Book Description

This introduction to topology provides separate, in-depth coverage of both general topology and algebraic topology. Includes many examples and figures.GENERAL TOPOLOGY. Set Theory and Logic. Topological Spaces and Continuous Functions. Connectedness and Compactness.Countability and Separation Axioms. The Tychonoff Theorem. Metrization Theorems and paracompactness. Complete Metric Spaces and Function Spaces. Baire Spaces and Dimension Theory. ALGEBRAIC TOPOLOGY. The Fundamental Group. Separation Theorems. The Seifert-van Kampen Theorem. Classification of Surfaces. Classification of Covering Spaces. Applications to Group Theory.For anyone needing a basic, thorough, introduction to general and algebraic topology and its applications. ... Read more

Reviews (23)

Standard Topology Text
Most people have a favorite color, fewer people have a favorite baseball team, and even fewer people have a favorite topology textbook. Granted I've only had an extensive relationship with this particular textbook, but given the reviews I've read and other recommendations I've recieved, I would have to go on record and vouch for this book.

When I took topology this text was recommended and our lectures were based on a book (which was required) compiled by the teacher. Often times, we found the lectures/required text to be lacking and were glad to have this text to refer to.

I've seen this book used for both point-set and algebraic topology courses, or some combination of the two. The coverage of point-set is fairly complete while the algebraic section covers introductory material (homotopy, fixed point theorem, lifts, fundamental groups, etc.). The breakdown of the material is approximately 65% Point-set and 35% algebraic thus making it a good choice for someone taking Point-set but personally motivated to glance ahead to some of the algebraic stuff.

Two particular strengths: A thorough introduction to basic concepts of analysis, and, because you don't see many of them around, a good introductory treatment of Algebraic Topology.

A Great Work
Taking a first course in topology could not be better complemented. This clear exposition of point set and algebraic topology is so well written that it could even be used for self study. Motivation from the professor is always helpful, but Munkres actually goes quite far in providing a reason for the topics in question. Furthermore, the examples clarify many of the presented concepts and even show some of the misconceptions a student may have.

Having a course in analysis would certainly make the book flow since otherwise it would just to be a mental exercise rather than an extension of familiar concepts.

The exercises are very well thought out and are meant to be solved by all students given that they have some diligence. They truly help in turning a fog of concepts into concrete understanding.

great!
Not much to add here... there are enough easy problems that I can get the hang of something, but also some really tough ones at the end of each problem section. The proofs and examples in the text are really good guides to doing the problems also. In some sections there are counterexamples for, say, the converse of a theorem which are always really pathological. At the beginning of each section there is some discussion on what to expect, why the stuff is important, what to do with it, etc. Even though I had a really good prof for the topology course I did this book was very helpful out of the classroom.

Excellent Topology Book
My introduction to Munkres was in an independent study of point set topology in my final semester of undergraduate work. A professor assigned me problems from the book, but my learning was largely self motivated. I found that it was an excellent book for independent study. The text was clear and readable and the exercises helped to cement the concepts that are introduced in the reading.

Later at graduate school, Munkres was also used in a topology class at the beginning graduate level. Highlights were taken from the first section (point set topology), and a large focus of the class was on the algebraic topology in the second section of the book. Sometimes I had difficulty following exactly what the professor was doing at the blackboard, but I could always understand what was going on when I consulted Munkres.

I would stress that this is only to be used as an introduction to algebraic topology, as there is nearly no development of homology groups and other algebraic concepts. However, it gives a very good presentation for the fundamental group. As a whole it would be a very good addition to your mathematical library.

Wonderful text in a poor binding
As far as contents is concerned, this is a wonderful textboot for self-studying topology. Full of examples and a bit slow-paced, it describes even the 'clever' proofs (like Tichonoff's theorem) so that it makes their core ideas come naturally. The selection of topics is superb (algebraic topology has a much wider coverage than in the 1st edition).

The only drawback, and it is a serious one, is the binding. For a well-selling book $[...] worth, one could expect a *decent* binding, but the outcome is a *shame*. With time, the covers of my copy got ridiculously bent outwards, quite like if was cooked in my oven (which I didn't, of course). ... Read more

Reviews (7)

Flat spheres and more
Highly stimulating and extremely hard to read, written for mathematicians in physics. However, the chapter on Riemannian Geometry can be worked through, up to a point, without any knowledge of exterior differential forms, and is notable if for only one fact alone: a simple calculation is provided that explains explicitly that spheres in four and eight dimensions (3-spheres and 7-spheres) are flat with torsion! I don't know another reference that a physicist without special background in math can consult to understand this highly nonintuitive fact.

Just a "better than nothing" book
It's not the best way to learn geometry / topology for physics. It's better than nothing, though, if you are familiar with the topics already. There are many "holes" in Nakahara's book, which you would spend much more time and hard working in a "big" library. than you should to fill in. It's not worth that money and struggle. It's the last one you should consider about owning.

If you are a physics graduate who needs a nice guide to "understand" the aspects and skills of geo / top, I would recommend the following: (1) Milnor's Topology from the Differentiable Viewpoint, and (2) Kreysig's Differential Geometry. The first one was old, and so it does not assume much knowledge about the topic. The latter is a kind-of-Bible for the topic, and all solutions are provided for the problems. These two books will help you a lot if you care about the meaning, not only for those classroom exams or just showing off that you know something about it. Frankel is the next to put on your bookshelf as a detailed and rigorous development for your preparation to be a theoretical physicist.

If you have only a rough idea about topology, Hocking and Steen are the best choices, and they are Dover!!

Anyway, if I could find a cheap used Nakahara, I would get it as a reference.

Best in its genre
I suppose I should preface this by saying that I read this book *after* reading similar books, so my ability to understand this book is probably better than others, but that said, I think that my comparative evaluation is free from this bias...

There seem to be a few books on the market that are very similar to this one: Nash & Sen, Frankel, etc. This one is at the top of its class, in my opinion, for a couple reasons:

(1) It's written like a math text that covers physics-related material, not a book about mathematics for physicists. I prefer this; you may not. As a consequence, this book is more rigorous than its alternatives, it relies less on physical examples, and it cuts out a lot of lengthy explanation that you may not need. Of course, there are drawbacks to all of these "features" -- you need to decide what you need and what's best for you.

(2) It's most comprehensive, with Frankel coming in second, and Nash & Sen least comprehensive (though they have quite a bit on Fibre bundles and related topics). Nakahara has a chapter on complex manifolds, which is absent from the other two. Nakahara also concludes with a nice intro to string theory, which is absent from the other two as well (though nothing you couldn't find in Polchinski or the like). Actually -- I modify this slightly. Frankel covers less subjects than Nakahara, but with more depth (though also more wordiness -- I quit Frankel about 2/3 through because it wasn't succinct enough and I got tired of it).

Depending on your tastes, I would recommend this book before the other two.

It presupposes that you have an understanding of algebra (groups, rings, fields, etc.) but it has an introduction to the necessary components of topology within. Frankel has presupposes both algebra and topology; Nash & Sen presupposes only algebra.

Excellent book
A very nice blending of rigor and physical motivation with well chosen topics. Plenty of examples to illustrate important points. Especially noteworthy is its description of actions of lie algebras on manifolds : the best I have read so far.

Most of the topics are intepreted in terms of their topological/geomtrical structure (and the interplay between those two), but that's what the title of the book says. So you will learn things again in new ways, and gain a powerful new set of tools. If nothing else, it gives you a nice warm fuzzy feeling when you read other field/string theory books that glosses over the mathematics.

One minor rant : the notation of the book can be better. I personally uses indices to keep track of the type of objects (eg. greek index=components of tensors, no index=a geometrical object etc..), but Nakahara drops indices here and there "for simplicity". But that's my personal rant.

Good book. Buy it.

A must for any theoretical physicist
With an excellent balance between mathematical rigor and pedagogical simplicity, Nakahara remarkably captures in a single volume much of the mathematics a physicist will ever need. (If he wrote a few chapters on group theory, 'much' might be replaced with 'all'). Containing as much as it does, it is not something to breeze through. Depending on your mathematical background, you may only want to read a few chapters (and if the Homology chapter is tripping you up, just keep moving). But invest the time with it, and you will be rewarded with a solid grasp of the mathematical pictures underlying most modern physics. And once you read it and see physics from this perspective, you'll be amazed you had ever thought you understood the physics it describes. It should be said, though, that some of the latter chapters, in particular 12, are horribly sloppy. There are dozens upon dozens of errors, many at a deep conceptual level. Nonetheless, it is a monumental text, and I recommend it heartily. ... Read more

Book Description

Colin Adams, well-known for his advanced research in topology and knot theory, is the author of this exciting new book that brings his findings and his passion for the subject to a more general audience. This beautifully illustrated comic book is appropriate for undergraduates, high school students in math clubs or honors math courses and is perfect for the lay math enthusiast.The Tangle is the ultimate tool for knot theory because knots are defined in mathematics as being "closed on a loop". Readers use the Tangle to complete the experiments throughout the brief volume. ... Read more

Reviews (4)

A clear, friendly introduction to topology
Is space finite or infinite? Does it have borders? What shape does it have? These are among the most pressing and interesting questions in astrophysics and cosmology today. To answer (or at least understand) these questions, one must possess an understanding of topology, a branch of mathematics dealing with properties of shapes that are not changed upon deformation.

This book is an ideal introduction to topology for beginners with little or no mathematical background. It introduces topological manifolds (especially 2- and 3-manifolds) and their applications to cosmology and the shape of space. It is filled with diagrams, examples and exercises with full solutions at the end of the book.

The book assumes almost no knowledge of mathematics or physics, and is thus suitable for high-school and beginning college students. It is a must read for students contemplating a career in pure mathematics or theoretical physics, and who want to get a taste of the applications of pure mathematics to the physical world.

For those wishing to go a step further on the subject of the shape of space, the author published a paper (Nature 425, 593 - 595, 09 October 2003) claiming that the universe is a dodecahedral 3-manifold, based on cosmic microwave background measurements. This book may be a nice introduction for this paper and for subsequent papers that will surely ensue, trying to describe the shape of space.

Topology for everyone
Jeffrey wrote this book with the high school student in mind, but even as a second year student in Mathematics I found this book quite informative. Most textbooks in Analysis or Topology do not give you an intuitive feel for the subject. I recommend this book for anyone taking a course in Topology, even Graduate students.
This book is well written with many illustrations and exercises to help you get an intuitive understanding of 3 Dimensional manifolds. This helped me a lot in my second year Analysis class as I had an intuitive notion of manifolds taught in class.
At the same time the book is easy enough for high school students who always wondered what a Mobius strip or a Klein bottle was but did not find any books on it. This book would make Topology interesting for everyone. I give it a five star rating.

Loads of fun
But this book can also be quite serious, although it may take someone with an extensive math background to see this. The book seems aimed primarily at high-schoolers, but graduate students in topology can definitely benefit from reading it.

Weeks starts out by explaining surfaces and the quotient space descriptions of the torus and klein bottle. Later chapters describe 3-manifolds, fibre bundles(!), and the 8 geometries relevant to Thurston's geometrization conjecture. The focus of the book is on applying these concepts to investigating the shape of our spatial universe. This is a particularly apt goal, given that many times in the book the reader is asked to imagine living in various kinds of spaces.

He has a very good set of exercises designed to increase one's visualization powers. For example, in the chapter on 3-manifolds, he has the reader color various covering space pictures of 3-manifolds like the 3-torus, according to some specifications; this really helps one understand how covering maps work.

As someone who was familiar with topology before reading the book, I can say that the book has definitely increased my understand of 3-manifolds, which is more than I can say for most topology books. In particular, I found the material on fibre bundles very enlightening.

Straight talk about curved space
What is the universe as a whole shaped like? Does it curve back on itself? Does it meet itself at the other side without curving? Is its Flatland analogy a plane, or a sphere, or a doughnut, or a Klein bottle? What other, stranger geometries become possible with the added dimension? And if the universe has one of these exotic shapes, how could astronomers ever know for sure?

Jeffrey Weeks, a MacArthur ("genius grant") fellow and a consultant to NASA on cosmological observations, believes that there's no reason why a liberal arts student or a high schooler shouldn't be able to have a solid understanding of the answers to these questions, even though some of them are at the edge of research in cosmology and three-manifolds, and others have traditionally not been part of the math curriculum before graduate school.

The math is presented at an elementary level, but it is genuine mathematics. Readers in the intended audience must be prepared to roll up their sleeves; there are exercises, and there are formulas, and their minds will be stretched. But there are no prerequisites other than a little first-year algebra, and the discussion stays at a vividly concrete level, with a plethora of diagrams to aid the swelling imagination. High schoolers will benefit from some guidance getting through it; it's appropriate for undergraduate self-study.

More mathematically sophisticated readers, even those who've taken a course in algebraic topology or differentiable manifolds, will find the book a lively read, but will still probably learn a thing or two. I, for one, was startled to be shown a Moebius strip that was two-sided! (The trick is to embed it in a non-orientable three-space.)

The payoff is in the final two chapters, which detail programs of astronomical observation that could well tell us the precise topology and geometry of the universe, and explain just how they would do it. One chapter is devoted to a technique based on correlating distances between galactic clusters, and the other to a statistical search for correlated arcs of great circles in the cosmic microwave background. Both observations will probably be completed within the next decade. It's an exciting prospect.

Buyers note: I believe the Amazon characterization of this as a paperback is in error. I bought the second edition in hardcover at the same list price. In its (successful) attempt to avoid intimidation, it uses a large typeface, so it would fill out some 200 pages in a more typical math format. ... Read more

Book Description

Theodore Frankel explains those parts of exterior differential forms, differential geometry, algebraic and differential topology, Lie groups, vector bundles and Chern forms essential to a better understanding of classical and modern physics and engineering. Key highlights of his new edition are the inclusion of three new appendices that cover symmetries, quarks, and meson masses; representations and hyperelastic bodies; and orbits and Morse-Bott Theory in compact Lie groups. Geometric intuition is developed through a rather extensive introduction to the study of surfaces in ordinary space. First Edition Hb (1997): 0-521-38334-X First Edition Pb (1999): 0-521-38753-1 ... Read more

Reviews (13)

over and over and over again
Having taken a course out of Frankel (over the first 7 chapters) and now having used it in my senior project (topology of circuit analysis) I have to say that I love this book more by the day.

Beforewarned it is not an easy text and you may have to read a section or a chapter over a hundred times. I have found that the material is dense and deep but in a way that welcomes effort. It is weak as far as rigor goes, but rigor can sometimes get in the way of understanding. Use this book alongside mathematics texts in topology, differential geometry and linear algebra and there is much to gain.

For an undergraduate in mathematical physics (which I am) I have come to love this book I highly recommend it to a serious student.

the geometry of physics
I just finished a class in mathematical physics, and the text we used was Bamberg & Sternberg. I found that books treatment muddled and shortsighted. I mean, most of the linear algebra in the book deals only with 2 dimensional vector spaces. And the book was entirely useless in teaching differential forms...

So i went looking for a better book to learn diferential forms. i didn t like flanders, it was too brief. this is the book for me. Don t expect to find any linear algebra here, but you d better know lin. alg. before you open this book.

it is a challenging book, mathematically speaking, to study on your own (for a senior ugrad phys major, anyway), but it s treatment of forms and tensors is comprehensive, thorough, and detailed. and it shows you all the applications to relativity and electrodynamics, etc... it also builds up all the theory in with a background of differential geometry and topology, which are developed in the first chapter (but wasn t i glad to have already studied those topics beforehand!)

this book prepared me for my mathematical physics class, plus gave me months of other material to study. it is difficult, so i read and reread each chapter.

Bad book.
Frankel's book is provbably the most confusing book I have ever looked into. As other readers noted, it is probably because of his approach not to define things properly. The book's style is extremely wordy, unnecessary wordy that is. The result - total confusion. Mr. Frankel probably thinks the readers are nearly morons, so he tries to re-express some (really simple) notions with words that supposedly will make things lucid. Well, he fails.
Alternative book by Nakahara is way better.I also recommend "Analysis, Manifolds and Physics" by Yvonne Cgiqyet-Bruhat, et al
2 stars for effort.

Good one, even if not the best, probably
This is a valuable reference for students pursuing a support or who want to get themselves deeper in the mathemathical part connected with QFT and GR. I particularly appreciated the first chapter about Manifolds and vector fields, the part about algebraic topology (chapter 13: chains, homology groups and De Rahm's theorem, Betti numbers) and the part about homotopy groups. On the other hand the first part about tensors, exterior forms, integration of differential forms and the Lie derivative seems to me a bit uneven compared to the one I've mentioned above. For this section I'd recommend: Aldrovandi - Pereira, "Introduction to geometrical Physics", or V.I. Arnold, "Classical Mechanics" (first part) which is not complete if compared to the other two books (this is a book about the symplectic formulation of CM and not strictly a matemathical book) but things that are contained are exposed in a beautiful way. Another valuable book is Nakahara (a classic one), but I still have to finish reading it so I'll leave a comment about it in the next. The level of T. Frankel is at last yr undergrad - 1st yr graduate.

There are better...
I have used this book in an independent study in Geometry of Differential Forms. It did not take me too long to start looking for other references. There is something about its content that makes it diffucult to follow. May be it's too wordy. There are several misprints in notation. After I few weeks of study, I turned to Morita's Geometry of Differential Forms. The mathematical presentation is much clear and it's only 300 pages. I really like Frankel's book mainly for its application to physics. But with respect to the math, I recommend Morita's and Thirring monographs. ... Read more

Book Description

Students will find all the information covered in the standard textbooks--and more--explained clearly and concisely in this powerful study tool. Unusually detailed, it elucidates all the most difficult-to-grasp concepts that class studies and texts sometimes gloss over. The hundreds of problems with fully explained solutions illuminate important points and teach students sound problem-solving skills. Ideal, also, for independent study.

Reviews (1)

Differential Geometry - A Schaum's Outline Series
As with all of the Schaum's Outline Series, this book is particularly useful if the readers intent is to gain a working knowledge of the subject. The subject of Differential Geometry is no exception. Dr. Lipschultz has done an excellent job of communicating the essential aspects of differential geometry to the reader. The book assumes a fairly low level of mathematical ability having calculus as the primary prerequisite. From this humble beginning, Dr. Lipschultz takes the reader through the necessary discussions of vector functions, curvature, fundamental forms, and tensor analysis. Given the theoretical nature of the subject, Dr. Lipschultz has included most of the theorems and associated proofs necessary for a general understanding of the subject. However, this book is not a substitute for a serious study of differential geometry. In addition most of the problems are limited to two dimensional surfaces and this reader would have enjoyed a more adventurous investigation of higher dimensional spaces. Like all Schaum's series, the text is chock full of problems and their solution. I recommend this book for anyone interested in quickly gaining a working knowledge of the subject. ... Read more

Book Description

This book introduces the tools of modern differential geometry--exterior calculus, manifolds, vector bundles, connections--and covers both classical surface theory, the modern theory of connections, and curvature. Also included is a chapter on applications to theoretical physics. The author uses the powerful and concise calculus of differential forms throughout.Through the use of numerous concrete examples, the author develops computational skills in the familiar Euclidean context before exposing the reader to the more abstract setting of manifolds. The only prerequisites are multivariate calculus and linear algebra; no knowledge of topology is assumed. Nearly 200 exercises make the book ideal for both classroom use and self-study for advanced undergraduate and beginning graduate students in mathematics, physics, and engineering. ... Read more

Reviews (5)

Not for everyone, flawed in basic ways
I disagree with reviewers who found this book useful for self-study. I would not recommend it for individuals first learning this material. The book is frankly contradictory in places, and frustratingly repetitive in others. In the early chapters it assumes concepts not yet explained, and introduces terminology and symbols that are nowhere defined.

If you already know quite a bit, you may find this approach enlightening. But if you're just beginning to master these concepts, I suggest you look elsewhere.

I also suggest that much tighter editing would do this book a world of good. Go with Kreiszig, or Lovelock and Rund instead.

Excellent book
This is a very modern, very concise, and very efficient book. By using vector bundles the curvature forms on semi-Riemannian manifolds are introduced. Definitions are given clearly and intuitively. Without spending tons of pages on digression to minimal surfaces, Hopf-Rinow thm, Gauss-Bonnet thm, etc., the book builds enough machinery to describe the gauge field theory in the last chapter. Most other differential geometry books either throw in too many applications to waste reader's time or give vague definitions (too bluntly abstract or not self-contained) to confuse the reader.

All exercise problems are interesting and important. Hints are given to some of them.

I found Warner's "Foundations of Differentiable Manifolds and Lie Groups" is a good complement to address the algebraic and topological side of differential geometry.

The ONLY book really suited for self study
I would just like to point out that Darling's book is the only book I've encountered which is suited for self study. It resembles someone's classroom notes - i.e., nothing fancy, no glossy color 3-d graphics or such - but it is very neatly organized, with many examples and helpful problems, and it is really, really suited for someone trying to study the subject by him/herself (me ... ). It is not very physically oriented - not many physical examples are provided throughout the text, and it is mathematical in nature, but don't let that deter you! In fact, the sharp distinction between mathematics and physics is pedagogically wise.
Another good thing about this book is that it does not begin with completely abstract definitions. First of all it develops exterior calculus and diff. manifolds in ordinary Euclidian space. This is a must for anyone studying on their own, believe me! No matter how mathematically mature you are, those things just don't make sense unless you've seen how they work in familiar settings. You don't have to worry, though - Darling keeps his notation clean; Darling tries as hard as he can to keep everything in pure geometrical language, referring to a specific basis only when absolutely necessary (or when it helps one understand).
I cannot say how good a classroom text this is, but do yourself a favor and check it out if you're thinking of studying this on your own! Darling is a clear and (equally important!) responsible teacher.

A must for both the physicist and mathematician
RWR Darling should be the first and foremost book for learning about differential geometry both for physicists and mathematicians. I have learned from numerous books on this subject, and while I can't say Darling includes everything one could want (I can't say anyone ever does), his text explains some very esoteric ideas in terms of linear algebra and vector calculus.

A notable departure this book makes is dispensing with the usual coordinate basis for tangent spaces which is commonly used by physicists. To the experienced physics reader, this may seem daunting, and unnecessarily abstract at first. However, the pay-off in the ability later on to discuss gauge theories and fiber bundles is huge.

This book is also suited for mathematicians less interested in physics. Darling does not always assume that a manifold has some metric, and discusses the subtle differences between vectors and co-vectors in modern mathematical language. Secondly, he provides a lot of motivation for the mathematical constructions and takes great care to present key definitions in extremely coordinate free ways.

Gauge theories in the mathematical way
The main difficulty found by physicists in the learning of modern differential geometry is topology. The various constructions introduced by Cartan and others, differential forms, connections, even fiber bundles, on the contrary, pose no difficulties: it is only a question of developing the appropriate muscles and reflexes. R. Darling wrote the ideal book to teach connections on a G-bundle (gauge theories, in the nomenclature of physicists), by refraining, as much as possible, to use explicit topology. As physicists are not a special kind of human beings, I believe what I said above is also true of (beginning) mathematicians. Otherwise, why would Darling choose such course (in the navigational sense). The book starts with Cartan calculus in Euclidean space, continues there up to surface theory, then introduces (intrinsic) manifolds. Perhaps the key concept of the book comes next: Vector Bundles. All previous constructions are extended to bundles, and the concept of conn! ections on vector bundles deserves a special chapter. The book ends with Applications to Gauge Field Theory (mathematics-wise, but quite accessible). There are many pedagogical virtues in this much welcome book. Finally a good alternative to Bishop-Goldberg`s "Tensor Calculus on Manifolds". ... Read more

Book Description

Many parallels between complex dynamics and hyperbolic geometry have emerged in the past decade. Building on work of Sullivan and Thurston, this book gives a unified treatment of the construction of fixed-points for renormalization and the construction of hyperbolic 3- manifolds fibering over the circle.

Both subjects are studied via geometric limits and rigidity. This approach shows open hyperbolic manifolds are inflexible, and yields quantitative counterparts to Mostow rigidity. In complex dynamics, it motivates the construction of towers of quadratic-like maps, and leads to a quantitative proof of convergence of renormalization.

Book Description

The topology optimization method solves the basic engineering problem of distributing a limited amount of material in a design space. The first edition of this book has become the standard text on optimal design, which is concerned with the optimization of structural topology, shape and material. This edition has been substantially revised and updated to reflect progress made in modelling and computational procedures. It also encompasses a comprehensive and unified description of the state of the art of the so-called material distribution method, based on the use of mathematical programming and finite elements. Applications treated include not only structures but also MEMS and materials. ... Read more

Book Description

This book is an introductory graduate-level textbook on the theory of smooth manifolds. Its goal is to familiarize students with the tools they will need in order to use manifolds in mathematical or scientific research--- smooth structures, tangent vectors and covectors, vector bundles, immersed and embedded submanifolds, tensors, differential forms, de Rham cohomology, vector fields, flows, foliations, Lie derivatives, Lie groups, Lie algebras, and more. The approach is as concrete as possible, with pictures and intuitive discussions of how one should think geometrically about the abstract concepts, while making full use of the powerful tools that modern mathematics has to offer. Along the way, the book introduces students to some of the most important examples of geometric structures that manifolds can carry, such as Riemannian metrics, symplectic structures, and foliations. The book is aimed at students who already have a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis. John M. Lee is Professor of Mathematics at the University of Washington in Seattle, where he regularly teaches graduate courses on the topology and geometry of manifolds. He was the recipient of the American Mathematical Society's Centennial Research Fellowship and he is the author of two previous Springer books, Introduction to Topological Manifolds (2000) and Riemannian Manifolds: An Introduction to Curvature (1997). ... Read more

Reviews (2)

Excellent, lucid book on manifolds
Topics are explained with exceptional clarity; portions of the book are well tied together; and the order of exposition flows very well. Lie groups are introduced quite early on, but their full power is not revealed until later in the book. I can't laud this book enough. I had a firm, well-developed basis of differential geometry after reading through this book for a course. The excersises are illuminating, as are the examples. Theorems and their proofs are clearly labeled. The motivational explanations prefacing theorems do an excellent job of conveying the intuition behind ideas.

I would recommend this book over Boothby any day. I haven't read Spivak, so I can't compare Lee to it, but Lee definitely seemed like an excellent choice for an intro grad class on differential geometry.

get the hardcover!!
I just finished a 20-week course from this book. It is well-written, with a healthy number of examples and many exercises (interspersed throughout the text) and problems (at the end of each chapter). The style is rather informal: this is good for the novice to this subject, which groans under the weight of its own notation. The presentation is well-organized, clear, and accessible. Dr. Lee maintains current errata for the book (some did make it into the problems unfortunately) at his website.

One thing I might suggest is that if you plan to use this book heavily (e.g., for a course rather than for reference or bedtime reading) you should consider investing in the hardcover version is possible. The book is lengthy and the binding tends to split. Mine is still in one piece, but only just. You have to be very gentle with this book to keep it intact. ... Read more

Book Description

This book develops the theory of global attractors for a class of parabolic PDEs that includes reaction-diffusion equations and the Navier-Stokes equations, two examples that are treated in detail.A lengthy chapter on Sobolev spaces provides the framework that allows a rigorous treatment of existence and uniqueness of solutions for both linear time-independent problems (Poisson's equation) and the nonlinear evolution equations which generate the infinite-dimensional dynamical systemss of the title. Attention then switches to the global attractor, a finite-dimensional subset of the infinite-dimensional phase space which determines the asymptotic dynamics. In particular, the concluding chapters investigate in what sense the dynamics restricted to the attractor are themselves "finite-dimensional." The book is intended as a didactic text for first year graduates, and assumes only a basic knowledge of Banach and Hilbert spaces, and a working understanding of the Lebesgue integral. ... Read more

Book Description

This book combining wavelets and the world of the spectrum focuses on recent developments in wavelet theory, emphasizing fundamental and relatively timeless techniques that have a geometric and spectral-theoretic flavor. The exposition is clearly motivated and unfolds systematically, aided by numerous graphics.

This self-contained book deals with important applications to signal processing, communications engineering, computer graphics algorithms, qubit algorithms and chaos theory, and is aimed at a broad readership of graduate students, practitioners, and researchers in applied mathematics and engineering. The book is also useful for other mathematicians with an interest in the interface between mathematics and communication theory. ... Read more

Reviews (6)

A masterpiece in wavelets
This is a very well written book on the theory of wavelets and its applications. The presentation is self-contained and may serve as an introduction for someone who wants to learn about this topic. It also is an excellent book for those who have an advanced graduate degree in applied mathematics since it demonstrates how to truly understand complex concepts. This book gives a general presentation of some recent developments in wavelet theory with an emphasis on techniques that have a geometric and spectral-theoretic flavor. It can be certainly used as a textbook for graduate students as well as as a reference book for the specialists and researchers. A very nice feature of the book are the tutorials at the beginning of each chapter and some sections, which serve as summaries of main ideas and guides through the background ideas and motivation. At the end of every chapter there are plenty of excercises of various level of difficulty. Overall, this is a great book and I recommend it highly to a motivated reader.

Distinguished link between math and computer science
This is an important mathematical reference written in excellent style. Wavelets have found applications in many areas of engineering and CS. The authors provide a detailed, rich and entertaining tour through this relatively young but important field for both math and CS/Eng. Connections are, e.g., made between advanced CS virtual-reality applications such as audio-systems processing, future applications such as quantum computing, and advanced math in functional analysis and operator theory.

A fresh look at wavelets
The Book by Bratteli and Jorgensen is a superb book on wavelet's theory. It is very well written and has new and a fresh point of view on the subject. Although there are several good books on wavelets, the book by Bratteli and Jorgensen covers an important niche that has not been covered before. In particular

1- The book covers the theory of wavelets from the point of view of operators and functional analysis and will appeal to a growing number of pure as well as applied mathematicians interested in the subject.

2- The writing of the book is very appealing: every chapter starts by a tutorial that gives motivation as well as intuition. It is then followed by a very clean mathematical development of the subject, together with many examples, figures, and applications from physics and engineering. A set of nice problems is provided at the end of each chapter. Thus this book can be used as a graduate textbook or for mathematical seminars in mathematics departments.

3- This book can even be used by experts in wavelet theory for learning about recent developments and new perspectives from operator theory and functional analysis.

I highly recommend this book.

an intriguing new wavelet book
This is a book about an important topic in applied mathematics by two authors with excellent credentials in both pure and applied areas. The reader will find many intriguing threads connecting wavelets to other parts of mathematics, including a wavelet index theorem, quantum computing, the ubiquitous C*-algebras O_n and, of course, spectral theory. The graphics are meticulously done.

I look forward to learning a lot from it.

An Intriguiging New Book
This is a book about an important topic in applied mathematics by two authors with excellent
credentials in both pure and applied areas.

The reader will find many intriguing threads connecting wavelets to other parts of mathematics, including a wavelet "index theorem", quantum computing, the ubiquitous Cuntz C*-algebras and, of course, spectral theory. The graphics are excellent. I look forward to learning a lot from it. ... Read more

Book Description

With more than 30 million copies sold, Schaum's are the most popular study guide on the planet. Mathematics students around the world turn to this clear and complete guide to general topology for its through introduction to the subject, includingeasy-to-follow explanations of topology of the line and plane and topological spaces. With 650 fully solved problems and hundreds more to solve on your own (with answers supplied), this guide can help you spend less time studying while you make better grades! ... Read more

Reviews (1)

If you are a good researcher, this is a good resource!
If you are taking an introductory Topology course, I recommend using this book. It is a little old, and some of the problems are not solved, but it will offer you an advantage when using your class text. You should use Lipschutz's book together with REA's Topology Problem Solver to give you full study advantage when taking on this most difficult subject. With some moderate study, you should do better on your assignements! ... Read more

Book Description

Reviews (7)

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.

An Outstanding Achievement.
As a matter of fact all the materials written by J. P. May are precise, concise and useful. He is not of kind of those people who write 1000 pages and reach at obvious matters. This book is really a good introduction to the modern aspects of algebraic topology. It has less than 250 pages. I liked the treatment very much and appreciate it for teaching me a lot of mathematics. I dare to say that if someone else wants to write a book including all materials treated in this book, then the book would consist of at least 1000 pages. There is more to this book than just classical algebraic topology.

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.

Excellent Modern Treatment of Algebraic Topology
One of the reasons that Algebraic Topology is difficult to learn is that often the more general constructions (which are algebraic) are difficult to motivate visually. In fact, I have often found that attempts at visuallizing lead to confusion. J. Peter May avoids confusing illistrations in this book. Constructions are motivated by the results they consort. Most importantly May employes a thoroughly modern point of view. For example: the language of cofibrations/fibrations is used throughout, the handy idea of fundamental groupoid is introduced early in the treatment of the fundamental groups, there are a couple of chapters dedecated to homological algebra intersperced, both homology and cohomology are developed from the axiomatic point of view. May concludes the text with introductions to several more advanced topics such as cobordism, K-theory, and characteristic classes. The list of books that May offers in the suggestions for further reading section at the end is fairily comprehensive.

[too much] for a book that will just sit on your bookself
this is not a bad book, but it isnt for real. the back of the book says: ...treatment is sophisticated, no prior knowledge of the subject is assumed.

i think not.

you better be armed with a few other books and be prepared to spend some hours if you want to "learn" from this book as a beginner. ... Read more

Book Description

This book is an introduction to manifolds at the beginning graduate level. It contains the essential topological ideas that are needed for the further study of manifolds, particularly in the context of differential geometry, algebraic topology, and related fields. Its guiding philosophy is to develop these ideas rigorously but economically, with minimal prerequisites and plenty of geometric intuition.

A course on manifolds differs from most other introductory mathematics graduate courses in that the subject matter is often completely unfamiliar. Unlike algebra and analysis, which all math majors see as undergraduates, manifolds enter the curriculum much later. It is even possible to get through an entire undergraduate mathematics education without ever hearing the word "manifold." Yet manifolds are part of the basic vocabulary of modern mathematics, and students need to know them as intimately as they know the integers, the real numbers, Euclidean spaces, groups, rings, and fields.

In his beautifully-conceived Introduction, the author motivates the technical developments to follow by explaining some of the roles manifolds play in diverse branches of mathematics and physics. Then he goes on to introduce the basics of general topology and continues with the fundamental group, covering spaces, and elementary homology theory. Manifolds are introduced early and used as the main examples throughout.

John M. Lee is currently Professor of Mathematics at the University of Washington in Seattle. In addition to pursuing research in differential geometry and partial differential equations, he has been teaching undergraduate and graduate courses on manifolds at U.W. and Harvard University for more than fifteen years. ... Read more

Reviews (2)

Review of a non-mathematician
Being a physicist I've always been fascinated with the use of manifolds and differential geometry in mechanics, field theory, etc ... Most differential geometry books I've encountered only devote about 1 chapter to manifolds and smooth manifolds at that. However this text takes its time to teach the reader what the author states he thinks is the minimum amount of general knowledge about topological manifolds (no discussion of smooth/analytic manifolds is included). The author takes his time developing everything from scratch, not even assuming any experience with (point set) topology, so this book is particularly suited for those who shy away from the subject just because they're not mathematicians and don't know topology. The only prerequisites are advanced calculus and linear algebra, nothing too fancy. The writing itself is very clear and while rigorous this book does not get lost in the boring lemma-theorem-proof vicious cycle so many other math books fall flat at. Throughout the book are scattered exercises for the reader to do (about 10-20 each chapter) and there are problems at the end of each chapter (no solutions/hints included). All-in-all I feel this text has offered me a much greater understanding of manifolds and the general theory dealing with them. Highly recommended.

A very readable text
An excellent text for a beginning graduate level class. This is NOT a comprehensive text covering the material in exhaustive detail, but it is an excellent overview of surfaces, simplicial complexes, homotopy, homology, and the briefest peek at cohomology. The sequence is efficient, and the author does a good job of motivating the discussions, rather than simply dumping an abstraction into your lap. As always, one should be familiar with point-set and groups before jumping in. If you are looking for a text at an undergraduate level, see Armstrong's Basic Topology or Kinsey's Topology of Surfaces. ... Read more

Book Description

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

Reviews (4)

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

exceptionally well organized
This is a lean fast introduction to topology at the third or fourth year level. Pure math types only. The book is terse but the topics are selected with care and one things leads to the next. The proofs are sufficiently detailed. Nearly every exercise has a solution in the back. The clearest exposition of the fundamental group I've seen.

Good grad school prep.
This is the usual text for introductory Topology at UCLA, where I took the course. The authors (who teach at UCLA) have "if you haven't chewed through every syllable you are not learning" mentality. In short, the book is terse and demands a lot from the reader. Looking back, this was great preparation for graduate school and is probably the best philosophy for the serious undergrad. The book contains all of the information one needs for an introductory course, but absolutely no more. Not a single character is wasted on "extraneous" explanation. Be ready for battle when opening this one, but it's worth it.

A solid introduction
This book is divided into two parts: an introduction to topology, using metric spaces to motivate the definition of a topological space, and the algebraic applications of topology (such as homotopy theory and Jordan's theorem). The good things about it is its strong geometrical flavor and short and to-the-point explanations, which are nevertheless very good. Another excellent thing is the large number of exercises, most of which have full solutions (!) and/or hints at the end of the book. The chapter on metric spaces is also very informative, and stresses the many uses these spaces have in mathematics. However, the book does not really go into detail, especially when dealing with topological spaces. This is fine if you're looking for an introduction to topology (as the name implies . . . ) but it makes the book unsuitable as a reference. All-in-all a very "cute" book - as cute as introduction-to-topology books get! ... Read more

Book Description

Knot theory now plays a large role in modern mathematics. This unparalleled text and reference describes the main concepts of modern knot theory with full proofs accessible to both beginners and professionals alike. It presents both classical and modern knot theory, as well as the most significant results from braid theory, including the full proof of Markov's theorem, and Alexander's and Vogel's algorithms. It includes valuable information on the theory of coding knots by d- diagrams, as well as the author's own results in virtual knot theory. The material is presented at a level suitable for advanced undergraduate students, and the text is ideal for a course on knot theory. ... Read more

Reviews (1)

Best of Knots
Knot Theory by Vassily Manturov (CRC Press) The aim of the present monograph is to describe the main concepts of modern knot theory together with full proofs that would be both accessible to beginners and useful for professionals. Thus, in the first chapter of the second part of the book (concerning braids) we start from the very beginning and in the same chapter construct the Jones two-variable polynomial and the faithful representation of the braid groups. A large part of the present title is devoted to rapidly developing areas of modern knot theory, such as virtual knot theory and Legendrian knot theory.
In the present book, we give both the "old" theory of knots, such as the fundamental group, Alexander's polynomials, the results of Dehn, Seifert, Burau, and Artin, and the newest investigations in this field due to Conway, Matveev, Jones, Kauffman, Vassiliev, Kontsevich, Bar-Natan and Birman. We also include the most significant results from braid theory, such as the full proof of Markov's theorem, Alexander's and Vogel's algorithms, Dehornoy algorithm for braid recognition, etc. We also describe various representations of braid groups, e.g., the famous Burau representation and the newest (1999-2000) faithful Krammer-Bigelow representation. Furthermore, we give a description of braid groups in different spaces and simple newest recognition algorithms for these groups. We also describe the construction of the Jones two-variable polynomial.
In addition, we pay attention to the theory of coding of knots by d-diagrams, described in the author's papers. Also, we give an introduction to virtual knot theory, proposed recently by Louis H. Kauffman. A great part of the book is devoted to the author's results in the theory of virtual knots.
Proofs of theorems involve some constructions from other theories, which have their own interest, i.e., quandle, product integral, Hecke algebras, connection theory and the Knizhnik-Zamolodchikov equation, Hopf algebras and quantum groups, Yang-Baxter equations, LD-systems, etc.
The contents of the book are not covered by existing monographs on knot theory; the present book has been taken a much of the author's Russian lecture notes book on the subject. The latter describes the lecture course that has been delivered by the author since 1999 for undergraduate students, graduate students, and professors of the Moscow State University.
The present monograph contains many new subjects (classical and modern), which are not represented in the author's earlier Russian version of this book.
While describing the skein polynomials we have added the Przytycky-Traczyk approach and Conway algebra. We have also added the complete knot invariant, the distributive grouppoid, also known as a quandle, and its generalisation. We have rewritten the virtual knot and link theory chapter. We have added some recent author's achievements on knots, braids, and virtual braids. We also describe the Khovanov categorification of the Jones polynomial, the Jones two-variable polynomial via Hecke algebras, the Krammer-Bigelow representation, etc.
The book is divided into thematic parts. The first part describes the state of "pre-Vassiliev" knot theory. It contains the simplest invariants and tricks with knots and braids, the fundamental group, the knot quandle, known skein polynomials, Kauffman's two-variable polynomial, some pretty properties of the Jones polynomial together with the famous Kauffman-Murasugi theorem and a knot table.
The second part discusses braid theory, including Alexander's and Vogel's algorithms, Dehornoy's algorithm, Markov's theorem, Yang-Baxter equations, Burau representation and the faithful Krammer-Bigelow representation. In addition, braids in different spaces are described, and simple word recognition algorithms for these groups are presented. We would like to point out that the first chapter of the second part (Chapter 8) is central to this part. This gives a representation of the braid theory in total: from various definitions of the braid group to the milestones in modern knot and braid theory, such as the Jones polynomial constructed via Hecke algebras and the faithfulness of the Krammer-Bigelow representation.
The third part is devoted to the Vassiliev knot invariants. We give all definitions, prove that Vassiliev invariants are stronger than all polynomial invariants, study structures of the chord diagram and Feynman diagram algebras, and finally present the full proof of the invariance for Kontsevich's integral. Here we also present a sketchy introduction to Bar-Natan's theory on Lie algebra representations and knots. We also give estimates of the dimension growth for the chord diagram algebra.
In the fourth part we describe a new way for encoding knots by d-diagrams proposed by the author. This way allows us to encode topological objects (such as knot, links, and chord diagrams) by words in a finite alphabet. Some applications of d-diagrams (the author's proof of the Kauffman-Murasugi theorem, chord diagram realisability recognition, etc.) are also described.
The fifth part contains virtual knot theory together with "virtualisations" of knot invariants. We describe Kauffman's results (basic definitions, foundation of the theory, Jones and Kauffman polynomials, quandles, finite-type invariants), and the work of Vershinin (virtual braids and their representation). We also included our own results concerning new invariants of virtual knots: those coming from the "virtual quandle", matrix formulae and invariant polynomials in one and several variables, generalisation of the Jones polynomials via curves in 2-surfaces, "long virtual link" invariants, and virtual braids.
The final part gives a sketchy introduction to two theories: knots in 3-manifolds (e.g., knots in RP3 with Drobotukhina's generalisation of the Jones polynomial), the introduction to Kirby's calculus and Witten's theory, and Legendrian knots and links after Fuchs and Tabachnikov. We recommend the newest book on 3-manifolds by Matveev.
At the end of the book, a list of unsolved problems in knot and link theory and the knot table are given.
The description of the mathematical material is sufficiently closed; the mono-graph is quite accessible for undergraduate students of younger courses, thus it can be used as a course book on knots. The book can also be useful for professionals because it contains the newest and the most significant scientific developments in knot theory. Some technical details of proofs, which are not used in the sequel, are either omitted or printed in small type. ... Read more

Book Description

In most major universities one of the three or four basic first-year graduate mathema