Product Description
This book is designed for the reader who wants to get a general view of the terminology of General Topology with minimal time and effort. The reader, whom we assume to have only a rudimentary knowledge of set theory, algebra and analysis, will be able to find what they want if they will properly use the index. However, this book contains very few proofs and the reader who wants to study more systematically will find sufficiently many references in the book.

Key features:

• More terms from General Topology than any other book ever published

• Short and informative articles

• Authors include the majority of top researchers in the field

• Extensive indexing of terms
... Read more

Customer Reviews (1)

A good and expensive reference
This, as it's name tells, is a book that you can use as a desk reference on general topology. There are no proofs and references to results are limited. Also, this book is way too expensive. I bought it because I have made my mind to being a topologist for the rest of my life. That means that you should buy this book if and only if your life is topology. I also think this book is for the college libraries to buy and have them for everybody at a university campus (or at least in the institute or school it belongs at). ... Read more

Customer Reviews (1)

An excellent reference book
I found Engelking a difficult book to study general topology from, but an excellent reference book. Engelking covers wide areas of General topology, starting from the basic definition. At times the exposition seemed to veer into too much technicalities, which seemed to distract one, if one was reading the book alone. What makes the book truly great, however, is its index, and bibliography. It's wealth of examples and counterexamples is very useful, as well.

Finally, the Russian translation seems to be enriched with the comments of translators and more bibliography. ... Read more

Product Description
This volume mainly focuses on various comprehensive topologicaltheories, with the exception of a paper on combinatorial topologyversus point-set topology by I.M. James and a paper on the history ofthe normal Moore space problem by P. Nyikos.The history of the following theories is given: pointfree topology,locale and frame theory (P. Johnstone), non-symmetric distances intopology (H.-P. Kunzi), categorical topology and topologicalconstructs (E. Lowen-Colebunders and B. Lowen), topological groups (M.G. Tkacenko) and finally shape theory (S. Marde J. Segal).Together with the first two volumes, this work focuses on the historyof topology, in all its aspects. It is unique and presents importantviews and insights into the problems and development of topologicaltheories and applications of topological concepts, and into the lifeand work of topologists. As such, it will encourage not only furtherstudy in the history of the subject, but also further mathematicalresearch in the field. It is an invaluable tool for topologyresearchers and topology teachers throughout the mathematical world. ... Read more

Product Description
Geometry and topology are strongly motivated by thevisualization of idealobjects that have certain specialcharacteristics. A clear formulation of a specific propertyor a logically consistent proof of a theorem oftencomesonly after the mathematician has correctly "seen" what isgoing on. These pictures which are meant to serve assignposts leading to mathematical understanding, frequentlyalso contain a beauty of their own.The principal aim of this book is to narrate, in anaccessible and fairly visual language, about some classicaland modern achievements of geometry and topology in bothintrinsic mathematical problems and applications tomathematical physics. The book starts from classicalnotionsof topology and ends with remarkable new results inHamiltonian geometry. Fomenko lays special emphasis uponvisual explanations of the problems and results anddownplays the abstract logical aspects ofcalculations. Asan example, readers can very quickly penetrate into thenewtheory of topological descriptions of integrableHamiltoniandifferential equations. The book includes numerousgraphicalsheets drawn by the author, which are presented inspecialsections of "Visual material". These pictures illustratethemathematical ideas and results contained in the book. Usingthese pictures, the reader can understand many modernmathematical ideas andmethods.Although "Visual Geometry and Topology" is aboutmathematics,Fomenko has written and illustrated this bookso that students andresearchers from all the naturalsciences and also artists and art students will findsomething of interest within its pages. ... Read more

Product Description
Topology, for many years, has been one of the most exciting and influential fields of research in modern mathematics.Although its origins may be traced back several hundred years, it was Poincaré who "gave topology wings" in a classic series of articles published around the turn of the century.While the earlier history, sometimes called the prehistory, is also considered, this volume is mainly concerned with the more recent history of topology, from Poincaré onwards.

As will be seen from the list of contents the articles cover a wide range of topics.Some are more technical than others, but the reader without a great deal of technical knowledge should still find most of the articles accessible.Some are written by professional historians of mathematics, others by historically-minded mathematicians, who tend to have a different viewpoint. ... Read more

Customer Reviews (2)

A lot of very good essays
This book collects 40 essays on various themes, events, and personalities in the history of topology. I have browsed half of them and carefully read 6 or 8.Plus I have used the book as a reference, and so gained isolated facts from many of the essays.

The essays assume various amounts of background, proportionate to their subjects.In generally they are very clear and if you have any knowledge at all of a topic you can read an essay on it here at least well enough to decide if you want to work harder on it.

The book concentrates more on manifolds, fixed point theorems, algebraic topology, and homological algebra, than on issues in general or point set topology. It covers Poincare, Brouwer, Weyl, and gets up to derived categories.It could not possibly cover everything in that range.For one, Lefschetz is mentioned often but he could well have deserved a whole article on himself.Or there could have been an article on the Princeton school. But incomplete we must all resign ourselves to be.

The book is a huge amount of information, very well organized and presented.

Absolutely Brilliant
A sublime account of the history of topology from it's earliest incarnation as analysis situs to contemporary research.

No home should be without one. ... Read more

Product Description
Translated and Revised by C. Cecilia Krieger. Critically acclaimed text presents detailed theory of Fréchet (V) spaces and a comprehensive examination of their relevance to topological spaces, plus in-depth discussions of metric and complete spaces. Numerous exercises reinforce teachings of each chapter. "...an elegant piece of work suitable for the beginning student and the mature mathematician."—Scripta Mathematica. 2nd ed.
... Read more

Product Description
This book gives the reader a thorough knowledge of the basic topological ideas necessary for studying differential manifolds. These topics include immersions and imbeddings, approach techniques, and the Morse classification of surfaces and their cobordism. The author keeps the mathematical prerequisites to a minimum; this and the emphasis on the geometric and intuitive aspects of the subject make the book an excellent and useful introduction for the student. There are numerous excercises on many different levels ranging from practical applications of the theorems to significant further development of the theory and including some open research problems. ... Read more

Customer Reviews (5)

Excellent intermediate-level text
While I agree with almost everything that reviewers Paul Thurston and Dr. Carlson say about this book, I would rate it a little higher, since this book fills a niche that not too many other books occupy: It is more advanced than truly introductory treatments such as Guillemin & Pollack's Differential Topology or Milnor's Topology from the Differentiable Viewpoint but more basic than Kosinski's Differential Manifolds, and far more comprehensive than specialized books such as Milnor's Lectures on the h-Cobordism Theorem. Thus it is useful for its coverage of a wide range of topics in differential topology - embeddings, vector bundles, transversality, degree and intersection numbers, cobordism, Morse theory, isotopies - which is rigorous yet still somewhat elementary.

Hirsch writes clearly with precise definitions using modern terminology (in contrast to, say, G&P or Milnor's definition of manifold as a subset of Euclidean space). The proofs are usually compact but easy to follow, and he often explains what he is going to do ahead of time. A lot is compressed into relatively few pages - his proof of the equivalence of C^r, r>=1, and smooth structures in Chapter 2 versus Munkres' 60 page proof (in Elementary Differential Topology) being a good illustration of this - and results are often proved in much generality.

The second chapter in particular stands out, which covers function spaces and approximations and contains a general theorem that immediately yields the denseness of diffeomorphisms, embeddings, immersions, submersions, proper maps, etc., in the strong C^r topology. There is an excursion into jets, which is not found much in the literature, and is not really my cup of tea either, but it is not used much elsewhere in the book. Analytic approximations are also mentioned, although a key result in this area is only cited rather than proved (since this is not a book on complex analysis).

Beginning with Chapter 3, transversality is rightly emphasized as a central concept, while Chapter 6 introduces the Morse theory that dominates the last few chapters. A complete proof of the Morse-Sard theorem is given, although only in the smooth case, with a more general theorem only being stated (but still, this is more than any of the other texts I mentioned). Standard results such as the easy Whitney embedding, the existence of collars and tubular neighborhoods, the Brouwer fixed point theorem, the hairy ball theorem, the Hopf theorem, the Morse lemma, and the Morse inequalities are proved in addition to more advanced theorems, such as the classification of vector bundles or Thom's "fundamental theorem of cobordism," although some results, such as the computation of some cobordism groups, are only stated.

The chapter on vector bundles is actually the longest in the book, and certainly seems sufficient, although I partially agree with Mr. Thurston that it is odd that the idea of tangent vectors as derivations is not even mentioned. (Tangent vectors are essentially defined as elements of a vector bundle that transform in a certain way under coordinate changes.)

It is good to see gluings handled properly without hand-waving about smoothings, and the technical theorems on isotopies and the characterization of the disk from a Morse-theoretic perspective are a welcome addition, but the treatment of cobordism is too brief and handles and surgery are only touched upon. The classification of compact surfaces at the end is kind of overkill with all this machinery as well, but proceeding to handle decompositions or the h-cobordism theorem would make this book much longer. Consider this, instead, as preparation for those topics, which can be learned from, say, Kosinski.

Note that this book contains nothing on differential forms, integration, Riemannian geometry, or Lie groups, as it is intended for students of topology itself, rather than those who wish to apply it to study analysis or physics on manifolds. It is certainly not geared toward physicists.

With one major type of exception, there aren't that many typos or other errors, but that persistent problem is that frequently the wrong letter is used for a variable, such as f is written for g, or f for F, or U for V, or 1 for 2, etc. Sometimes words or subscripts are omitted, too, and in a few places a cross-reference is given to the wrong theorem number. The proof of Theorem 9.2.1 is a bit sloppy, though, with 3 mathematical typos, an unproven assumption, an unnecessary step, an unmentioned restriction, a couple of paragraphs in reversed order, and a reference to a theorem (8.1.9) that wasn't established in enough generality to be applied here, but no other proof is remotely like this.

There are sometimes historical remarks at the ends of sections that contain references to significant extensions. The chapters also begin with relevant quotes from mathematicians, including Whitehead's remark, '"Transversal" is a noun; the adjective is "transverse."' (Someone should tell Lang.) Most sections end with exercises, with many of them being rather challenging. In fact, this book probably has more exercises than any of the other works I cited. Many important results are contained within the exercises, too, although these are not cited elsewhere in the book.

Differential topology
This is an introductory and coincise text in differential topology. While for beginner or advanced students, this text should probably be accompanied by more descriptive texts to developper this matter.

Good Book
I originally bought this book because I wanted to see a more modern treatment of Morse theory, as contained in Milnor's book. Hirsch provides just that, and much more. It's really readable, and doesn't require too much differential geometry to understand.

My one comeplaint with this book, and the reason it didn't get 5 stars, is that Hirsch uses a bit too much functional analysis in his book. That's just my taste, andI could be wrong, but I'm not too keen on learning what a "jet" is in the mathematical sense. I don't even really like to fly.

The book is very nice in appearence. I'm sure it looks very impressive on my shelf. Many women have commented on it's nice burnt orange color, though it does not match my older springer GTM books.

Excellent Foundational Text, but Stops Short
Hirsch has assembled a very fine text which is suitable for a second year graduate mathematics course in differentiable manifolds. The development and presentation of the material is quite accessible.

The prerequisites or co-requisites for this book are a solid background in general topology. I highly recommend Munkres' Topology (2nd Edition). You'll also need a good understanding of analysis as you might find in Rudin's Principles of Mathematical Analysis. It's helpful to know the basics about vector spaces and their duals, and I recommend Hungerford's Algebra for this. Finally, for the material on intersection numbers and Euler characteristic, you might like to know something about homology and cohomology. A good basic reference for this is Munkres' Elements of Algebraic Topology.

The author begins his study with an introduction of differential manifolds and maps in Chapter 1. Hirsch introduces the tangent space rather intuitively, but his formal definition in terms of equivalences class of triples is not intuitive. This may also seem strange to readers who think of a tangent vector as a first order differential operator. The highlight of the first chapter is a very nice, geometrically motivated prove that smooth compact n-manifolds can be smoothly embedded in 2n+1 dimensional Euclidean space.

The main aim of Chapter 2 is to study when a smooth map of manifolds can be approximated by an embedding. This is technical, but crucial material needed for the important study of transversality. To achieve this, the author carefully studies the topology of the spaces of maps between differential manifolds. In order to show that (weakly) closed subspace of maps of manifolds is a Baire space, the concept of an r-jet and the r-jet space are introduced.

Chapter 3 is a short chapter which focuses on the all-important concept of transversality and general position. The key to showing that differential maps can be approximated by transverse maps is the Morse-Sard Theorem. The author gives a very nice treatment of this and is carefully only to introduce the simple notion of a set of measure zero and not get side-tracked by a discourse of Lebesgue measure theory.

Up next, the notions of tubular neighborhoods and collars are studied by introducing the general concept of a vector bundle in Chapter 4. Basic properties and construction techniques are introduced before covering orientability. The tubular neighborhood is defined and shown to exist. Basic isotopy properties of the tubular neighborhood are studied. The chapter concludes with a study of collars on manifold boundaries.

Chapter 5 - 8 are really tantalizing chapters, especially for readers of Rourke and Sanderson's Introduction To Piecewise Linear Topology. Chapter 5 covers some basic algebraic topology concepts such as the degree of a map and intersection numbers. Chapter 6 covers Morse theory and attaching handles. Chapter 7 introduces the notion of a cobordism and Chapter 8 studies the isotopy extension theorem.

Equipped with the basic ingredients of the last 4 chapters (plus the Whitney Trick), the reader is now fully equipped to understand Smale's proof of the H-Cobordism Theorem and the proof of the Poincare Conjecture for homotopy spheres of dimensions > 5. This material could easily serve as a springboard into the modern research literature.

Unfortunately, Hirsch chose not to pursue the H-Cobordism Theorem and instead gives a handle-theoretic proof of the classification of compact surfaces. This seems like a waste of all of the beautiful theory that has been developed. It is definitely a re-hash if the reader has gone through Munkres' Topology (2nd Edition) text and studied the very accessible basic proof of surface classification contained there.

Each section concludes with a nice set of exercises which often continues the thread of discussion. Exercises have difficulty levels marked by a "*". Two-star problems are important results that are usually published in the research literature. Three-star problems are open, unsolved problems at the time of printing (6th printing, 1997).

Nice introduction to differential topology
This book introduces the basic concepts in differential topology, a fieldthat has taken on particular importance in medical imaging, game theory,and network optimization. Although written for mathematicians, andtherefore somewhat formal, a good course in multivariable calculus shouldprepare the reader for this book. The most difficult chapter is probablyChapter 2, where Hirsch studies manifolds by means of function spaces andjets. He does do a good job in this chapter thoughof explaining theorigin and need for partitions of unity and gives examples. He also givesthe reader good insight into why analytic maps are more difficult to handlethan the C-r case, and wets the readers appetite for further reading on theanalytic case. The important notion of transversality is discussed inChapter 3, which would be good reading for one interested in applicationsof differential topology to dynamical systems. A more detailed discussionof vector bundles would have been helpful in Chapter 4, which discussesthese important objects and the idea of a tubular neighborhood. Sringtheorists or those learning the mathematics should get a lot out of Chapter5, wherein intersection theory in differential topology is discussed. Themost important chapter of the book is Chapter 6, which discusses Morsetheory. The applications of Morse theory are immense, and cover not onlymathematics, but physics via quantum field theory and string theory,economics, and even computer graphics. A short chapter on cobordismfollows, which is very nicely written, but a few more words would have beennice on this topic. After discussing isotopy in Chapter 8, Hirsch gives agood proof of the classification for surfaces in the last chapter of thebook. A nice book to have for reference if one is interested in thesubject for its own sake or for its many applications. It should prepareone for further advanced reading in differential topology, such as the workof Freedman and Smale on the Poincare conjecture in dimesions 4 and above.Those interested in applications of differential topology will be amplyprepared to apply these results to the relevant areas, which are many. ... Read more

Product Description
This book presents a geometric introduction to the homology of topological spaces and the cohomology of smooth manifolds. The author introduces a new class of stratified spaces, so-called stratifolds. He derives basic concepts from differential topology such as Sard's theorem, partitions of unity and transversality. Based on this, homology groups are constructed in the framework of stratifolds and the homology axioms are proved. This implies that for nice spaces these homology groups agree with ordinary singular homology. Besides the standard computations of homology groups using the axioms, straightforward constructions of important homology classes are given. The author also defines stratifold cohomology groups following an idea of Quillen. Again, certain important cohomology classes occur very naturally in this description, for example, the characteristic classes which are constructed in the book and applied later on. One of the most fundamental results, Poincare duality, is almost a triviality in this approach. Some fundamental invariants, such as the Euler characteristic and the signature, are derived from (co)homology groups. These invariants play a significant role in some of the most spectacular results in differential topology. In particular, the author proves a special case of Hirzebruch's signature theorem and presents as a highlight Milnor's exotic 7-spheres. This book is based on courses the author taught in Mainz and Heidelberg. Readers should be familiar with the basic notions of point-set topology and differential topology. The book can be used for a combined introduction to differential and algebraic topology, as well as for a quick presentation of (co)homology in a course about differential geometry. ... Read more

Product Description
Thorough coverage is given to the fundamental conceptsof topology, axiomatic set theory, mappings, cardinal numbers, ordinalnumbers, metric spaces, topological spaces, separation axioms,Cartesian products, the elements of homotopy theory, and other topics.A comprehensive study aid for the graduate student and beyond. ... Read more

Customer Reviews (2)

A good resource for Topology and Analysis study!
This book is especially helpful for studying subjects such as countable & uncountable sets, cardinal & ordinal numbers, metric spaces, and compactness, connectedness, and completeness of topological spaces. Keep in mind that this book will cover the most common type of questions that you will find in a Topology text. If you are using Munkres or Mendelson with your course, the Topology Problem Solver will not answer every question in those texts, but it will familiarize you with the concepts you will need to understand. Plus it gives you a good style to model your assignment proofs. I recommend using three books in a introductory Topology course: your own class text (preferably Munkrers or Mendelson), the Topology Problem Solver by REA and Shaum's General Topology Outline by Lipschutz. With all these resources, and some moderate study, you should be ahead of most of the other students in your class.

Don't waste your money
This is one of the poorest books on topology have have ever had the misfortune to purchase. There are several problems with it.

1. The typesetting is horrible 2. The notation is poor 3. The choice of material is very questionable (a good portion of the material has nothing whatsoever to do with topology, but rather with set theory and analysis). It goes without saying that there is very little focus on algebraic topology. 4. The "problems" mainly consist of "Prove this..", where "this turns out to be a standard theorem proved in most any text or "Describe that..", where "that" turns out to be a standard definition defined in most any text.

A "real" topology problem solver would include solutions to the types of problems found in Munkres/Lee/Massey. ... Read more

Customer Reviews (1)

Efficient.
This is an oustanding book on introductory topology (point set topology). The book is short and the author does not get into basic algebraic topology as Munkres, for example, does. However, it is solid and complete and the proofs presented by Dixmier are surprisingly optimised: very concise but always clear. Not only is this convenient while studying from it, but the style is also a model on how one should write one's proofs.

For those able to read mathematics in French, I'd strongly recommend the original version, if available. ... Read more

Product Description
Topology of Surfaces, Knots, and Manifolds offers an intuition-based and example-driven approach to the basic ideas and problems involving manifolds, particularly one- and two-dimensional manifolds. A blend of examples and exercises leads the reader to anticipate general definitions and theorems concerning curves, surfaces, knots, and links--the objects of interest in the appealing set of mathematical ideas known as "rubber sheet geometry." The result is a book that provides solid coverage of the mathematics underlying these topics. ... Read more

Customer Reviews (3)

Fun, but not very substantive.
This is a relatively fun romp through some very interesting concepts, but it lacks rigor.The book could have been much stronger if the author had simply developed some of the basic concepts (compactness, connectedness, homeomorphisms, homotopy, etc) rather than do a little hand-waving around a nice illustration.As it stands, this book is only 140 pages long, and does not develop any of its topics (manifolds, surfaces, graphs, knots) adequately.This book is far too weak to serve as a good text.Kinsey's TOPOLOGY OF SURFACES is much stronger, and costs less.Or look as something like Gamelin's INTRO TO TOPOLOGY.Or even Schaum's outline GENERAL TOPOLOGY, which deals with the basics, but is highly readable and rigorous.

Very Misleading Title, Quite Thin, No Rigor, and Overpriced
This book is subtitled "A First Undergraduate Course" but is certainly below undergraduate level.A high school student could easily follow this--which might be a good thing in certain cases--but the rigor is lacking.In fact, there is barely a hint of any rigor whatsoever.It is mostly intuitive arguments and the author often says things like "but we won't bother worrying about mathematical technicalities".It does get you to be able to visualize certain things well, but the visualization techniques can be found in other books also.The book is very thin and a quick read--hardly worth the money they are trying to get for it.If you're really at the undergraduate level and want to learn some topology, try something like Mendelson's "Introduction to Topology" by Dover or one of the excellent topology books in the series "Undergraduate Texts in Mathematics" by Springer.Munkres is also a classic.If you're not an undergraduate in a math related field and just want to know about the ideas behind topology or perhaps see some visualization techniques, try something like "The Shape of Space" by Weeks.Overall I was very disappointed with this text.If you could purchase this book for under $20 it might be worth it, but even then I think the other books I quoted are better in both price and substance.

No math library is complete without this book
This book presents the topology of surfaces, manifolds and knots in a manner that is reachable for undergraduate students with only a knowledge of calculus.Some linear algebra might be helpful.The text is written in a style that is easy to follow and there are superfluous examples.The exercises in the text are well thought out and are not extremely difficult.The exercises complement the text very well.The text makes clear a lot of difficult concepts such as isotopic surfaces as opposed to homeomorphic surfaces.I particularly enjoyed the manner in which the topology of knots was explained.After reading this text, the reader should be able to better visualize the projective plain and even the Klein bottle as it exists in 4-dimensional space.I have not read a text on topology that I enjoyed reading as much since Munkres.This text is a must have for any topologist. ... Read more

Product Description
This book is written as a textbook on algebraic topology. The first part covers the material for two introductory courses about homotopy and homology. The second part presents more advanced applications and concepts (duality, characteristic classes, homotopy groups of spheres, bordism). The author recommends starting an introductory course with homotopy theory. For this purpose, classical results are presented with new elementary proofs. Alternatively, one could start more traditionally with singular and axiomatic homology. Additional chapters are devoted to the geometry of manifolds, cell complexes and fibre bundles. A special feature is the rich supply of nearly 500 exercises and problems. Several sections include topics which have not appeared before in textbooks as well as simplified proofs for some important results. Prerequisites are standard point set topology (as recalled in the first chapter), elementary algebraic notions (modules, tensor product), and some terminology from category theory. The aim of the book is to introduce advanced undergraduate and graduate (master's) students to basic tools, concepts and results of algebraic topology. Sufficient background material from geometry and algebra is included. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. ... Read more

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Discover a unique and modern treatment of topology employing a cross-disciplinary approach

Implemented recently to understand diverse topics, such as cell biology, superconductors, and robot motion, topology has been transformed from a theoretical field that highlights mathematical theory to a subject that plays a growing role in nearly all fields of scientific investigation. Moving from the concrete to the abstract, Topology and Its Applications displays both the beauty and utility of topology, first presenting the essentials of topology followed by its emerging role within the new frontiers in research.

Filling a gap between the teaching of topology and its modern uses in real-world phenomena, Topology and Its Applications is organized around the mathematical theory of topology, a framework of rigorous theorems, and clear, elegant proofs.

This book is the first of its kind to present applications in computer graphics, economics, dynamical systems, condensed matter physics, biology, robotics, chemistry, cosmology, material science, computational topology, and population modeling, as well as other areas of science and engineering. Many of these applications are presented in optional sections, allowing an instructor to customize the presentation.

The author presents a diversity of topological areas, including point-set topology, geometric topology, differential topology, and algebraic/combinatorial topology. Topics within these areas include:

• Open sets
• Compactness
• Homotopy
• Surface classification
• Index theory on surfaces
• Manifolds and complexes
• Topological groups
• The fundamental group and homology

Special "core intuition" segments throughout the book briefly explain the basic intuition essential to understanding several topics. A generous number of figures and examples, many of which come from applications such as liquid crystals, space probe data, and computer graphics, are all available from the publisher's Web site. ... Read more

Customer Reviews (1)

Needs corrections?
A student came by my office yesterday with a question about an exercise from this book that asked the reader to show that the product topology and the standard metric topology on R^2 are equivalent.The student thought he had a counterexample (a disc containing part of its boundary), and he was right as far Basener's faulty definition of the product topology was concerned.Basener defines a set to be open in the product topology if and only if its image under each projection is open.

I scoured the Internet for mention of this error and came across the Zentralblatt review (which is omitted from Amazon's list of editorial reviews for this book).That review lists this error and others and states in summary:"The book is absolutely terrible."

That's obviously a strong assertion, and, not having read the book, I'm in no position to confirm or deny it.Still, I felt it was important to post this note to warn potential purchasers/readers of problems with this text, and to encourage the author and publisher to fix things.Neither the publisher website nor the (apparently broken) author website nor a Google search yielded a list of errata.

[8/19/2008 Update:The author's website has been fixed, and it has a list of errata.Furthermore, the author assures me that the error in the definition of the product topology has been fixed in printings after the first.]
... Read more

Product Description
Since it was first published in 1967, Simplicial Objects in Algebraic Topology has been the standard reference for the theory of simplicial sets and their relationship to the homotopy theory of topological spaces.J. Peter May gives a lucid account of the basic homotopy theory of simplicial sets (discrete analogs of topological spaces) which have played a central role in algebraic topology ever since their introduction in the late 1940s.

"Simplicial Objects in Algebraic Topology presents much of the elementary material of algebraic topology from the semi-simplicial viewpoint.It should prove very valuable to anyone wishing to learn semi-simplicial topology.[May] has included detailed proofs, and he has succeeded very well in the task of organizing a large body of previously scattered material."--Mathematical Review ... Read more

Customer Reviews (1)

Fast delivery
The book was in the same conditions as described when I bought it and it came in time. ... Read more

Product Description
Geometry is central to many branches of mathematics and physics, and offers a complete range of views on the universe. This introduction includes many simple explanations and examples. With minimal prerequisites, the book provides a first glimpse of many research topics in modern algebra, geometry and theoretical physics. The book is based on many years' teaching experience, and is thoroughly class-tested. There are copious illustrations, and each chapter ends with exercises. Further teaching material is available via the web, including assignable problem sheets with solutions. ... Read more

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

Customer Reviews (2)

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

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

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

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

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
The amount of algebraic topology a graduate student specializing in topology must learn can be intimidating. Moreover, by their second year of graduate studies, students must make the transition from understanding simple proofs line-by-line to understanding the overall structure of proofs of difficult theorems.To help students make this transition, the material in this book is presented in an increasingly sophisticated manner. It is intended to bridge the gap between algebraic and geometric topology, both by providing the algebraic tools that a geometric topologist needs and by concentrating on those areas of algebraic topology that are geometrically motivated.Prerequisites for using this book include basic set-theoretic topology, the definition of CW-complexes, some knowledge of the fundamental group/covering space theory, and the construction of singular homology. Most of this material is briefly reviewed at the beginning of the book.The topics discussed by the authors include typical material for first- and second-year graduate courses. The core of the exposition consists of chapters on homotopy groups and on spectral sequences. There is also material that would interest students of geometric topology (homology with local coefficients and obstruction theory) and algebraic topology (spectra and generalized homology), as well as preparation for more advanced topics such as algebraic $K$-theory and the s-cobordism theorem.A unique feature of the book is the inclusion, at the end of each chapter, of several projects that require students to present proofs of substantial theorems and to write notes accompanying their explanations. Working on these projects allows students to grapple with the "big picture", teaches them how to give mathematical lectures, and prepares them for participating in research seminars.The book is designed as a textbook for graduate students studying algebraic and geometric topology and homotopy theory. It will also be useful for students from other fields such as differential geometry, algebraic geometry, and homological algebra. The exposition in the text is clear; special cases are presented over complex general statements. ... Read more

Customer Reviews (1)

Godd for the second step.
This book is nicely written to explain "tools" of algebraic topology in a small number of pages. However, this is by *no* means a book for beginners, as it assumes its readers to have coverd a basic course.
For beginners I would reommend Hatcher "Algebraic Topology" or Bredon "Topology and Geometry" instead. ... Read more