Product Description
Topology is a relatively young and very important branch of mathematics, which studies the properties of objects that are preserved through deformations, twistings, and stretchings. This book deals with the topology of curves and surfaces as well as with the fundamental concepts of homotopy and homology, and does this in a lively and well-motivated way. This book is well suited for readers who are interested in finding out what topology is all about. ... Read more

Customer Reviews (1)

Gentle topology book without heart
This short, elementary survey of topology is meant to be accessible for the most part to high school students and beginning undergraduates. I hope that such unspoiled souls will have the courage to be dissatisfied at least with the first chapter, since it cares only about concepts (continuity, homeomorphism, etc.) while offering little substance, and also there is the usual overemphasis of the Jordan curve theorem and pathological curves. Young people should not be tricked into thinking that topology has been built around such silly things. But in the other two chapters we get to actual topology, and all the usual stuff is here: Euler characteristic, classification of surfaces, knots, the fundamental group, homology, etc. Each topic is treated in a relatively sensible, swift manner; rather too swift towards the end, I think, when there seems to be a race to include as many topological concepts as possible, with little concern for what would be the most natural or interesting way to proceed (of course this does not have to be a bad thing if one is using the book as a down-to-earth complement to a formal textbook). ... Read more

Product Description
Applications from condensed matter physics, statistical mechanics and elementary particle theory appear in the book. An obvious omission here is general relativity--we apologize for this. We originally intended to discuss general relativity. However, both the need to keep the size of the book within the reasonable limits and the fact that accounts of the topology and geometry of relativity are already available, for example, in The Large Scale Structure of Space-Time by S. Hawking and G. Ellis, made us reluctantly decide to omit this topic. ... Read more

Customer Reviews (6)

Excellent overview and graphical explanation
This book shows you the geometric view of some advanced mathematical topics. It can greatly assist your intuition of what is going on in a mathematical setting when reading a true mathematics book. Armed with this book the other advanced text in Topology, Algebraic Geometry and Differential Geometry make more sense from a Physics point of view.

Excellent overview and graphical explanation
This book shows you the geometric view of some advanced mathematical topics. It can greatly assist your intuition of what is going on in a mathematical setting when reading a true mathematics book. Armed with this book the other advanced text in Topology, Algebraic Geometry and Differential Geometry make more sense from a Physics point of view.

Good attempt
When reading this book one can both admire these authors and feel sympathy with them. They have made an honest effort to explain the conceptsof differential geometry and topology in a way that is understandable and appreciated by the physicist reader. But the book falls short in many places, although there are some places where they do a fine job. They have taken on a very difficult project in this book, for it is quite straightforward to expound on the formalism of mathematics, but explaining it in a way that grants insight into its conceptual meaning is another matter altogether. Many physicists complain, with justification, that the way mathematics is presented in textbooks is not sufficient for giving them a deep appreciation of the underlying ideas involved. This, they argue, is what is needed for devising new physical theories and results based on these ideas. Physicists must assimilate very complex mathematical ideas very quickly in order to formulate these theories in a reasonable time frame. This is especially true in high energy physics, which in the last two decades has used mathematics like it has never been used before. Indeed, the mathematical complexity of high energy physics is dizzying, and if progress is going to be made in this field by the students of the 21st century, they are going to need mathematics books and documents that are more than just formal expositions. But, again, writing these kinds of books is very hard to do, and has yet to be done in a book to this date, although there are helpful discussions scattered throughout the mathematical literature.

Some of the concepts that need more in-depth explanation include: the theory of characteristic classes, sheaf theory, the theory of schemes in algebraic geometry, and spectral sequences in algebraic topology. There are of course many others, and some of the ones that the authors do a fairly good job of explaining in this book include: 1. the reason that the continuity of a function is defined in terms of inverses of open sets; 2. The orientability of a manifold; 3. The fundamental group and its relation with the first homology group. 4. The discussion on Morse theory.

Covers a lot of ground . . . but not always well
Unlike many physics students, I grant a lot of leeway to books on mathematics for physicists.I think it's all right for an author to engage in hand-waving arguments if this enhances physical intuition or even to make the occasional statements without proof if this allows more ground to be covered.However, if a proof actually is presented, I expect this proof to be correct.In this book, proofs are sometimes only for special cases of theorems stated more generally and often contain logical errors.

flawed and incomplete
Nash's book commits the sin many mathematical physics textbooks out there commit: "oh, we're writing for dimwit physicists, lets just give them a few scrawny examples and assure them everything else works alright." I'm sorry but writing for physicists is NOT an excuse for writing a sloppy textbook. Would you feel alright not knowing how an integral is defined? Would you use a numerical evaluation software to calculate integrals in serious research without understanding the algorithm it uses? If you do then you're a pretty shoddy physicist. I'm not saying this out of some "macho" sentiment many purist physicists have - I'm simply saying this because I feel the way this book teaches you diff. geometry is wrong - it teaches you to draw pictures and go by the pictures. When the pictures run out, so does your understanding.

This book is supposed to teach differential geometry. However, very little can be learned from it unless one already knows differential geometry: definitions are sometimes not general and sometimes not present at all, theorems are often stated only for special cases and even more often than that not proved at all. Sure, the book offers nice geometrical intuition, but this is not enough. An example: the book "proves" Stoke's theorem around page 40. Now, even a rigorous and condensed book would have problems doing that, considering the amount of "machinery" one needs to build up for it (tensors, differential forms, manifolds and so forth). This means the book makes a mess of it - big time.
There are many fine diff. geometry books out there, some for physicists, some not, which you should check out - Nakahara's text is so much better. For geometrical intuition I suggest picking up Schutz's book. Several books from the GTM (Graduate texts in mathematics series, the yellow ones) are really very accessible, such as Introduction to Topological Manifolds/Smooth Manifolds. Another good one is Allen Hatcher's Algebraic Topology for homotopy, homology and cohomology. For a good and responsible exposition, do yourself a favor and look for something else. ... Read more

Product Description
This significantly expanded second edition of Riemann, Topology, and Physics combines a fascinating account of the life and work of Bernhard Riemann with a lucid discussion of current interaction between topology and physics, The author, a distinguished mathematical physicist, takes into account his own research at the Riemann archives of Göttingen University and developments over the last decade that connect Riemann with numerous significant ideas and methods reflected throughout contemporary mathematics and physics.Special attention is paid in part one to results on the Riemann–Hilbert problem and, in part two, to discoveries in field theory and condensed matter such as the quantum Hall effect, quasicrystals, membranes with nontrivial topology, "fake" differential structures on 4-dimensional Euclidean space, new invariants of knots and more. In his relatively short lifetime, this great mathematician made outstanding contributions to nearly all branches of mathematics; today Riemann’s name appears prominently throughout the literature."The book is highly recommendable—for students and scientific workers—not only for the valuable information in it, but also for its spirit: history and higher mathematics are not dry here; they become alive and motivate further studies."—ZAA"This is a new translation of a book first published in English in 1987... Translated from Russian...it consists of two separate but related works. The first is an account of the life and work of Riemann, the second an account of several different topics in physics which are illuminated by the introduction of topological ideas. The discussion of Riemann is even better in the new edition. The mathematical account is richer and various errors have been corrected... The second half has been revised in a similar fashion... It has also been enriched by a new chapter which starts with von Neumann algebras and the work of Vaughn Jones... The book does three things very well: it reminds us of the range and depth of Riemann’s interests, which are emblematic of what the author values in mathematical physics; describes some of the many successes of Russian mathematicians and physicists; and it provides a lucid account of some modern work in which topology is genuinely applied. Books like this are vital for the health of mathematics and it is to be hoped that more will be written."---Mathematical Reviews ... Read more

Customer Reviews (2)

Some nice insights, but very uneven (3.5 stars)
This book is worth reading for its second (stand-alone) half, about topology and physics. Michael Monastyrsky (MM) relies mainly on words to convey the big picture of many connections between the two topics. I had a number of "aha" moments while reading it, which had eluded me despite having struggled through some more formal introductions to T&P. Chapter 9's discussion of the relationship between homology and homotopy and later chapters about liquid crystals and topological particles were especially enlightening.

That said, I think it would be difficult to have those moments if you hadn't already had at least an introduction to algebraic topology (e.g., Michael Henle's wonderfully clear "A Combinatorial Introduction to Topology," from Dover). The treatment gets more abstruse in the later chapters, where it feels like MM was rushing. E.g., the chapter on braids and knots, which should be relatively intuitive to understand, struck me as quite abstract even though I'd already read a couple of books on the subject. And the discussions of magnetohydrodynamics and "What's next?" (at the end) were more like catalogues of topics than anything you could learn from. MM is also sometimes quite rambling, such as in his long chapter on gauge fields, inmore than 80% of which he doesn't discuss topology at all.

The rambling tendency is even more evident in the Riemann part of the book. E.g. Dirichlet doesn't make an appearance in the chapter entitled "Riemann and Dirichlet" until, again, 80% of the way through the chapter; and in the 2-page chapter "Last Years", MM jumps from Riemann's marriage and illness in 1862 to June 28, 1866, then to early 1866, then to June 14 and finally to Riemann's death on July 20 (@75-76). Even though Riemann's accomplishments are extremely interesting, there's something flat and dull about the writing style in this part of the book, so much so that I'd lost interest on two previous attempts to read it. But the main flaw in this section is that MM seems unsure of what level of readership he's aiming for. E.g., on one hand MM feels readers need to be told that "shock waves are formed when high-speed aircraft break the sound barrier, when atomic bombs explode, and so forth" (@69), but on the other hand if you don't have any prior background in complex analysis you will be lost. (It also doesn't hurt to have encountered monodromy mappings previously (@56)).

A few more diagrams would have been helpful especially in the Riemann section, and throughout the book it would have been nice if the diagrams had been re-drawn from the Russian edition: some of them are quite murky, and one or two of them don't seem to match the revised text. All in all, I give "Riemann" barely a 3, and "T&P" a 4, for an average of 3.5 stars.

Excellent
This is a translation of two quite small Russian books. The first is a biography of Riemann, the second a review of "Geometrical" physics, recently updated. Even the biography includes some mathematicswhich I did not fully understand, but it is a great introduction, and it'sgood to get your feet a little wet! The book is small and very readable. Irecommend it to students interested in understanding the relationshipsbetween physics, mathematics, and geometry. A gentle introduction to the"technical" journey. ... 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
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 most major universities one of the three or four basic first-year graduate mathematics courses is algebraic topology. This introductory text is suitable for use in a course on the subject or for self-study, featuring broad coverage and a readable exposition, with many examples and exercises.The four main chapters present the basics: fundamental group and covering spaces, homology and cohomology, higher homotopy groups, and homotopy theory generally.The author emphasizes the geometric aspects of the subject, which helps students gain intuition.A unique feature is the inclusion of many optional topics not usually part of a first course due to time constraints: Bockstein and transfer homomorphisms, direct and inverse limits, H-spaces and Hopf algebras, the Brown representability theorem, the James reduced product, the Dold-Thom theorem, and Steenrod squares and powers. ... Read more

Customer Reviews (19)

Terrible textbook
This book is horrible if regarded as a mathematics book. Like previous reviewers I feel there is a total lack of clarity and rigor. Definitions are lacking, perhaps the author feels it is better to provide a "intuitive" feel for the material, than just definingg things. He fails miserably. The fact that what we are really dealing with in this subject are functors((co)homology, homotopy ) is nearly absent from the text. Instead drawings and pictures that are meant to provide "geometric" feel are supplanted.
I would state that this book attempts to teach how to compute in and use the theory than have you understand how the theory is built. It is a book for using the oven, not understanding how it works.

More Hand-Waving Than an Orchestral Conductor
In the TV series "Babylon 5" the Minbari had a saying: "Faith manages."If you are willing to take many small, some medium and a few very substantial details on faith, you will find Hatcher an agreeable fellow to hang out with in the pub and talk beer-coaster mathematics, you will be happy taking a picture as a proof, and you will have no qualms with tossing around words like "attach", "collapse", "twist", "embed", "identify", "glue" and so on as if making macaroni art.

To be sure, the book bills itself as being "geometrically flavored", which over the years I have gathered is code in the mathematical community for there being a lot of cavalier hand-waving and prose that reads more like instructions for building a kite than the logical discourse of serious mathematics.Some folks really like that kind of stuff, I guess (judging from other reviews).Professors do, because they already know their stuff so the wand-waving doesn't bother them any more than it would bother the faculty at Hogwarts.When it comes to Hatcher some students do as well, I think because so often Hatcher's style of proof is similar to that of an undergrad:inconvenient details just "disappear" by the wayside if they're even brought up at all, and every other sentence features a leap in logic or an unremarked gap in reasoning that facilitates completion of an assignment by the due date.

Some will say this is a book for mature math students, so any gaps should be filled in by the reader en route with pen and paper.I concede this, but only to a point.The gaps here are so numerous that, to fill them all in, a reader would be spending a couple of days on each page of prose.It is not realistic.Some have charged that this text reads like a pop science book, while others have said it is extremely difficult.Both charges are true.Never have I encountered such rigorous beer-coaster explanations of mathematical concepts.Yet this book seems to get a free ride with many reviewers, I think because it is offered for free.In the final analysis is it a good book or a bad book?Well, it depends on your background, what you hope to gain from it, how much time you have, and (if your available time is not measured in years) how willing you are to take many things on faith as you press forward through homology, cohomology and homotopy theory.

First, one year of graduate algebra is not enough, you should take two. Otherwise while you may be able to fool yourself and even your professor into thinking you know what the hell is going on, you won't really.Not right away.Ignore this admonishment only if you enjoy applying chaos theory to your learning regimen.

Second, you better have a well-stocked library nearby, because as others have observed Hatcher rarely descends from his cloud city of lens spaces, mind-boggling torus knots and pathological horned spheres to answer the prayers of mortals to provide clear definitions of the terms he is using.Sometimes when the definition of a term is supplied (such as for "open simplex"), it will be immediately abused and applied to other things without comment that are not really the same thing (such as happens with "open simplex") -- thus causing countless hours of needless confusion.

Third: yes, the diagram is commutative.Believe it.It just is.Hatcher will not explain why, so make the best of it by turning it into a drinking game.The more shots you take, the easier things are to accept.

In terms of notation, if A is a subspace of X, Hatcher just assumes in Chapter 0 that you know what X/A is supposed to mean (the cryptic mutterings in the user-hostile language of CW complexes on page 8 don't help).It flummoxed me for a long while.The books I learned my point-set topology and modern algebra from did not prepare me for this "expanded" use of the notation usually reserved for quotient groups and the like.Munkres does not use it.Massey does not use it.No other topology text I got my hands on uses it.How did I figure it out?Wikipedia.Now that's just sad.Like I said earlier:one year of algebra won't necessarily prepare you for these routine abuses by the pros; you'll need two, or else tons of free time.

Now, there are usually a lot of examples in each section of the text, but only a small minority of them actually help illuminate the central concepts.Many are pathological, being either extremely convoluted or torturously long-winded -- they usually can be safely skipped.

One specific gripe. The development of the Mayer-Vietoris sequence in homology is shoddy.It's then followed by Example 2.46, which is trivial and uncovers nothing new, and then Example 2.47, which is flimsy because it begins with the wisdom of the burning bush: "We can decompose the Klein bottle as the union of two Mobius bands glued together by a homeomorphism between their boundary circles." Oh really?(Cue clapping back-up chorus: "Well, ya gotta have faith...")That's the end of the "useful" examples at the Church of Hatcher on this important topic.

Another gripe. The write-up for delta-complexes is absolutely abominable. There is not a SINGLE EXAMPLE illustrating a delta-complex structure.No, the pictures on p. 102 don't cut it -- I'm talking about the definition as given at the bottom of p. 103.A delta-complex is a collection of maps, but never once is this idea explicitly developed.

A final gripe.The definition of the suspension of a map...?Anyone?Lip service is paid on page 9, but an explicit definition isn't actually in evidence.I have no bloody idea what "the quotient map of fx1" is supposed to mean. I can make a good guess, but it would only be a guess. Here's an idea for the 2nd edition, Allen: Sf([x,t]) := [f(x),t]. This is called an explicit definition, and if it had been included in the text it would have saved me half an hour of aggravation that, once again, only ended with Wikipedia.

But still, at the end of the day, even though it's often the case that when I add the details to a one page proof by Hatcher it becomes a five page proof (such as for Theorem 2.27 -- singular and simplicial homology groups of delta-complexes are isomorphic), I have to grant that Hatcher does leave just enough breadcrumbs to enable me to figure things out on my own if given enough time.I took one course that used this text and it was hell, but now I'm studying it on my own at a more leisurely pace.It's so worn from use it's falling apart.Another good thing about the book is that it doesn't muck up the gears with pervasive category theory, which in my opinion serves no use whatsoever at this level (and I swear it seems many books cram ad hoc category crapola into their treatments just for the sake of looking cool and sophisticated).My recommendation for a 2nd edition:throw out half of the "additional topics" and for the core material increase attention to detail by 50%.Oh, and rewrite Chapter 0 entirely. Less geometry, more algebra.

Really bad as a "readable" texbookbut good reference
I am not able to understad why people seems to love this book my feelings, beeing mixed, are perhaps closer to hate.

The book is OK if (and only if) you previously know the matter but the lack of clear definitions, the excessive reliance in reader geometrical intuition, the conversational style of demos the long paragraphs describing obscure geometric objects, etc make it very difficult to follow if it is your first approach to AT.

On the other hand has useful insigths if you already know the matter.

If the purpouse of the author has really been to write a "readable" book (as he told us repeatedly) I think the attemp is a complete failure.

On the other handthe "Table of contents" is excellent and is a very good book for teachers,I think this is the reason of itspopularity.

If you can afford the cost, purchase J Rotman "An introduction to Algebraic Topology" and you really will get a "readable" book

amazing book, but caveat emptor
I think that Allen Hatcher has given us all something very valuable in this book.If you are like me, you've had those moments when reading in a math book when you read a sentence, and your eyes shoot open and you suddenly feel like someone has been standing behind you that you never knew was there.There are lots of those kinds of sentences in this book.On the other hand, I view it as a supplement to a book like Munkres or Bredon that provides the rigor necessary to allow the learner to figure out the topologist's geometric language.I have used these three and found them to compliment one another well.

excellent modern introduction
This is an excellent introduction to the subject. It's affordable, well-written, and the topics are well chosen. The presentation is modern, but includes enough intuition that the fairly naive reader (e.g., me) can see the point of things. I needed to (re)learn topology for a research project I was part of in the intersection of math/CS/statistics and this book was a big help. I wish that he had included simplicial sets in the topics, because I like the way he writes and would like to have a more elementary exposition tied to the rest of the book (I eventually found an expository paper that did a pretty good job, but worked out examples would still help with that topic), but it can't include everything. I highly recommend this book to anyone trying to get started in this fascinating subject. It will just scratch the surface, but it does a good job of that. ... Read more

Product Description
Thorough, modern treatment, essentially from a homotopy theoretic viewpoint. Topics include homotopy and simplicial complexes, the fundamental group, homology theory, homotopy theory, homotopy groups and CW-Complexes and other topics. Each chapter contains exercises and suggestions for further reading. 1980 corrected edition.
... Read more

Customer Reviews (3)

Solid
A very interesting book which I enjoyed. I particularly found useful the "crash review" in algebra and analysis which functioned as a useful reference throughout the book.

Shouldn't be your first text in algebraic topology.
It is a decent book in algebraic topology, as a reference.At first, I found this textbook rather hard to read. Too manylemmas, theorems, etceteras. Three suggestions:

1. Needs more pictures, especially for the simplicialhomology Chapter.

2. CW complexes should be covered before duality and not after.

3. Needs more examples and exercises.

Overall, the book is very good, if you have already someexperience in Algebraic Topology. I found that the Croom'sbook "Basic concepts of Algebraic Topology" is an excellent first textbook. Too bad it is out of print, since it is very popular, every time I get it from the library, someone else recalls it. The combination of these two books probablyis the right thing to have: Maunder's book picks up whereCroom has left you.

Not bad.
Maunder's text may not be the "best" book on algebraic topology, but I still recommend this one to those who find other more advanced texts like Spanier rather inaccessible. Warning: the chapter on cohomology andduality is not very well-organaized (compared to other chapters), so youmay want to consult Bredon's book instead. ... Read more

Product Description
Following their introduction in the early 1980s, o-minimal structures have provided an elegant and surprisingly efficient generalization of semialgebraic and subanalytic geometry. This book gives a self-contained treatment of the theory of o-minimal structures from a geometric and topological viewpoint, assuming only rudimentary algebra and analysis. It starts with an introduction and overview of the subject. Later chapters cover the monotonicity theorem, cell decomposition, and the Euler characteristic in the o-minimal setting and show how these notions are easier to handle than in ordinary topology. The remarkable combinatorial property of o-minimal structures, the Vapnik-Chervonenkis property, is also covered. This book should be of interest to model theorists, analytic geometers and topologists. ... Read more

Product Description
Like any books on a subject as vast as this, this book has to have a point-of-view to guide the selection of topics. Naber takes the view that the rekindled interest that mathematics and physics have shown in each other of late should be fostered, and that this is best accomplished by allowing them to cohabit. The book weaves together rudimentary notions from the classical gauge theory of physics with the topological and geometrical concepts that became the mathematical models of these notions. The reader is asked to join the author on some vague notion of what an electromagnetic field might be, to be willing to accept a few of the more elementary pronouncements of quantum mechanics, and to have a solid background in real analysis and linear algebra and some of the vocabulary of modern algebra. In return, the book offers an excursion that begins with the definition of a topological space and finds its way eventually to the moduli space of anti-self-dual SU(2) connections on S4 with instanton number -1. ... Read more

Customer Reviews (6)

An Introduction with Mathematical Integrity
Gregory Naber is to be commended for writing a thorough introduction to gauge field theory in which the mathematics is presented with clarity and rigor.For the professional mathematician who is interested in physics, or for the graduate student who prefers to see the mathematics "done right" in advanced applications to physics, Naber's wonderful two-volume set stands apart from its major competitors, nearly all of which were written by physicists, for physicists.

Despite the attention to mathematical rigor, it is clear that Naber intended his books to be accessible to a dual audience of physicists and mathematicians. For the physicists, he has included gentle introductory chapters on topological spaces, homotopy groups, principal bundles, manifolds and Lie groups, and differential forms. For mathematicians, the chapters on physical motivation, gauge fields and instantons, Yang-Mills-Higgs theory, Spinor structures, etc., provide unusually accessible introductions to some difficult physics materials.

Chapter 0 of the first volume is worth the price of both books, as it leads the reader, in 26 succinct pages, to a compelling appreciation ofthe natural "fit" of the Hopf Bundle to the task of providing a quantum mechanical analysis of the exterior of a single magnetic monopole.For outsiders who have become incredulous about the increasingly sophisticated uses of topology and geometry in theoretical physics, this example provides some much-needed assurance.As the reader quickly learns, the use of connections on principal fiber bundles is neither gratuitous nor mathematical overkill:indeed, the bundle machinery emerges quite NATURALLY as the simplest and best mathematical tool, perfectly fitted to the special problem at hand.

Any serious reader will want to buy both volumes of this set:Topology, Geometry, and Gauge Fields:Foundations (volume 1), and Topology, Geometry, and Gauge Fields:Interactions (volume 2).These books take their place alongside the work of authors such as Jerrold Marsden, Theodore Frankel, Barrett O'Neill, and Walter Thirring, all of whom write about modern mathematical physics in a way that does not obscure the true role of the mathematics.

correction to dost
The review "Easy reading, complete proofs, plenty of exercises, October 29, 2005 by Rehan Dost is of the first volume, Foundations, not this volume which is Interactions. Naber's books are crafted to bridge physics, undergraduate mathematics and graduate mathematics. This is one more of his beautiful volumes in applied mathematics.

Easy reading, complete proofs, plenty of exercises
This text is by far the best introductory text marrying basic concepts of physics with pure mathematics.

Some background in the basic concepts of vector calculus, linear algebra, complex numbers and group theory is required.

The author begins by motivating the mathematics by the pursuit of finding a vector potential to represent a magnetic monopole. We see that the topology of R3-0 precludes such a vector potential from existing. We see here a simple example of how the topology of a space affects the physics associated with it.
The importance of the vector potential as something other than a convenient computational tool is highlighted by a reference to essential inclusion in quantum mechanics. Thus we NEED such a potential.

The author now asks whether there is a "trick" or device to get around this difficulty. The device are principal bundles and connections. For example the potentials noted above must keep track of the phase of a charged test particle as it moves thru the field of a magnetic monopole. We need a "bundle" of circles ( representing the phase at each point ) over S2 ( the author explains why we need only consider S2 instead of R3-0, briefly we need only keep track of 2 of the 3 spherical co-ordinates ).
Thus a curve in S2 thought of as the particles trajectory will have to be "lifted" to the bundle space by a lifting procedure called a connection.
In a more general setting elementary particles have an internal structure ( spin etc ) which becomes apparent during interactions although may not be apparent in uniform motion thru a vacuum. Since the phase of the particle does not alter the modulus when calculating probabilities these do not change. However, when the particles interact phase differences are important. We need to keep track of such phases as the particles interact.

Thus we need a "bundle" over a 4-manifold ( keeps track of the particles space-time path ) to keep track of such internal states. One sees we also need a group to transform states into one another ( usually incorporated into the bundle ). Connections then model physical phenomena which mediate changes in the internal states.
We see that some connections satisfy the Yang-Mills equations and using the appropriate equivalence relation form Moduli spaces.

Now that may seem like alot to digest with only a spattering of mathematical maturity.

The beauty of the book is that the author starts from FIRST principles.

Chapter 1 introduces topological concepts of topology, continuity, quotient topology, projective spaces, compactness, connectivity, covering spaces and topological groups.

Chapter 2 introduces concepts of path lifting, fundamental groups, contractability, simple connectedness, covering homotopy theorem, higher homotopy groups

Chapter 3 introduces principle bundles, transition functions, bundle maps and principle bundles over spheres.

Chapter 4 introduces manifolds, derivatives on manifolds, tangent/cotangent spaces, submanifolds, vector fields, matrix lie groups, vector valued 1- forms, 2 forms and Riemann metrics

Chapter 5 gets to some physics with gauge fields and connections, curvature, Yang-Mills functional, moduli spaces, Hodge dual , matter fields and covariant derivatives.

At each step the author carefully provides complete proofs and easy exercises to ensure understanding.

It was a pleasure to read the book and complete the exercises. At no point did I feel frustration or boredom.

Don't waste your money
This review refers only to the book printing quality not to the contents.

I had purchased some books from Springer in the past (Like Arnold Mathematical Methods of Classical Mechanics, Lang Algebra etc..) and found them beautifully edited: good binding, paper etc..

And to my surprise I was very disappointed with the overall quality of this book, poor binding -glued instead of sewn- bad quality paper -forming waves at the binding spine, etc..

You pay for a quality item, a book you can use for years, and you get a hardbound crap that you can not left open in a table without holding it tight risking to lose the pages after a few days of use in the process.

I find this unacceptable in books costing 60$+. Sadly I find this to occur very often, publishers should be more careful with their printings and custumers should demand a better quality.

Don't waste your money.

A reader.

MATH AND TOPOLOGY
Topology is very important scince in the fields of mathematics. And it using in many of another sinceis. ... Read more

Product Description
In recent years topology has firmly established itself as an important part of the physicist's mathematical arsenal. It has many applications, first of all in quantum field theory, but increasingly also in other areas of physics. The main focus of this book is on the results of quantum field theory that are obtained by topological methods. Some aspects of the theory of condensed matter are also discussed. Part I is an introduction to quantum field theory: it discusses the basic Lagrangians used in the theory of elementary particles. Part II is devoted to the applications of topology to quantum field theory. Part III covers the necessary mathematical background in summary form. The book is aimed at physicists interested in applications of topology to physics and at mathematicians wishing to familiarize themselves with quantum field theory and the mathematical methods used in this field. It is accessible to graduate students in physics and mathematics. ... Read more

Product Description
This Introduction to Topology, which is a thoroughlyrevised, extensively rewritten, second edition of the work firstpublished in Russian in 1980, is a primary manual of topology. Itcontains the basic concepts and theorems of general topology andhomotopy theory, the classification of two-dimensional surfaces, anoutline of smooth manifold theory and mappings of smooth manifolds.Elements of Morse and homology theory, with their application to fixedpoints, are also included. Finally, the role of topology inmathematical analysis, geometry, mechanics and differential equationsis illustrated.
Introduction to Topology contains many attractive illustrationsdrawn by A. T. Frenko, which, while forming an integral part of thebook, also reflect the visual and philosophical aspects of moderntopology. Each chapter ends with a review of the recommendedliterature.
Audience: Researchers and graduate students whose work involvesthe application of topology, homotopy and homology theories.
... Read more

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This book arose from courses taught by the authors, and isdesigned for both instructional and reference use during and after afirst course in algebraic topology. It is a handbook for users whowant to calculate, but whose main interests are in applications usingthe current literature, rather than in developing the theory. Typicalareas of applications are differential geometry and theoreticalphysics. We start gently, with numerous pictures to illustrate the fundamentalideas and constructions in homotopy theory that are needed in laterchapters. We show how to calculate homotopy groups, homology groupsand cohomology rings of most of the major theories, exact homotopysequences of fibrations, some important spectral sequences, and allthe obstructions that we can compute from these. Our approach is tomix illustrative examples with those proofs that actually developtransferable calculational aids. We give extensive appendices withnotes on background material, extensive tables of data, and a thoroughindex. Audience: Graduate students and professionals in mathematics andphysics. ... Read more

Product Description
Differential geometry and topology have become essential tools for many theoretical physicists. In particular, they are indispensable in theoretical studies of condensed matter physics, gravity, and particle physics. Geometry, Topology and Physics, Second Edition introduces the ideas and techniques of differential geometry and topology at a level suitable for postgraduate students and researchers in these fields.

The second edition of this popular and established text incorporates a number of changes designed to meet the needs of the reader and reflect the development of the subject. The book features a considerably expanded first chapter, reviewing aspects of path integral quantization and gauge theories. Chapter 2 introduces the mathematical concepts of maps, vector spaces, and topology. The following chapters focus on more elaborate concepts in geometry and topology and discuss the application of these concepts to liquid crystals, superfluid helium, general relativity, and bosonic string theory. Later chapters unify geometry and topology, exploring fiber bundles, characteristic classes, and index theorems. New to this second edition is the proof of the index theorem in terms of supersymmetric quantum mechanics. The final two chapters are devoted to the most fascinating applications of geometry and topology in contemporary physics, namely the study of anomalies in gauge field theories and the analysis of Polakov's bosonic string theory from the geometrical point of view.

Geometry, Topology and Physics, Second Edition is an ideal introduction to differential geometry and topology for postgraduate students and researchers in theoretical and mathematical physics. ... Read more

Customer Reviews (13)

Excellent review of math for (particle) physicists
I bought this book to supplement my knowledge of mathematics which frequently is involved in understanding Particle physics concepts. The book is terse, but peppered with examples and insights about the definitions, and so far it is really fun to read. Seems like a good investment.

Too many errors to be useful for study
Reading all the glowing reviews of this book, I wonder whether the reviewers actually tried to use the book to understand the material, or just checked the table of contents. There are so many misprints, throughout, that one wonders if the book was proofread at all. Some of the mistakes will be obvious to every physicist - for example, one of the Maxwell equations on page 56 is wrong - others are subtle, and will confuse the reader. The careful reader, who wants to really understand the material and tries to fill in the details of some of the derivations, will waste a lot of time trying to derive results that have misprints from intermediate steps which have different misprints! Some chapters are worse than others, but the average density of misprints seems to be more than one per page.
The book might be useful as a list of topics and a "road map" to the literature prior to 2003, but that hardly justifies the cost (or the paper) of a whole book.

Geometry Topology and Physics: A condesed view
This book provide a complete and useful review of geometrical instuments of mathematical physics from the beginnig to the most advanced topics of interest. It can be used by students at the beginnig of thei studies in this topics, and it's found to be a useful gallery for higher level students (or scholar).

An excellent book
This is the best book of its type, that is, a book that contains almost all if not all the advance mathematics a theoretical physicist should know. I have studied chapters 2-9 and it has the perfect balance between rigorous presentation of topics and practical uses with examples. The level is for advance graduate students. The range of topics covered is wide including Topology topics like Homotopy, Homology, Cohomology theory and others like Manifolds, Riemannian Geometry, Complex Manifolds, Fibre Bundles and Characteristics Classes. I believe this book gives you a solid base in the modern mathematics that are being used among the physicists and mathematicians that you certainly may need to know and from where you will be in a position to further extent (if you wish) into more technical advanced mathematical books on specific topics, also it is self contained but the only shortcoming is that it brings not many exercises but still my advice, get it is a superb book!

A great reference book.
No doubt, the interplay of topology and physics has stimulated phenomenal research and breakthroughs in mathematics and physics alike.

Unfortunately, there is so much mathematics to master that the average graduate physics student is left bewildered.....until now.

The text is an excellent reference book. I emphasize reference. The book presupposes an acquaintance with basic undergraduate mathematics including linear algebra and vector analysis.

The author covers a wide range of topics from tensor analysis on manifolds to topology, fundamental groups, complex manifolds, differential geometry, fibre bundles etc.

The exposition in necessarily brief but the main theorems and IDEAS of each topic are presented with specific applications to physics. For example the use of differential geometry in general relativity and the use of principal bundles in gauge theories, etc.

Unfortunately, there are very few exercises necessitating the use of supplementary texts. However, to the author's credit appropriate supplementary texts are provided. The author goes to great lengths to show which texts inspired the chapters and follows the same line of presentation.

Perhaps the greatest attribute of the text is to take disparate branches of mathematics and coallate them under one text with applications to physics. In doing so one gains a better grasp of how the fields of mathematics interact in the domain of physics. ... Read more

Product Description
An advanced textbook in mathematics, offering a coherent and thorough treatment of the configuration spaces of Euclidian spaces and spheres. Requires a minimal background in classical homotopy theory and algebraic topology. Covers a variety of advanced topics, including a geometric presentation of the classical pure braid group. DLC: Configuration space. ... Read more

Product Description
This monograph provides an introduction to the theory oftopologies defined on the closed subsets of a metric space, and on theclosed convex subsets of a normed linear space as well. A unifyingtheme is the relationship between topology and set convergence on theone hand, and set functionals on the other. The text includes for thefirst time anywhere an exposition of three topologies that over thepast ten years have become fundamental tools in optimization,one-sided analysis, convex analysis, and the theory of multifunctions:the Wijsman topology, the Attouch--Wets topology, and the slicetopology. Particular attention is given to topologies on lowersemicontinuous functions, especially lower semicontinuous convexfunctions, as associated with their epigraphs. The interplay betweenconvex duality and topology is carefully considered and a chapter onset-valued functions is included. The book contains over 350 exercisesand is suitable as a graduate text.
This book is of interest to those working in general topology,set-valued analysis, geometric functional analysis, optimization,convex analysis and mathematical economics.
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Customer Reviews (9)

If U want generalization out to infinity, this is it for you, in algebraic topology basics.
This former professor, and sometime head of the math department at U of Chicago, is quite a fellow. He is so DEEP that I many times didn't have a clue about some of his books. But this one seems more down at my level of intelligence, even though it is a whirlwind romance so to speak with algebraic topology basics.

Have at it, if you like the whirlwind!

The Title Says it All
I have always believed that the "goodness" of a mathematical textbook is inversely proportional to its length. J. P. May's book "A Concise Course in Algebraic Topology" is a superb demonstration of this. While the book is indeed extremely terse, it forces the reader to thoroughly internalize the concepts before moving on. Also, it presents results in their full generality, making it a helpful reference work.

The opposite of Hatcher
This book is clear, and direct.It tells you want you want to know.

Lucid and elegant, but not for beginners
This tiny textbook is well organized with an incredible amount of information. If you manage to read this, you will have much machinery of algebraic topology at hand. But, this book is not for you if you know practically nothing about the subject (hence four stars). I believe this work should be understood to have compiled "what topologists should know about algebraic topology" in a minimum number of pages.

A Unique and Necessary Book
Ones first exposure to algebraic topology should be a concrete and pictorial approach to gain a visual and combinatorial intuition for algebraic topology. It is really necessary to draw pictures of tori, see the holes, and then write down the chain complexes that compute them. Likewise, one should bang on the Serre Spectral Sequence with some concrete examples to learn the incredible computational powers of Algebraic Topology. There are many excellent and elementary introductions to Algebraic Topology of this type (I like Bott & Tu because of its quick introduction of spectral sequences and use of differential forms to bypass much homological algebra that is not instructive to the novice).

However, as Willard points out, mathematics is learned by successive approximation to the truth. As you becomes more mathematically sophisticated, you should relearn algebraic topology to understand it the way that working mathematicians do. Peter May's book is the only text that I know of that concisely presents the core concepts algebraic topology from a sophisticated abstract point of view. To make it even better, it is beautifully written and the pedagogy is excellent, as Peter May has been teaching and refining this course for decades. Every line has obviously been thought about carefully for correctness and clarity.

As an example, ones first exposure to singular homology should be concrete approach using singular chains, but this ultimately doesn't explain why many of the artificial-looking definitions of singular homology are the natural choices. In addition, this decidedly old-fashioned approach is hard to generalize to other combinatorial constructions.

Here is how the book does it: First, deduce the cellular homology of CW-complexes as an immediate consequence of the Eilenberg-Steenrod axioms. Considering how one can extend this to general topological spaces suggests that one approximate the space by a CW-complex. Realization of the total singular complex of the space as a CW-complex is a functorial CW-approximation of the space. As the total singular complex induces an equivalence of (weak) homotopy categories and homology is homotopy-invariant, it is natural to define the singular homology of the original space to be the homology of the total singular complex. Although sophisticated, this is a deeply instructive approach, because it shows that the natural combinatorial approximation to a space is its total singular complex in the category of simplicial sets, which lets you transport of combinatorial invariants such as homology of chain complexes. This approach is essential to modern homotopy theory. ... Read more

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Product Description
Combining concepts from topology and algorithms, this book delivers what its title promises: an introduction to the field of computational topology. Starting with motivating problems in both mathematics and computer science and building up from classic topics in geometric and algebraic topology, the third part of the text advances to persistent homology. This point of view is critically important in turning a mostly theoretical field of mathematics into one that is relevant to a multitude of disciplines in the sciences and engineering. The main approach is the discovery of topology through algorithms. The book is ideal for teaching a graduate or advanced undergraduate course in computational topology, as it develops all the background of both the mathematical and algorithmic aspects of the subject from first principles. Thus the text could serve equally well in a course taught in a mathematics department or computer science department. ... Read more

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The single most difficult thing one faces when one begins to learn a new branch of mathematics is to get a feel for the mathematical sense of the subject. The purpose of this book is to help the aspiring reader acquire this essential common sense about algebraic topology in a short period of time. To this end, Sato leads the reader through simple but meaningful examples in concrete terms. Moreover, results are not discussed in their greatest possible generality, but in terms of the simplest and most essential cases.

In response to suggestions from readers of the original edition of this book, Sato has added an appendix of useful definitions and results on sets, general topology, groups and such. He has also provided references.

Topics covered include fundamental notions such as homeomorphisms, homotopy equivalence, fundamental groups and higher homotopy groups, homology and cohomology, fiber bundles, spectral sequences and characteristic classes. Objects and examples considered in the text include the torus, the Möbius strip, the Klein bottle, closed surfaces, cell complexes and vector bundles. ... Read more

Customer Reviews (5)

"Intuition" more a prerequisite than a result
This was my first crack at algebraic topology, self-studying long after my university days. I thought I'd read this book as a warm-up for Bott & Tu. The book is written in the laid-back discursive style that is one of the more charming attributes of Japanese math books. It's also short, and the author has provided solutions or hints for most of the modest exercises. At first glance it looks like a pleasant way to spend a few afternoons in a cafe.

But appearances can be deceiving. The intuitions referred to are not those of a typical beginner. No less disingenuous is the occasional advice saying it's OK to skip a chapter: the concepts and definitions are inevitably used in later ones. These are what Japanese call "tatemae" -- the stuff that's said just for the sake of making a good (or at least better) impression.

The reviewers who suggested that the book supplements more advanced texts are closer to the mark. I found myself resorting to Bott & Tu and Hatcher to clear up concepts presented in this one, when I'd expected the reverse. E.g., Sato's explanation of exact sequences was ultra-concise and rather puzzling, while the two books I mentioned and even Wikipedia are quite helpful about them. B&T also uses many more diagrams when it counts, including in some clear and beautiful proofs about homotopies that Sato presents in a drier style. Nor does Sato do a good job of motivating why cohomology is more useful than homology; for all its shortcomings (including lack of coverage of De Rham cohomology), even 1970's-vintage Maunder does a better job at this. (The first few pages of Hatcher's Chapter 3 are even better on that point, but that's what one would expect from such a humongous book.) And the diagrams accompanying the description of fiber bundles don't even indicate a fiber; there are many more "intuitive" explanations of this topic elsewhere.

This may be a good tool for reinforcing material you have learnt or are learning from another source. But you might not find it as suitable for a free-standing introduction as the title and a casual inspection might suggest. I give it 3.5-4 stars instead of 3 as a handicap, considering my own amateurism, and also because of the good range of topics touched on.

Excelent Start
In my opinion, this is a great little book to take with you to a park or on a trip to read before you start tackling a more serious book such as the one by Allen Hatcher.This book will give you a great over view of many major topics in Algebraic Topology; for a serious reader, you might want to read this book in parallel with Hatcher, Massey and Munkres (Topology, 2nd Edition).I find that these three books compliment one another very well if you are trying to learn this beautiful subject on your own.I use Sato's book to read about general ideas; once I understand the surface of the concepts I then reference the latter two books to dive deeper into the machinery.It's working well for me; however, do not be fooled, nothing replaces a great teacher!

Excellent accompaniment to Hatcher
As a student just wading into the realm of Algebraic Topology, this book has been a wonderful companion. If you are looking for a book that will lay out precise proofs of theorems and get down to the nity-gritty, this book is not for you. However, if you are new to A.T. as I am, and want a book that will give you a nice easy to follow introduction to a topic before wading into your thicker text, then this book will help you tremendously. For instance, reading the chapters regarding CW-complexes and Homotopy in Sato, although thin and easy to follow (you will have to do a little bit of lifting, but not too much), helped me to more easily digest what was to come in Chapters 0 and 1 of Hatcher (which I also highly recommend, incidentally). It always helps to read material taken from a different person's perspective, and Sato has truly made Algebraic topology more transparent in this brief overview.

Good Supplementary Reading
This modest 118-page book would best accompany one of the standard graduate texts -- Spanier, Dold, Switzer, Massey, Husemoller,Maunder, Munkres, Bott and Tu, Bredon, or Greenberg and Harper. It can't be used as a text.

The book presents the most basic ideas pertaining to homotopy, homology, cohomology, fibre bundles, spectral sequences, and characteristic classes. The emphasis is on simple examples and simple calculations to demonstrate what is going on. Rigorous definitions, proofs, and even frequently even the statements of theorems, are avoided.

One good aspect of the treatment is the axiomatic presentation of homology and cohomology a la Eilenberg and Steenrod. Some of the essential material is also presented, e.g. the cup product that gives a ring structure to the cohomology group, the Kunneth theorem, the Universal Coefficient theorem, and so on.

The book would afford a bird's-eye view, a conspectus, to a bright undergraduate or beginning graduate student. It goes without saying, of course, that this is for motivation, and it doesn't replace the hard technical grind required to master the subject.

The book suffers in comparison to the one by Fomenko, Fuchs and Gutenmacher (Homotopic Topology), but that, alas, can't be had for love or money.

algeblaic topology
there are much examples. so good to understand. ... Read more

Product Description

The authors present introductory material in algebraic topology from a novel point of view in using a homotopy-theoretic approach. This carefully written book can be read by any student who knows some topology, providing a useful method to quickly learn this novel homotopy-theoretic point of view of algebraic topology.