work of Poincare, Brouwer, and Alexander. He proves the Jordan curve theorem, classifies all compact surfaces, and the relationship with vector fields. The homology groups are defined and used.

There are excellent examples, clear writing, and humour. An outstanding introduction.

One nice feature is that he bases his notions of continuity on "nearness" not epsilon-delta.

An excellent read
Ignore those that suggest this book is too "elementary". This is a wonderful text that concretizes the more abstract notions of algebraic topology. True, it should not be your only text on algebraic topology, andthe proofs are not as rigorous as a pedant might want, but it clearlyconveys the geometric underpinnings of topology and deserves a space on anytopologist's bookshelf.

Not for resolute students of algebraci/diff. topology.
I believe the two existing reviews are over-ratng. True, the book is accessible to anyone without prior knowledge of topology/algebra, but the treatment is too "elementary".For example, the author doesn'teven introduce the word "mod 2 homology".If you are resolutelyto study algebraic (or differential) topology, this is NOT the book to"study". Try Bredon or Fomenko-Novikov or May. For the subjectcovered, look for the book by Stillwell. ... Read more

Product Description
This textbook on elementary topology contains a detailed introduction to general topology and an introduction to algebraic topology via its most classical and elementary segment centered at the notions of fundamental group and covering space. The book is tailored for the reader who is determined to work actively. The proofs of theorems are separated from their formulations and are gathered at the end of each chapter. This makes the book look like a pure problem book and encourages the reader to think through each formulation. A reader who prefers a more traditional style can either find the proofs at the end of the chapter or skip them altogether. This style also caters to the expert who needs a handbook and prefers formulations not overshadowed by proofs. Most of the proofs are simple and easy to discover. The book can be useful and enjoyable for readers with quite different backgrounds and interests. The text is structured in such a way that it is easy to determine what to expect from each piece and how to use it. There is core material, which makes up a relatively small part of the book. The core material is interspersed with examples, illustrative and training problems, and relevant discussions. The reader who has mastered the core material acquires a strong background in elementary topology and will feel at home in the environment of abstract mathematics. With almost no prerequisites (except real numbers), the book can serve as a text for a course on general and beginning algebraic topology. ... Read more

Customer Reviews (1)

good concept, too many typos
I like the concept of a problem textbook and this one has a great deal of good stuff.
Unfortunately the editing is too poor to give a correct rate. I urge the authors to rewrite the part on covering spaces which should not have been published with so many mistakes and to improve the solutions given to the problems at the back of the book. Also serious work is needed in reducing the number of typos. With these improvements I would be willing to consider a different rate. ... Read more

Product Description
This volume presents an array of topics that introduce the reader to key ideas in active areas in geometry and topology. The material is presented in a way that both graduate students and researchers should find accessible and enticing. The topics covered range from Morse theory and complex geometry theory to geometric group theory, and are accompanied by exercises that are designed to deepen the reader's understanding and to guide them in exciting directions for future investigation. ... Read more

Product Description
This book provides historical background and a complete overview of the qualitative theory of foliations and differential dynamical systems. Senior mathematics majors and graduate students with background in multivariate calculus, algebraic and differential topology, differential geometry, and linear algebra will find this book an accessible introduction. Upon finishing the book, readers will be prepared to take up research in this area. Readers will appreciate the book for its highly visual presentation of examples in low dimensions. The author focuses particularly on foliations with compact leaves, covering all the important basic results. Specific topics covered include: dynamical systems on the torus and the three-sphere, local and global stability theorems for foliations, the existence of compact leaves on three-spheres, and foliated cobordisms on three-spheres. Also included is a short introduction to the theory of differentiable manifolds. ... Read more

Product Description
Topology, the foundation of modern analysis, arose historically as a way to organize ideas like compactness and connectedness which had emerged from analysis. Similarly, recent work in dynamical systems theory has both highlighted certain topics in the pre-existing subject of topological dynamics (such as the construction of Lyapunov functions and various notions of stability) and also generated new concepts and results (such as attractors, chain recurrence, and basic sets). This book collects these results, both old and new, and organizes them into a natural foundation for all aspects of dynamical systems theory. No existing book is comparable in content or scope. Requiring background in point-set topology and some degree of ``mathematical sophistication'', Akin's book serves as an excellent textbook for a graduate course in dynamical systems theory. In addition, Akin's reorganization of previously scattered results makes this book of interest to mathematicians and other researchers who use dynamical systems in their work. ... Read more

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Customer Reviews (3)

I just couldn't read this.
Maybe I'm retarded, but this is a tough book for me to read.I have a basic knowledge of set theroy, but I couldn't get through chapter two of this book.

This book just blazes through stuff.Its proofs use as little writing as possible, which makes understanding the stuff more difficult.It takes me about 10 minutes to understand one sentence.At that rate it would probably take me 20 years to finish the book.You need to be part of a very, very special audience to enjoy this book.Get this only if your IQ is close to 200.

Hey, at least the price ain't bad compared to other college textbooks.

Excellent for Self-Teaching
I am teaching myself mathematics and I love this book. Not being an expert at math, I can't speak for everyone when I say that this book is at times very challenging. But, in my case, that is part of why I find it so enjoyable; unlike all of the other (math) books I presently own, its difficulty really inspires me to read further.

An oldy but a goody
A good succinct presentation of the basics of topology with topological groups folded in.The writing is easy to follow, but the book is definitely not light on rigor.

May not work well for a one-semester course, as the topology and group theory are interwoven fairly substantially.One does not need to know group theory before reading the book, but one cannot easily escape it and still get at all the topology. ... 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
Algebraic and Differential Topology presents in a clear, concise, and detailed manner the fundamentals of homology theory. It first defines the concept of a complex and its Betti groups, then discusses the topolgoical invariance of a Betti group. The book next presents various applications of homology theory, such as mapping of polyhedrons onto other polyhedrons as well as onto themselves. The third volume in L.S. Pontryagin's Selected Works, this book provides as many insights into algebraic topology for today's mathematician as it did when the author was making his initial endeavors into this field. ... Read more

Product Description

Application of the concepts and methods of topology and geometry have led to a deeper understanding of many crucial aspects in condensed matter physics, cosmology, gravity and particle physics. This book can be considered an advanced textbook on modern applications and recent developments in these fields of physical research. Written as a set of largely self-contained extensive lectures, the book gives an introduction to topological concepts in gauge theories, BRST quantization, chiral anomalies, sypersymmetric solitons and noncommutative geometry. It will be of benefit to postgraduate students, educating newcomers to the field and lecturers looking for advanced material.

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
How many dimensions does our universe require for a comprehensive physical description? In 1905, Poincaré argued philosophically about the necessity of the three familiar dimensions, while recent research is based on 11 dimensions or even 23 dimensions. The notion of dimension itself presented a basic problem to the pioneers of topology. Cantor asked if dimension was a topological feature of Euclidean space. To answer this question, some important topological ideas were introduced by Brouwer, giving shape to a subject whose development dominated the twentieth century. The basic notions in topology are varied and a comprehensive grounding in point-set topology, the definition and use of the fundamental group, and the beginnings of homology theory requires considerable time. The goal of this book is a focused introduction through these classical topics, aiming throughout at the classical result of the Invariance of Dimension. This text is based on the author's course given at Vassar College and is intended for advanced undergraduate students. It is suitable for a semester-long course on topology for students who have studied real analysis and linear algebra. It is also a good choice for a capstone course, senior seminar, or independent study. ... Read more

Customer Reviews (3)

My First Book in Topology
John McCleary, the author of "A First Course in Topology: Continuity and Dimension", attempted to answer the question -- What is topology?--in his book. According to Wikipedia, topology is the study of those properties of objects that do not change when homeomorphisms are applied. John McCleary thinks that the central concept of topology is continuity, defined for the functions between the sets equipped with a notion of nearness (topological spaces). Since a coffee mug can be continuously deformed into a donut, the topological spaces of the coffee mug and donut are homeomorphic. The two spaces are the same from a topological view point.

John McCleary, the author, revealed that he spent too much time on developing the definitions when he first taught his undergraduate courses in topology. In fact, his book contains many definitions but a few examples. His strategy on presenting topology is that: "The first chapter reviews the set theory, so that the problem of topological invariance of dimension can be posed. The next five chapters treat the basic point-set notions of topology. The next two chapters treat the fundamental group of a space (equivalent spaces lead to isomorphic groups). The last two chapters focus on the combinatorial theme (simplicial complexes). We then associate the homology of the simplicial complexes, a sequence of vector spaces. This eventually leads to a proof of the topological invariance of dimension through homology."

There are hardly examples on the book. Thus the readers have hard time on applying the concepts. On the other hand, the book comes with proofs on well-known results. For example, the well-known proofs include (1) the fundamental theorem of algebra and (2) merely five Platonic solids exist.

$35 - This book is FREE online!
This is an awesome book, however, $35 dollars is a bit high.You can download the book online for free.If the book was $20 or less, I would say sure, but $35 for a thin soft cover......download it and review it yourself.

Elegant introduction to topology.
Being a mathematics student, I find McCleary as one of the finest introduction to topology for a beginner. ... Read more

Product Description
In this book structural principals of proteins are reviewed and analysed from a geometric perspective with the aim on revealing the underlying regularities in their construction. Computer methods for structure analysis and the automatic comparison and classification of these structures are reviewed with an analysis of the statistical significance of comparing diferent shapes. Folloiwng an analysis of the current state of the classification of proteins, more abstract geometric and topological representations are explored, including the occurrence of knotted topologies. The book concludes with a consideration of the origin of higher-level symmetries in protein structure. ... Read more