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Leading experts present a unique, invaluable introduction tothe study of the geometry and typology of fluid flows. From basicmotions on curves and surfaces to the recent developments in knots andlinks, the reader is gradually led to explore the fascinating world ofgeometric and topological fluid mechanics. Geodesics and chaotic orbits, magnetic knots and vortex links,continual flows and singularities become alive with more than 160figures and examples. In the opening article, H. K. Moffatt sets the pace, proposing eightoutstanding problems for the 21st century. The book goes on to provideconcepts and techniques for tackling these and many other interestingopen problems. ... Read more

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Geometry With An Introduction To Cosmic Topology Is Motivated By Questions That Have Ignited The Imagination Of Stargazers Since Antiquity.What Is The Shape Of The Universe? Does The Universe Have An Edge?Is It Infinitely Big? Dr. Hitchman Aims To Clarify This Fascinating Area Of Mathematics And Focuses On The Mathematical Tools Used To Investigate The Shape Of The Universe. The Text Follows The Erlangen Program, Which Develops Geometry In Terms Of A Space And A Group Of Transformations Of That Space. This Approach To Non-Euclidean Geometry Provides Excellent Material By Which Students Can Learn The More Sophisticated Modes Of Thinking Necessary In Upper-Division Mathematics Courses.This Unique Text Is Organized Into Three Natural Parts:Chapter 1 Introduces The Geometric Perspective Taken In The Text And The Motivation For The Material That Comes From Cosmology.Chapters 2-7 Contain The Core Mathematical Content Of The Text, Developing Hyperbolic Elliptic, And Euclidean Geometry From The Complex Plane And Subgroups Of Mobius Transformations. Other Topics Include The Topology And Geometry Of Surfaces And Dirichlet Domains.Finally, Chapter 8 Explores The Topic Of Cosmic Topology Through The Geometry Learned In The Preceding Chapters. ... Read more

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This text on contact topology is the first comprehensive introduction to the subject, including recent striking applications in geometric and differential topology: Eliashberg's proof of Cerf's theorem via the classification of tight contact structures on the 3-sphere, and the Kronheimer-Mrowka proof of property P for knots via symplectic fillings of contact 3-manifolds. Starting with the basic differential topology of contact manifolds, all aspects of 3-dimensional contact manifolds are treated in this book. One notable feature is a detailed exposition of Eliashberg's classification of overtwisted contact structures. Later chapters also deal with higher-dimensional contact topology where the focus mainly on contact surgery, but other constructions of contact manifolds are described, such as open books or fibre connected sums. ... Read more

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

Customer Reviews (2)

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

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

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

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

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In recent years noncommutative geometry has been a richtopic of research with discoveries leading to an increasing number ofapplications in mathematics and theoretical physics.Very little hasappeared in book form since Alain Connes' work in the early 90s todeal with this subject."Elements of Noncommutative Geometry" fillsan important gap in the literature.

Key features of the work include: * unified and comprehensive presentation of core topics and key research results drawing from several branches of mathematics
* rigorous, well-written, nearly self-contained exposition of noncommutative geometry and some of its useful applications to quantum theory
* excellent exposition of introductory material; main topics covered repeatedly in the text at gradually more demanding levels of difficulty
* many applications to diverse fields: index theory, foliations, number theory, particle physics, and fundamental quantum theory
* rich in proofs, examples and exercises
* comprehensive bibliography and index

This text is an introduction to the language and techniques ofnoncommutative geometry at a level suitable for graduate students, andalso provides sufficient detail to be useful to physicists andmathematicians wishing to enter this rapidly growing field.It mayalso serve as a reference text on several topics that are relevant tononcommutative geometry. ... Read more

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right complement
Pepe and Joe have been involved in NCG research
since the first physical calculations. And
the book happened to be published in a very
adequate time range, so it contains up-to-date
information on all the developments, including
the finding of Connes-Kreimer renormalization
algebra (hopf algebras and butcher groups).

It is a compulsory complement to Connes's book.

right complement
Pepe and Joe have been involved in NCG research
since the first physical calculations. And
the book happened to be published in a very
adequate time range, so it contains up-to-date
information on all the developments, including
the finding of Connes-Kreimer renormalization
algebra (hopf algebras and butcher groups).

It is a compulsory complement to Connes's book. ... Read more

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This elegant book by distinguished mathematician John Milnor, provides a clear and succinct introduction to one of the most important subjects in modern mathematics. Beginning with basic concepts such as diffeomorphisms and smooth manifolds, he goes on to examine tangent spaces, oriented manifolds, and vector fields. Key concepts such as homotopy, the index number of a map, and the Pontryagin construction are discussed. The author presents proofs of Sard's theorem and the Hopf theorem. ... Read more

Customer Reviews (9)

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

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

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

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

Exactly would it should be
I would suggest to use this book as a companion to more serious books on topology. Weighing in at a mere 51 pages, this book accomplishes what it needs to: a brief, succinct introduction to topology mostly based on the work of Brouwer. There is a nice mixture of topics, ranging from Sard's theorem to Poincare-Hopf theorem. The proofs and ideas are not fully rigorous or developed, but that would be quite a bit to expect from such a short exposition.

best math book ever written

Despite the lovely subject matter covered in this book, it more importanty gives one a taste of Mathematics as an intellectual discipline. It in outline shows how a mathematical theory - in this case Differential Topology -is constructed and consquently what mathematicians actually do and think about.
Anyone who would like to appreciate Mathematics as a field of study rather than just learn some math should open this book.

Better still, the prerequisite is only multivariate calculus!I have long thought this book should be the third year of calculus rather than differential equations or complex analysis.

Additionally, for the novice it is the only entry I know of into the mysteries of high dimensional geometry, that amazing almost unbelieveable accomplishment of the human mind.

There is a Star Trek episode in which a blind woman wears a dress of sensors which enable her to know more about her environment than a person can know from seeing. She knows exact distances and dimensions, can detect minute movements, can process the complete spectrum of light. In some sense she sees better. Modern topology and geometry are like that sensor dress for seeing higher dimensions. While we can not visualize the sphere in 5 dimensions, we know more about it from these mathematical theories than a five dimensionally sighted being ever could.

Today, mathematics is often considered to be just a practical tool - like a spread sheet - or a toaster oven. We forget its power to widen our imagination, to frame the unimaginable. This book reminds us of this and shows why Mathematics is the Queen of Sciences.

Compact and useful
This book packs a lot of interesting material into a small volume. E.g., I picked up another book recently that started talking about cobordisms right off the bat; despite my having a couple of shelves full of well-known Dover, Springer, Cambridge UP etc. books on topology, differential geometry, mathematical physics, etc., Milnor's tiny book was the only one I found that could help me understand what cobordisms are right away. The book also uses many illustrations to help understanding.

I demote this to 4 stars only because Princeton UP's price is a bit high; many years ago I was lucky enough to find a used copy of the old U. Virginia edition, and paid much less. ... Read more

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This book is a well-informed and detailed analysis of the problems and development of algebraic topology, from Poincaré and Brouwer to Serre, Adams, and Thom. The author has examined each significant paper along this route and describes the steps and strategy of its proofs and its relation to other work. Previously, the history of the many technical developments of 20th-century mathematics had seemed to present insuperable obstacles to scholarship. This book demonstrates in the case of topology how these obstacles can be overcome, with enlightening results.... Within its chosen boundaries the coverage of this book is superb. Read it!

—MathSciNet

Customer Reviews (1)

More than a mere "history".
This book painstakingly describes and explains algebraic topology in the chronological order of its development. I quite agree with Glen Bredon's remark in his "Geometry and Topology" that goes like "this is more than a history and should be in the bookshelf of every student of topology"(not word-for-word, as the citation is done offhand). ... Read more

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This text examines topological methods in algebraic geometry. ... Read more

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This book introduces the important ideas of algebraic topology emphasizing the relation of these ideas with other areas of mathematics. Rather than choosing one point of view of modern topology (homotropy theory, axiomatic homology, or differential topology, say) the author concentrates on concrete problems in spaces with a few dimensions, introducing only as much algebraic machinery as necessary for the problems encountered. This makes it possible to see a wider variety of important features in the subject than is common in introductory texts; it is also in harmony with the historical development of the subject. The book is aimed at students who do not necessarily intend on specializing in algebraic topology.

The first part of the book emphasizes relations with calculus and uses these ideas to prove the Jordan curve theorem. The study of fundamental groups and covering spaces emphasizes group actions. A final section gives a taste of the generalization to higher dimensions. ... Read more

Customer Reviews (3)

A book of ideas
This book is an introduction to algebraic topology that is written by a master expositor. Many books on algebraic topology are written much too formally, and this makes the subject difficult to learn for students or maybe physicists who need insight, and not just functorial constructions, in order to learn or apply the subject. Anyone learning mathematics, and especially algebraic topology, must of course be expected to put careful thought into the task of learning. However, it does help to have diagrams, pictures, and a certain degree of handwaving to more greatly appreciate this subject.

As a warm-up in Part 1, the author gives an overview of calculus in the plane, with the intent of eventually defining the local degree of a mapping from an open set in the plane to another. This is done in the second part of the book, where winding numbers are defined, and the important concept of homotopy is introduced. These concepts are shown to give the fundamental theorem of algebra and invariance of dimension for open sets in the plane. The delightful Ham-Sandwich theorem is discussed along with a proof of the Lusternik-Schnirelman-Borsuk theorem. I would like to see a constructive proof of this theorem, but I do not know of one.

Part 3 is the tour de force of algebraic topology, for it covers the concepts of cohomology and homology. The author pursues a non-traditional approach to these ideas, since he introduces cohomology first, via the De Rham cohomology groups, and these are used to proved the Jordan curve theorem. Homology is then effectively introduced via chains, which is a much better approach than to hit the reader with a HOM functor.Part 4 discusses vector fields and the discussion reads more like a textbook in differential topology with the emphasis on critical points, Hessians, and vector fields on spheres. This leads naturally to a proof of the Euler characteristic.

The Mayer-Vietoris theory follows in Part 5, for homology first and then for cohomology.

The fundamental group finally makes its appearance in Part 6 and 7, and related to the first homology group and covering spaces. The author motivates nicely the Van Kampen theorem. A most interesting discussion is in part 8, which introduces Cech cohomology. The author's treatment is the best I have seen in the literature at this level. This is followed by an elementary overview of orientation using Cech cocycles.

All of the constructions done so far in the plane are generalized to surfaces in Part 9. Compact oriented surfaces are classified and the second de Rham cohomology is defined, which allows the proof of the full Mayer-Vietoris theorem.

The most important part of the book is Part 10, which deals with Riemann surfaces. The author's treatment here is more advanced than the rest of the book, but it is still a very readable discussion. Algebraic curves are introduced as well as a short discussion of elliptic and hyperelliptic curves.

The level of abstraction increases greatly in the last part of the book, where the results are extended to higher dimensions. Homological algebra and its ubiquitous diagram chasing are finally brought in, but the treatment is still at a very understandable level.

For examples of the author's pedagogical ability, I recommend his book Toric Varieties, and his masterpiece Intersection Theory.

This is one of the great algebraic topology books!
This is a book for people who want to think about topology, not just learna lot of fancy definitions and then mechanically compute things. Fulton hasput the essence of Algebraic Topology into this book, much in the way MikeArtin has done with his "Algebra". In my opinion, he should winsome sort of expository award for it.

Probably better as a 2nd (or 3rd) course rather than 1st
Most mathematicians, I suspect, can relate to the "colloquium experience": the first minutes of a lecture go easily, followed by twenty or thirty of real edification, concluded by ten to fifteen of feeling lost.I regret to say that this was pretty much my experience with the book.Fulton writes with unusual enthusiasm and the first two- thirds of the book is a joy to read, even while it is real work.I imagine that he must be a remarkable teacher in person.He has some threads such as winding numbers and the Mayer-Vietoris Sequence that continue throughout the book, bringing unity to a wide selection of topics.There are a number of applications of the subject to other areas, such as complex analysis (Riemann surfaces) and algebraic geometry (the Riemann-Roch Theorem), to name only two.There are particularly interesting illustrations of the Brouwer Fixed Point Theorem and related results.Unfortunately, there are two rather major reservations I have about the book.The first, already alluded to, is that it seemed to me to become precipitously difficult towards the end.The second is that this book would be excellent for a second or perhaps third course in the subject rather than a first.While the topics he covers are interesting in their own right, I still favor a more "standard" approach covering simplicial complexes, homology, CW complexes, and homotopy theory with higher homotopy groups, such as in the books by Maunder, Munkres, or Rotman (the last two of which I recommend unreservedly).It is true that Fulton has some coverage these topics, and a particularly extensive discussion of group actions and G-spaces, but he presupposes a background or ability that the novice to algebraic topology is unlikely to have.I would like to recommend this book, as I found it very edifying, but it seems better suited for one with some prior acquaintance to the subject. ... Read more

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Customer Reviews (9)

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

Have at it, if you like the whirlwind!

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

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

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

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

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

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

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

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

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This book is an introduction to the theory of toposes, as first developed by Grothendieck and later developed by Lawvere and Tierney. Beginning with several illustrative examples, the book explains the underlying ideas of topology and sheaf theory as well as the general theory of elementary toposes and geometric morphisms and their relation to logic. This is the first text to address all of these various aspects of topos theory at the graduate student level. ... Read more

Customer Reviews (2)

Excellent
Topos theory now has applications in fields such as music theory, quantum gravity, artificial intelligence, and computer science. It has been viewed by some as being excessively abstract and difficult to learn, and this is certainly true if one attempts to learn it from the research literature. The use of this book to learn topos theory certainly puts this view to rest, as the authors have given the readers an introduction to topos theory that is crystal clear and nicely motivated from an historical point of view. Indeed the prologue to the book gives the reader a deep appreciation of the origins of the subject, and could even serve as an introduction to a class on algebraic geometry.

An understanding of sheaf theory and category theory will definitely help when attempting to learn topos theory, but the book could be read without such a background. Readers who want to read the chapters on logic and geometric morphisms will need a background in mathematical logic and set theory in order to appreciate them. Topos theory has recently been used in research in quantum gravity. A reader interested in understanding how topos theory is used in this research should concentrate on the chapter on properties of elementary topoi, the one on basic categories of topoi, and the chapter on localic topoi.

The authors introduce topos theory as a tool for unifying topology with algebraic geometry and as one for unifying logic and set theory. The latter application is interesting, especially for readers (such as this reviewer), who approach the book from the standpoint of the former. Indeed, the authors discuss a fascinating use of topos theory by Paul Cohen in his proof of the independence of the Continuum Hypothesis in Zermelo-Fraenkel set theory.

The prologue for this book is excellent, and should be read for the many insights and motivations for the subject of topos theory. The elementary category theory needed is then outlined in the next section. A "topos" is essentially a category that allows the construction of pullbacks, products, and so on, with the philosophy being that objects are to be viewed not only as things but as also having maps (functors) between them. In the section on categories of functors, this viewpoint becomes very transparent due to the many examples of categories that are also topoi are discussed. These examples are presented first so as to motivate the general definition of topos later on. Some of these categories are very familiar, such as the category of sets, the category of all representations of a fixed group, presheaves, and sheaves. Of particular interest in this section is the discussion of the propositional calculus, and its representation as a Boolean algebra. Replacing the propositional calculus with the (Heyting) intuitionistic propositional calculus results in a different representation by a Heyting algebra. From the standpoint of ordinary topology, the Heyting algebra is significant in that the algebra of open sets is not Boolean, i.e. the complement (or "negation") of an open set is closed and not open in general Instead it follows the rules of a Heyting algebra. This type of logic appears again when considering the subobjects in the sheaf category, which have a "negation" which belong to a Heyting algebra. Thus topos theory is one that follows more than not the Brouwer intuitionistic philosophy of mathematics. Recently, research in quantum gravity has indicated the need for this approach, and so readers interested in this research will find the needed background in this part of the book.

After a straightforward overview of how sheaf theory fits into the topos-theoretic framework, the authors also discuss the role of the Grothendieck topology in sheaf theory. This involves thinking of an open neighborhood of a point in a space as more than just a monomorphism of that neighborhood into the space (all the open neighborhoods thus furnishing a "covering" of the space). This need was motivated by certain constructions in algebraic geometry and Galois theory, as the authors explain in fair detail. A covering of a space by open sets is replaced by a new covering by maps that are not monomorphisms. Starting with a category that allows pullbacks, an indexed family of maps to an object of this category is considered. If for each object in this category one uses a rule to select a certain set of such families, called the coverings of the object under this rule, then ordinary sheaf theory can be used on these coverings. If one desires to drop the requirement that the category have pullbacks, this can be done by introducing a category that comes with such "covering families." This is the origin of the Grothendieck topologies, wherein the indexed families are replaced by the sieves that they generate. A Grothendieck topology on a category is thus a function that assigns to each object in the category a collection of sieves on the object (this function must have certain properties which are discussed by the authors). Several examples of categories with the Grothendieck topologies are discussed, one of these being a complete Heyting algebra. Another example discussed comes from algebraic topology, via its use of the Zariski topology for algebraic varieties. The discussion of this example is brilliant, and in fact could be viewed as a standalone discussion of algebraic geometry.

When considering the notion of the Grothendieck topology, the authors define the notion of a `site', which is essentially a (small) category along with a Grothendieck topology on the category. They then show how to define sheaves on a site, which then form a category. A `Grothendieck topos' is then a category which is equivalent to the category of sheaves on some site. The authors then show, interestingly, that a complete Heyting algebra can be realized as a subobject lattice in a Grothendieck topos.

Clear explicit descriptions
This book is written in the best Mac Lane style, very clear and very well organized. It also benefits from Moerdijk's extensive work organizing the theory of Grothendieck toposes by elementary means. The reader should havebasic graduate knowledge of algebra and topology. The book is long becauseit gives very explicit descriptions of many advanced topics--you can learna great deal from this book that, before it was published, you could onlylearn by knowing researchers in the field. ... Read more

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This book is intended to serve as a textbook for a course in algebraic topology at the beginning graduate level.The main topics covered are the classification of compact 2-manifolds, the fundamental group, covering spaces, singular homology theory, and singular cohomology theory.These topics are developed systematically, avoiding all unecessary definitions, terminology, and technical machinery.Wherever possible, the geometric motivation behind the various concepts is emphasized.The text consists of material from the first five chapters of the author's earlier book, ALGEBRAIC TOPOLOGY: AN INTRODUCTION (GTM 56), together with almost all of the now out-of-print SINGULAR HOMOLOGY THEORY (GTM 70).The material from the earlier books has been carefully revised, corrected, and brought up to date. ... Read more

Customer Reviews (2)

Good for newbies
Very nice way to start learning Alg topology. I am reading it for a class and it's been quite pleasant.

Excellent text on algebraic topology
The text contains material from the author's earlier two books; Algebraic Topology: An Introduction (GTM 56), and Singular Homology Theory (GTM 70). The book starts with an introductory chapter on 2-manifolds and thencontinues with the fundamental group; which is conceptually easier thanhomology, with which some books on algebraic topology start. The onlyprerequisite for this book is a basic knowledge of general topology; andthe book is easily accessible to anyone studying on his own. In short, Irecommend the book to anyone interested in algebraic topology. ... Read more

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This book presents the classical theorems about simply connected smooth 4-manifolds: intersection forms and homotopy type, oriented and spin bordism, the index theorem, Wall's diffeomorphisms and h-cobordism, and Rohlin's theorem. Most of the proofs are new or are returbishings of post proofs; all are geometric and make us of handlebody theory. There is a new proof of Rohlin's theorem using spin structures. There is an introduction to Casson handles and Freedman's work including a chapter of unpublished proofs on exotic R4's. The reader needs an understanding of smooth manifolds and characteristic classes in low dimensions. The book should be useful to beginning researchers in 4-manifolds. ... Read more

Customer Reviews (1)

Excellent
For those genuinely interested in understanding the proof of the 4-dimensional Poincare conjecture, and for those who need a more geometric, intuitive view of some of the main results in topological 4-manifolds, rather than one based on the heavy machinery of algebraic topology, this book is an excellent beginning. The author endeavors in this book to be as clear as possible, and he does not hesitate to use diagrams to get the point across. Rigor however, is not sacrificed. One of the main goals in the book is to get a more geometric proof of Rohlin's theorem, which states that cobordism ring in 4-dimensions over the special orthogonal group and over the spin group is the integers.

The author starts the book with an overview of handlebody theory, noting that for the case of interest, 4-dimensional toplogical manifolds must be smooth in order for them to be handlebodies. Smooth handlebody decompositions can be described by Morse theory, and one smooth handlebody decomposition can be related to another via an isotopy of attaching maps and creation or annihilation of handle pairs. The author visualizes handlebodies in four dimensions by drawing their attaching maps in the 3-sphere. This results in the use of framed links to model the attaching maps, with examples of the 3-torus, the Poincare homology 3-sphere, and a homotopy 4-sphere, the latter of which is homeomorphic to the 4-sphere and is a double cover of an exotic smooth structure on 4-d real projective space. The author also gives a brief but interesting discussion on why the methods of this chapter are difficult to do in three dimensions.

The theory of intersection forms appears in chapter two, with the author proving first that for a closed, smooth, oriented, 4-d manifold M any element of the second integer homology group is represented by a smoothly imbedded oriented surface. Any two such surfaces can be joined by smooth oriented 3-manifold imbedded in M. The isomorphism between the second homology and cohomology groups (over the integers) modulo torsion is the famous "intersection pairing". The author then proves that two simply-connected, closed, oriented 4-manifolds are homotopy equivalent if and only if their intersection forms are isometric. The proof emphasizes the geometric connection between homotopy type and intersection forms. A brief review of symmetric bilinear forms and characteristic classes is then given, as preparation for the classification results given later in the book.

The author treats classification theorems in chapter three, which he describes as deciding which forms, whether symmetric, integral, or unimodular, can be represented by simply connected closed 4-manifolds. The relation between forms and homotopy type makes this implicitly a classification for the homotopy type of the manifold. Rohlin's theorem was historically the first major result in this problem, but the author delays its proof until chapter eleven. The author briefly discusses the work of Freedman in the topological case, and Donaldson, in the smooth case.

Spin structures are discussed in chapter four and several examples are given. The author also shows how to relate spin structures on the boundary of a manifold to spin structures on the manifold itself, to set up later discussions on cobordism. Chapter five then concentrates on the Lie group spin structure of the 3-torus T3(Lie) and the surface constructed by taking the nine-fold direct sum of complex 2-d projective space and its reverse orientation. The latter is a complex analytic projection, which is a smooth fiber bundle with fiber the two-torus except for a finite number of singular fibers. The author shows in detail how to use this object to obtain a spin manifold with spin boundary T3(Lie).

Chapter six is devoted to showing how to immerse closed, smooth, oriented 4-manifolds in Euclidean 6-d space. This involves the calculation of a characteristic class in the second integral cohomology group. Then as a warm-up to showing that a spin 4-manifold with index zero spin bounds a spin 5-manifold, the author proves in chapter 7 that every orientable 3-manifold is spin, bounds an orientable 4-manifold, and if spin bounds a spin 4-manifold with only 0-handles.

In chapter eight, the author proves that a closed, smooth, connected, and orientable 4-manifold is the boundary of a smooth 5-manifold if the first Pontryagin class is 0. If the 4-manifold is spin, and the first Pontryagin class is 0, then there exists a smooth, spin 5-manifold whose boundary is the 4-manifold, where both manifolds are considered as spin manifolds. Chapter nine proves the Hirzebruch index theorem in dimension 4, and the author shows that the cobordism ring for SO and Spin is the integers. Chapter ten is devoted to a proof of Wall's theorem and the h-cobordism theorem in dimension 4. The geometric proof of Rohlin's theorem promised by the author is finally done in chapter eleven.

Casson handles, so important in the proof of the 4-d Poincare conjecture, are discussed in chapter twelve. The author shows the role of the Whitney trick in dimensions 5 or more, and how its failure in dimension 4 results in the use of Casson handles, which are constructed using the famous "finger moves". He gives an explicit handlebody description of the simplest Casson handle, and then relates it to the Whitehead continuum.

The most fascinating part of the book is chapter thirteen, which outlines briefly Freedman's proof of the 4-dimensional Poincare conjecture. The proof makes use of 4-dimensional handlebody theory and decomposition space theory. Casson handles are decomposed via an imbedding of a Cantor set of Casson handles inside them. The "Big Reimbedding theorem" of Freedman, which points to the existence of an exotic smooth structure on the 3-sphere cross the real line, is quoted but not proved. The book ends with chapter fourteen being a brief discussion of exotic structures, their existence following from the non-smoothness of Casson handles. ... Read more

Product Description
This volume covers the proceedings of an international conference held in Oxford in June 2002. In addition to articles arising from the conference, the book also contains the famous as yet unpublished article by Graeme Segal on the Definition of Conformal Field Theories. It is ideal as a view of the current state of the art and will appeal to established researchers as well as to novice graduate students. ... Read more

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

Customer Reviews (3)

A readable alternative to Hatcher

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

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

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

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

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

Product Description
Designed to be an introduction to some of the basic ideas in the field of algebraic topology. Devoted to the foundations and applications of homology theory. DLC: Homology theory. ... Read more

Customer Reviews (3)

(Co)Homology the Way it Should Be
This was the textbook for the first third of a year-long algebraic topology sequence at Oregon State in 1973-4. We were told by the prof that Vick was a student of Stong and that the book was essentially Stong's course written up with his blessing. It's hard to ask for a better pedigree than that, as Stong was a legend for his teaching (as well as his research).

Although there are some minor quibbles (noted in the 4-star reviews), I still haven't found a better treatment of the key results, nor a more direct path. The proof of Poincare duality in particular is that of Hans Samelson, another legend in the field for both teaching and research.

Checking the references, one finds that this was not the only such example where Vick sought out what was then regarded as the best proof available for beginners. It is also noteworthy that community consensus on which are best has not changed much, if any, since then.

The plethora of typos may be a "feature" of the reprint, since I don't recall that many in the original Acad. Press edition we used, and I still have.

As should be clear, this one is a real keeper.

For more modern/advanced study, continue with Switzer and Brayton Gray. By then the journals should be reasonably accessible.

Has the good and bad
This is a terrific book on homology theory, covering all the standard topics, plus some nice topics that are hard to find in other introductory books. The motivation for theory is presented in both algebraic/categorical and geometric flavors. The structure of the book is mostly solid, getting straight to the point with singular homology instead of wasting time with simplicial homology and its results (a rarity with algebraic topology books). My only complaints are that the book is riddled with typos and chapter 5 (on products in homology and cohomology) is quite messy.

Masterful
This introduction to singular homology combines a strong historical sense with an easy mastery of modern methods. The massive contributions of Poincare and Brouwer are credited, and their geometrical motivations are clear. At the same time the book neither minimizes nor apologizes for modern algebraic machinery, but treats categories and acyclic models and more as natural means to simplify the subject. The book goes through Poincare duality and a good account of the Lefschetz fixed point theorems. It is at once very visual and algebraically slick.The only problem with this approach is that the author seems a bit uncomfortable descending into the nuts and bolts of the longer proofs of two key results (the acyclic model theorem, and the duality theorem). He handles the details unevenly and makes some actual mis-statements. Here the reader needs the experience and confidence to make some corections. ... Read more

Product Description
At the present time, the average undergraduate mathematics major finds mathematics heavily compartmentalized. After Calculus, students take courses in analysis and algebra, and depending on their interest, they take courses in special topics. If the student is exposed to topology, it is usually straightforward point set topology; if the student is exposed to geometry, it is usually classical differential geometry.

These notes are an attempt to break up this compartmentalization, at least in topology-geometry. What the student has learned in algebra and advanced calculus are used to prove some fairly deep results relating geometry, topology and group theory. The material studied includes De Rhams's theorem, the Gauss-Bonnet theorem for surfaces, the functional relation of fundamental group to covering space, and surfaces of constant curvature as homogeneous spaces. ... Read more

Customer Reviews (2)

Nice overview of Point-Set, Algebraic, and Differential Topology
I believe that one of the authors, I. Singer is a Fields Medalist, so this is co-written by one of the masters in the subject.It is one of the few books I'm aware of that covers point-set, algebraic, and differential topology.However, this is not in any way an exhaustive text.It is very spare and to the point.This is a small book packed with information.No exercises are included.It is not particularly advance in any one area of topology, so keep in mind that this is just an overview of all the main areas.However, just because it is not particularly advanced, does not mean it is easy to read for someone completely unfamiliar with the material.I tried reading the early chapters as a first introduction to topology, and didn't know what was going on.Reading of it got better when I read it side-by-side with the books by Munkres and Armstrong.This book is pretty abstract and dense, so it will read slowly.A lot of the finer details in the subjects are found in other texts.

Concise and modern
First, a small carp at Amazon: The book has two authors and since John Thorpe is the junior and by far the less famous, I suspect that he actually did most of the writing. Please give him his due!

This is a very dense book. While this makes for rough sledding for the first timer, it's also an exciting introduction to modern topology and geometry and a good first step for those interested in such things in physics as gauge theories and superstrings. It's worth the effort.

Starting with the basics of set theory,the first couple chapters take the reader through point set topology. The next couple chapters introduce algebraic topology. The rest of the book is about the algebraic topology of differentiable manifolds and a very clean, modern introduction to the classical differential geometry of surfaces. The only caveat is, as Spivak says, "a weird proof of the de Rham theorem" in Chapter 6. I'm torn about this. The proof in Warner's "Foundations of Differentiable Manifold and Lie Groups" is much cleaner and better lends itself to other applications, but involves lots of machinery. The proof in Singer and Thorpe is a lot less elegant, using the lowest level tools possible. This makes the learning curve shorter and may make the theorem more clear, but may also obscure the big picture. Much of the important work in algebraic topology over the next 20 years and theoretical physics up to now is related to this result. Though much of this work was developed by Singer with his collaborater Michael Atiyah, their approach is closer to Warner's than to that in Singer and Thorpe.

For any particular topic in this book, you can find sources that you'll undoubtably find more digestible. This is the only book that brings them all together. It's an audacious effort. ... Read more

Product Description
During the week of May 18-26, 1992, a conference on low-Dimensional Topology was held at the University of Tennessee, Knoxville. The Conference was devoted to a broad spectrum of topics in Low-Dimensional Topology. However, special emphasis was given to hyperbolic and combinatorial structures, minimal surface theory, negatively curbed groups, group actions on R-trees, and gauge theoretic aspects of 3-manifolds. Recent results in these topics are published here. A special attempt was made to make this conference accessible and worthwhile for young researchers in the field. This volume is the most complete and current compilation of research in the field of Low-Dimensional Topology. ... Read more

Product Description
The recent introduction of the Seiberg-Witten invariants of smooth four-manifolds has revolutionized the study of those manifolds. The invariants are gauge-theoretic in nature and are close cousins of the much-studied SU(2)-invariants defined over fifteen years ago by Donaldson. On a practical level, the new invariants have proved to be more powerful and have led to a vast generalization of earlier results. This book is an introduction to the Seiberg-Witten invariants.

The work begins with a review of the classical material on Spin c structures and their associated Dirac operators. Next comes a discussion of the Seiberg-Witten equations, which is set in the context of nonlinear elliptic operators on an appropriate infinite dimensional space of configurations. It is demonstrated that the space of solutions to these equations, called the Seiberg-Witten moduli space, is finite dimensional, and its dimension is then computed. In contrast to the SU(2)-case, the Seiberg-Witten moduli spaces are shown to be compact. The Seiberg-Witten invariant is then essentially the homology class in the space of configurations represented by the Seiberg-Witten moduli space. The last chapter gives a flavor for the applications of these new invariants by computing the invariants for most Kahler surfaces and then deriving some basic toological consequences for these surfaces. ... Read more

Customer Reviews (2)

the first book on Seiberg-Witten gauge theory, but not for beginners
This was the first book published on Seiberg-Witten gauge theory in 1996 (but written in early 1995). As such, while it gives an introduction to the field, it doesn't include any of the multitude of results and extensions discovered since then, so this would hardly be suitable as a reference on the subject that would allow one to understand current research. On the other hand, Prof. Morgan was mainly writing for mathematicians who were familiar with the older Yang-Mills/Donaldson gauge theory, so he doesn't explain the techniques and motivations behind many of the proofs that a beginner in gauge theory would fail to grasp.

There have been 4 books (that I'm aware of) devoted to SW gauge theory - this one, John Moore's Lectures on Seiberg-Witten Invariants, Nicolaescu's Notes on Seiberg-Witten Theory, and Marcolli's Seiberg-Witten Gauge Theory. Morgan and Moore's books came out at about the same time and largely cover the same material, whereas Marcolli and Nicolaescu's works came 4-6 years later and encompass much more of the breadth of the field. But having said that, each book has some advantages over the others and none could be said to be strictly better than any other in all respects. The advantages of this book are that (1) it is relatively concise (126 pg),as compared to Nicolaescu, which rambles, (2) it contains a complete proof of the properties of the moduli space that one needs to define invariants (unlike Moore) and gives much more detail than Marcolli, (3) it proves results in more generality than Moore, and at a higher level, with at times more insight, and also includes some results that Moore omits, and (4) it explains the background material on Clifford algebras, spinor bundles, and Dirac operators much better than Marcolli. The disadvantages are that, as mentioned above, (1) this includes only results from Witten's original 1994 paper (fleshed out and proved rigorously), (2) Morgan is implicitly assuming that the reader has some knowledge of YM gauge theory as can be found in, e.g., Freed and Uhlenbeck's Instantons and Four-Manifolds, and (3) more knowledge of algebraic topology, Sobolev spaces, differential geometry, the index theorem, etc., is assumed than is the case for Nicolaescu or Moore.

For those unfamiliar with Seiberg-Witten gauge theory, or mathematical gauge theory in general, I'll give a brief introduction. Gauge theory (in mathematics) is the study of the spaces of solutions to certain differential equations on sections of bundles over a given manifold that originally came from gauge theory in physics. The space of such solutions modulo automorphisms of the bundles (the group of gauge transformations) is called a moduli space. Gauge theory consists in a set of techniques used to study these moduli spaces and prove that under appropriate conditions on the underlying manifold the moduli spaces have various properties, such as, smoothness, compactness, finite-dimensionality, and orientability. If a moduli space possesses all these nice properties, one can define smooth invariants for the underlying manifold that allow one to study its differential topology. The first equation for which this was done was the Yang-Mills equation, the fundamental equation of particle physics, for which Donaldson won a Fields Medal for his work in the '80s in defining the invariants that bear his name and applying them to the study of the smooth topology of 4-manifolds. The proofs in this field tend to be long and very technical, requiring a broad background in differential geometry, emphasizing in particular principal bundles, nonlinear and functional analysis, particularly Sobolev spaces and elliptic operators, Lie groups and algebras, some complex analysis, index theory, and algebraic topology, including characteristic classes, so it usually takes a couple of years for graduate students to master the material necessary to even begin studying the field. The Seiberg-Witten equations, which were introduced in 1994, had some properties (namely, an abelian gauge group U(1) and compact moduli spaces) that allowed the proofs in Donaldson theory to be derived with far less effort, thus revolutionizing the field.

This book was written immediately after Witten (who also won a Fields Medal, in part due to this work) published his groundbreaking paper that launched the field and led to an explosion in results on smooth and symplectic 4-manifolds in the span of a couple of years. I was just entering graduate school in math at the time (having switched from physics) and Prof. Morgan was my new advisor. There was a lot of excitement in the air and a rush to be the first to publish on this new field, so when he wrote this book, he had more in mind an audience of mathematicians who were already familiar with the old Donaldson theory and wanted to apply the techniques they had learned to the new SW theory. Consequently, the book is careful to develop the Clifford algebra and spinor theory that is the new feature in the SW equations, but doesn't spend much time explaining things such as the rationale behind the method of proof for transversality or the fact that the configuration space is a Banach manifold since those techniques had already been developed for the YM gauge theory.Nowadays any mathematics student learning gauge theory for the first time is likely to start with SW theory, or both simultaneously, so few students fit the profile of the intended readership of this book. I also have to confess that Prof. Morgan was one of the most difficult lecturers to understand that I have ever come across, and he sometimes writes that way, too, as he demands a lot from the reader, as one can see from some of his asides, e.g., on the homotopy type of the quotient space, or his proof of the existence of liftings for Spin-c bundles.

Chapters 2 and 3 form a nice introduction to Clifford algebras, spinors, and Dirac operators. A good reference for more detail (and also the index theorem) is Lawson and Michelson's Spin Geometry. Then the Seiberg-Witten equations are introduced in Chpt. 4 and properties of the equations (gauge invariance and ellipticity on slices), quotient space (smoothness away from reducibles and Hausdorffness, missing in Moore), and moduli space (it's formal dimension) are proved. In Chpt. 5 more background on curvature identities for spin connections is presented and then applied to prove the a priori bounds that are one of the biggest advantages of SW theory over Donaldson theory. This immediately leads to the vanishing theorem and a proof of compactness, without assuming simply connectedness as Moore does. Morgan's boot strapping argument for compactness is needlessly complicated, involving more Sobolev inequalities than is really necessary - Moore's and Nicolaescu's proofs are much more streamlined. In Chpt. 6, perturbations are introduced to the equations allowing the Sard-Smale theorem to be applied to achieve smoothness for "generic" moduli spaces, which are also shown to generically lack reducible solutions (although Morgan appeals to a result of Taubes; see Marcolli for a simpler, self-contained explanation). Compactness is then reestablished for the perturbed moduli spaces (this should've been organized better to avoid having to prove compactness twice), a generic metrics theorem is proven (missing in Moore), and finally the moduli space is shown to be orientable (probably the most boring part of any gauge theoretic proof), giving all the ingredients necessary to define the Seiberg-Witten invariants (it only took 99 pages!), the purpose of the book. The chapter concludes with some more advanced topics (an involution in the theory and a wall-crossing formula) that Moore lacks.

The final chapter calculates the Seiberg-Witten invariants for various Kaehler manifolds. However, these computations are not actually used to derive any results about the topology of smooth 4-manifolds, so the title of the book is misleading, although the application of these results is straightforward (i.e., if 2 manifolds have different SW invariants then they are not diffeomorphic). The final paragraph of the book mentions the work of Taubes in connecting SW invariants to symplectic geometry and Gromov invariants that became the focus of the theory over the next few years (no mention of this in Moore, but not that much on it in Marcolli or Nicolaescu either; Taubes papers, some of which were collected in a book, Seiberg - Witten and Gromov Invariants for Symplectic 4-Manifolds, are the best reference).

There is a preponderance of typos in the book (it was poorly edited), but they are mostly harmless. The bootstrapping in the proof of Lemma 4.5.3 wasn't done correctly, but the reader should easily be able to fix it. The last sentence of Remark 4.5.6 should read "onto a neighborhood of x," not "onto a neighborhood of the orbit through x." Corollary 4.5.7 should say, "The fixed points FORM the tangent space...," not "for the tangent space." In the proof of Lemma 5.3.1 there's a mix up with exponentials; to wit, alpha1 = alpha0 + 2s0 (not sigma0) and near the end of the proof it should read "(det exp(phi))," not "(det phi)," in a couple of equations. None of the other dozens of typos should impede the understandability of the book.

For someone who wants to learn SW theory, Nicolaescu is the most complete book, but since it is poorly organized and written, it is worthwhile to read Moore's and then Morgan's books first (or concurrently) to get a feel for the subject. It seems that the definitive book on the subject has yet to be written.

Fairly good book on the subject
This book is a pretty good introduction to the main results that caused a flurry of excitement in the mathematical community in the mid 1990's. The mathematical constructions involved here are interesting mostly to those in the area of the differential topology of 4-manifolds. The Seiberg-Witten invariants as they are now called, have been widely discussed since then, but mostly now in the context of symplectic geometry. After a brief overview of spin geometry and Clifford algebras, the author discusses the complex spin representation. This sets up the discussion of spin bundles in the next chapter, and, even though it is really not the place for it, the author does not prove that a principal SO(V) bundle lifts to a principal Spin(V) if and only if the second Stiefel-Whitney class is equal to zero. There are many different proofs of this in the literature, but I have not discovered in any of these proofs any real, sound insight as to why this result is true. The chapter continues its very formal treatment with an overview of spin bundles and the Dirac operator. The next chapter then moves immediately to the Seiberg-Witten equations and they are viewed as nonlinear generalizations of elliptic partial differential equations in the sense that the linearization of both the Seiberg-Witten equations and the gauge group action is shown to be an ellipic complex. The next chapter shows that the moduli space of solutions to the Seiberg-Witten equations is compact. This is the most technical of the chapters and requires attentive reading. The Seiberg-Witten invariant for complex spin structures is discussed in the next chapter. Again one must pay close attention to the details of the arguments. The actual calculation of a Seiberg-Witten invariant is performed in the context of Kahler manifolds in the last chapter of the book. This sets up the reader nicely for the current work on symplectic manifolds. The book will be of interest to mathematicians wanting an understanding of this area of four-dimensional topology and to high-energy physicists who are interested in the low energy behavior and duality in SU(2) supersymmetric gauge theories. The constructions of Seiberg and Witten in quantum field theory are what led to the invariants outlined in this book. All in all a fascinating area of mathematics and its consequences are sill being worked out with diligence. ... Read more