Product Description

The basics of differentiable manifolds, global calculus, differential geometry, and related topics constitute a core of information essential for the first or second year graduate student preparing for advanced courses and seminars in differential topology and geometry. Differentiable Manifolds is a text designed to cover this material in a careful and sufficiently detailed manner, presupposing only a good foundation in general topology, calculus, and modern algebra. This second edition contains a significant amount of new material, which, in addition to classroom use, will make it a useful reference text. Topics that can be omitted safely in a first course are clearly marked, making this edition easier to use for such a course, as well as for private study by non-specialists.

Customer Reviews (1)

Couldn't be better! I'll bow down on this one !!!!!
It is SO HARD to create a precise yet not too-many-pages book on a lot of these higher math subjects. This guy really knows how to pick his topics from within the field which he is surveying. The font is beautiful as well. I wish that I had had him as a teacher when I learned this material -- I would have learned it SO much better the first time. As it was, it took about 3 times for it all to sink in. He should get a gold metal from someone for this! ... Read more

Product Description
"The book gives an excellent overview of 4-manifolds, with many figures and historical notes. Graduate students, nonexperts, and experts alike will enjoy browsing through it."

-- Robion C. Kirby, University of California Berkeley

This is a panorama of the topology of simply-connected smooth manifolds of dimension four.

Dimension four is unlike any other dimension; it is large enough to have room for wild things to happen, but too small to have room to undo them. For example, only manifolds of dimension four can exhibit infinitely many distinct smooth structures. Indeed, their topology remains the least understood today.

The first part of the book puts things in context with a survey of higher dimensions and of topological 4-manifolds. The second part investigates the main invariant of a 4-manifold--the intersection form--and its interaction with the topology of the manifold. The third part reviews complex surfaces as an important source of examples. The fourth and final part of the book presents gauge theory. This differential-geometric method has brought to light the unwieldy nature of smooth 4-manifolds; and although the method brings new insights, it has raised more questions than answers.

The structure of the book is modular and organized into a main track of approximately 200 pages, which are augmented with copious notes at the end of each chapter, presenting many extra details, proofs, and developments. To help the reader, the text is peppered with over 250 illustrations and has an extensive index. ... Read more

Customer Reviews (1)

best simultaneous introduction to both topological and smooth 4-manifolds
Scorpan's "Wild World of 4-Manifolds" is to my knowledge a unique book in that it covers extensively just about everything that one would need to know to study 4-manifold topology, both the topological (i.e., continuous) and smooth theories, which involve widely different techniques. However, it doesn't actually contain all the information one needs but rather is only a survey, that informs the reader of the subject matter, sketches the ideas behind the major proofs, and points to the relevant literature for the details. This should be read by graduate students, with the standard background in algebraic and differential topology that one acquires in 1st-year graduate courses, who would like to enter the vast field of 4-manifold topology but don't know where to begin. The book emphasizes the most important concepts, with much effort to illustrate them intuitively before presenting formalism, but at the same time more than half the book consists of "notes" that present the more technical aspects.

Dimension 4 is special in that it is large enough to have interesting phenomena but not large enough that they can be undone (by, e.g., the Whitney trick). Thus that is the only dimension where, e.g., the smooth Poincare conjecture is still open or there exist (uncountably infinite) different smooth structures on Euclidean 4-space. For the study of topological 4-manifolds, first techniques that were successful in the study of higher dimensional manifolds were applied, but these were not sufficient, until the work of Freedman in the early 1980s, who classified (simply connected) top 4-manifolds completely using Casson handles (and capped gropes). For smooth 4-manifolds, aside from Rokhlin's theorem, no real progress was made until the gauge-theoretic approach of Donaldson in the mid-'80s and continuing up the present with the study of the Seiberg-Witten equations.Scorpan's monograph is rare in that it attempts to treat both kinds of manifolds, whereas almost every other book concentrates on either one or the other. For the topological side in particular this book is invaluable to students, as the standard references, Freedman & Quinn's Topology of 4-Manifolds or Kirby's The Topology of 4-Manifolds (but see also A la Recherche de la Topologie Perdue, to which the author is heavily indebted), are difficult to follow for the uninitiated.

The book begins with higher-dimensional manifolds, illustrating the handle surgery techniques that were successful in proving the h-cobordism theorem, the central technical theorem in proving other results (cf. Milnor's Lectures on the h-Cobordism Theorem or Kosinski's Differential Manifolds for the h-cobordism theorem; Gompf & Stipcisz's 4-Manifolds and Kirby Calculus or Kirby for handlebodies). The difficulties in applying these techniques to 4-manifolds are discussed, and Freedman's success in overcoming them is explained very clearly. The book is probably worth reading for the explanation of the proof of Freedman's theorem alone.

The book next explains how the intersection form is central to the homotopic, topological, and smooth study of 4-manifolds, including the classification of such forms and 2 nice proofs of Whitehead's theorem, and adding notes on such important topics as spin structures (an overarching theme, the intuitive description being perhaps the best feature of the book), Cech cohomology, Stiefel-Whitney classes, obstruction theory, classifying spaces, cobordism groups, the Pontrjagin-Thom construction, the Kirby-Siebenmann obstruction to smoothing, plumbings, etc. The basics of almost all of these topics are explained in a way that is readily comprehensible.

The author takes an excursion through complex surfaces (complex manifolds of 4 real dimensions), giving just enough of the theory to provide examples that will be used throughout the latter parts of the book on gauge theory. Here is the weakest part of the book, as it largely consists of definitions and a list of theorems, without explaining enough to help the reader actually absorb any of it. To his credit, though, he recognizes this deficiency, with self-deprecating humor, but this doesn't compensate for the shortfall in the exposition.

The last part of the book treats the smooth theory, which is built around gauge theory. Here again he does an excellent job of explaining the concepts behind the proofs, but unlike the case for topological 4-manifolds, for this part at least there exist alternatives (e.g., Nicolaescu's Notes on Seiberg-Witten Theory, Morgan's The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds, or Moore's Lectures on Seiberg-Witten Invariants) that do introduce the subject to novices. His approach to spin-C structures and the Seiberg-Witten equations using quarternions, while it does have some advantages (in, e.g., representing the structure groups of bundles via their transition functions), misses some advantages that come from using the more standard approach via Clifford algebras. The way in which he brushes off the all the Sobolev space techniques that are so important for actually proving anything in gauge theory is also unsatisfactory.

His applications of gauge theory near the end of the book, however, are excellent, and give a much fuller and up-to-date illustration of the applications of the theory than any of the other (earlier) works that I cited above. In particular, the connections with symplectic manifolds and Gromov invariants, the results on the minimum genus of embedded surfaces (including an outstanding explanation of the Arf invariant), the Fintushel-Stern theorem relating gauge theory to knot invariants are not found in any other book remotely this elementary.

Don't be fooled by the title, which is more indicative of the author's (ever present) sense of humor than any lack of mathematical depth. Other than the aforementioned content, some other nice features of the book include many casual remarks that serve to clear up confusion that beginners often have, and the numerous cross references throughout the book to specific pages in earlier and later chapters; I don't think I've ever seen so many in any book, and it really helps a lot. The bibliographic notes at the end of chapters are also outstanding - virtually every work at all related to 4-manifold topology is discussed. If one were to consult those books and papers after (and while) reading this book, one really would be well-prepared to do research in this field.

The book does have some major problems, though, to the point that I considered giving it only 4 stars at times. There's a large amount of repetition (e.g., the exact same figure, Fig. 4.19, appears about 6 times, even on consecutive pages), partially by design since the notes that accompany the book can be read independently of the main text, but still, it can be irritating. There's also a wide range in the level of the book - in some places very basic topological definitions (e.g., homotopy groups) are given that the reader should really know already, while in other, sometimes earlier, places such material will be assumed without comment. It does seem that the book could've been organized a lot better, and probably some of it should've been condensed. The fact that almost all of the proofs are incomplete is also a drawback, even though one is warned about this at the beginning and throughout the text, since the author doesn't always indicate when he is sweeping technical details under the rug - you should operate under the assumption that none of these proofs should be accepted as complete on its own, but rather should use them as a guide in helping to understand the proofs of the primary sources.

My biggest complaint about the book is the many errors therein. Most of them are of the mathematical typo variety - mistakes in signs, directions of inequalities, indices, etc. - but they seem to result from the fact that the author is often just citing results and intermediate steps of proofs rather than actually working them out, so even though an equation will have an error, it's not showing up in later equations because he isn't really using it. Examples of persistent mistakes include writing homology classes as connected sums (3[CP^1] is not the same thing as #3CP^1) and factors of 2 and i in the quarterionic equations. Many of the proofs are overly long and inefficient, too, a good example of this being that of Feedman and Kirby's theorem on pp. 512-14, which actually contains an unrecognized and much shorter proof within itself and moreover contains references to versions of Wu's formula not covered in the book, and also on pp. 428-433, which contains several errors in the variables and glosses over an important point (cf. Algebraic and Geometric Topology (Proceedings of Symposia in Pure Mathematics Volume XXXII, Part 2) from which the proof was taken) despite being about 4 times longer than the original reference and much more fully explained. The worst case is in the only proof in the entire book that is claimed to be original, that on pp. 186-188 purporting to demonstrate the equivalence between 2 definitions of spin structures. In it he makes several embarrassing factual misstatements (bottom of p. 186 and top of p. 188), misuses the word "trivial," has almost a whole page (187) of unnecessary material, makes 2 important assumptions that contain the essence of what he is trying to prove (top of p. 188), and gives a proof of half the result (bottom of p. 188) that if replicated could've correctly proved the whole thing much simpler.

Despite the fact that the exposition should've been streamlined, this still is an essential book for those seeking to enter the field. I wish something like it had been available when I was a graduate student. ... Read more

Product Description
The second edition of this text has sold over 6,000 copies since publication in 1986 and this revision will make it even more useful. This is the only book available that is approachable by "beginners" in this subject. It has become an essential introduction to the subject for mathematics students, engineers, physicists, and economists who need to learn how to apply these vital methods. It is also the only book that thoroughly reviews certain areas of advanced calculus that are necessary to understand the subject.



Line and surface integrals
Divergence and curl of vector fields ... Read more

Customer Reviews (5)

Great book
Great introductory differential geometry text!I used this book to help me pass my qualifying exam.Yay Boothby!

This is a book for REAL mathematicians
This book is an wonderful introduction to Differential Geometry for the serious student of mathematics. However, it is not aimed at engineers, physicists or even applied mathaticians.
The author assumes the reader has an extensive knowledge of abstract algebra and at least one course in analysis. Likewise, he has chosen to emphasis applications of the subject to Lie Groups, homotopy theory, and group actions, rather than the physical applications that applied mathematicians are looking for. But, for the student of pure mathematics, this text is a great starting point into the rich world of differential geometry.
Also, while this book is an introduction and requires no previous knowledge of the subject, it covers enough ground to be followed up by such topics as the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet's Theorem, or Morse Theory.

When accountants and soldiers take interest in geometry.....
One day, accountants and soldiers may take an interest in differential geometry. If and when such a day comes to pass, this book will have a role to play. Until then, engineers, physicists and mathematicians alike have better alternatives, such as the inspiring texts, with complementary qualities, by Burke, "Applied Differential Geometry"; by do Carmo, "Riemannian Geometry", or by Spivak, "A Comprehensive Introduction to Differential Geometry".

Even more advanced books such as Lang's or Petersen's are more readable: in them the extra formalism brings the reward of more powerful results. Here the retentive attention to the trees at the expense of the forest is merely a barrier to entry for the uninitiated. This text's popularity in some areas of engineering must have played a role in the slow acceptance of Riemannian geometric methods.

Manuel Tenide

great introductory text
My first course on manifolds was based on this book,and I believe that it is the best introduction to the subject (especially for beginners). I thoroughly enjoyed it! I should also recommend Conlon's 'Differentiable Manifolds' (2ed, Birkhauser), as it is the perfect follow-up to Boothby. --A

Very Nice Nontrivial Introduction
This book is a careful treatment of the subjects in the title. It is an introduction, but it manages to cover quite a bit of ground with lots of examples to illustrate. One of it's distinguishing pointsis the way inwhich the concrete, coordinate based calculations are emphasized even whileusually presenting the more abstract, coordinate free approach as well.

The book does a good job at stimulating those studying it to developintuition. I found the book helpful when I was first studying the subject. ... Read more

Product Description
Provides counter examples to the Osserman conjuncture in generic semi-Riemannian signature provided. The properties of semi-Riemannian Osserman manifolds are investigated. Softcover. ... Read more

Product Description

Foundations of Differentiable Manifolds and Lie Groups gives a clear, detailed, and careful development of the basic facts on manifold theory and Lie Groups. Coverage includes differentiable manifolds, tensors and differentiable forms, Lie groups and homogenous spaces, and integration on manifolds. The book also provides a proof of the de Rham theorem via sheaf cohomology theory and develops the local theory of elliptic operators culminating in a proof of the Hodge theorem.

Customer Reviews (5)

Worthless
I used this for a grad class and it sucked! Language is bad, no examples, seemed to give no real insight, i highly recommend getting a different book

Great book, but not perfect or ideal for every purpose
This is a solid introduction to the foundations (and not just the basics) of differential geometry. The author is rather laconic, and the book requires one to work through it, rather than read it. It presupposes firm grasp of point-set topology, including paracompactness and normality. The basics (Inverse and Implicit Function Theorems, Frobenius Theorem, orientation, and rudiments of de Rham cohomology) are covered in about 100 pages (Chapters 1, 2, and 4). This is not really suitable for an undergraduate course in differential geometry, but is great for a graduate course.

Chapter 3, 5, and 6 (self-contained introductions to Lie Groups, Sheaf Theory, and Hodge Theory, all from a geometric viewpoint) are a really nice feature. The book can't be covered in one semester, but these chapters are great for selft-study. In fact, the organization of Chapter 5 is more suitable for self-study than for being taught in class (lots of theory developed first, with all applications delayed until the end). The real jewel of the book is Chapter 6, a very clean introduction to Hodge Theory, with immediate applications.

The main drawback of the book in my view is that the author avoids vector bundles like the plague. These could have been very nicely incorporated into the book. No mention is made of Mayer-Vietoris or Kunneth formula, even though the former follows easily from the section on cochain complexes in Chapter 5 and the latter with some effort from Chapter 6. There is no mention of manifolds with boundary either, except as regular domains of manifolds for the purpose of Stokes Theorem.

The organization of the book could have been better as well. In particular, the section on cochain complexes could have been incorporated in the rather short de Rham Cohomology Chapter 4, so that MV could have been proved and used to compute the cohomology of spheres (beyond the circle). Some subsections, including in Chapter 1, appear out of order to me. There is a shortage of exercises in my view. Some of the author's notation (for tangent spaces, tangent bundles) is rather non-standard.

However, all-in-all, I can't think of a better differential geometry text for a graduate course. Spivak and Lee are quite wordy and do not have the same breadth. Either book would be preferable to Warner for an undergraduate course though. The price is a relative bargain too.

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

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

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

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

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

Don't waste your money.

A reader.

Good, as long as you have enough background
I read this book at the very beginning of my studying in differential geometry and was striked. The definitions and methods used in this book seemed totally incomprehensible to me. However, after some development in this field, I found that this book is very concise. It is a very good surey on differential geometry but not a good book to start with. Definitions are given from the most "down to bottom" one. It is a very good attitude, yet, if you do not have much background in differential geometry, this book may takes you several days in order to understand the concept of tensor and exterior algebra.

A good book if you have some background
This book is a good introduction to manifolds and lie groups.
Still if you dont have any background ,this is not the book to start with.The first chapter is about the basics of manifolds:vector fields,lie brackts,flows on manifolds and more,
this chapter can help one alot as a second book on the subject.
The second chapter is about tensors, this introduction can be very hard for someone who didnt met the notion of tensors ,since the book tends to take a very general line with out down to earth examples.the 3ed chapter is about lie groups.It is avery good introduction ,to my point of view ,one of the best there is.
The 4th chapter is about integration on manifolds and is very good too.Chapters 5and6 are about De Rham cohomology theory and the hodge theorem.
If you have some knowledge on allthe above subjects this book can serve as a very good overview on the subject. ... Read more

Product Description
Differential geometry began as the study of curves and surfaces using the methods of calculus. In time, the notions of curve and surface were generalized along with associated notions such as length, volume, and curvature. At the same time the topic has become closely allied with developments in topology. The basic object is a smooth manifold, to which some extra structure has been attached, such as a Riemannian metric, a symplectic form, a distinguished group of symmetries, or a connection on the tangent bundle. This book is a graduate-level introduction to the tools and structures of modern differential geometry. Included are the topics usually found in a course on differentiable manifolds, such as vector bundles, tensors, differential forms, de Rham cohomology, the Frobenius theorem and basic Lie group theory. The book also contains material on the general theory of connections on vector bundles and an in-depth chapter on semi-Riemannian geometry that covers basic material about Riemannian manifolds and Lorentz manifolds. An unusual feature of the book is the inclusion of an early chapter on the differential geometry of hypersurfaces in Euclidean space. There is also a section that derives the exterior calculus version of Maxwell's equations. The first chapters of the book are suitable for a one-semester course on manifolds. There is more than enough material for a year-long course on manifolds and geometry. ... Read more

Customer Reviews (1)

FINALLY, AN EXCELLENT BOOK IN MODERN DIFFERENTIAL GEOMETRY!
Finally (!), an excellent book in Modern Differential Geometry that is here to stay! For people who struggled with the old-fashioned style and notations of Kobayashi and Nomizu, or with the happy-go-lucky style of Spivak's book, this new book is a must read! You will find it very well written, and you will regret that a book like this was not available 10 years ago! It leaves the pre-existing graduate textbooks behind, thanks to several attributes: 1. it is methodical and full of rigor, but at the same time easy to read; 2. it builds up all the concepts gradually, from a beginner's first-year graduate level .... up to the advanced level of a researcher who wants to learn about Lorentz manifolds and the Geometry of Physics; 3. it is smart, modern and user-friendly. As a professor and a researcher, I consider this book to be one of the best choices for your graduate students with an interest in differential geometry, topology and related areas. ... Read more

Product Description

The study of CR manifolds lies at the intersection of three main mathematical disciplines: partial differential equations, complex analysis in several complex variables, and differential geometry. While the PDE and complex analytic aspects have been intensely studied in the last fifty years, much effort has recently been made to understand the differential geometric side of the subject.

This monograph provides a unified presentation of several differential geometric aspects in the theory of CR manifolds and tangential Cauchy–Riemann equations. It presents the major differential geometric acheivements in the theory of CR manifolds, such as the Tanaka–Webster connection, Fefferman's metric, pseudo-Einstein structures and the Lee conjecture, CR immersions, subelliptic harmonic maps as a local manifestation of pseudoharmonic maps from a CR manifold, Yang–Mills fields on CR manifolds, to name a few. It also aims at explaining how certain results from analysis are employed in CR geometry.

Motivated by clear exposition, many examples, explicitly worked-out geometric results, and stimulating unproved statements and comments referring to the most recent aspects of the theory, this monograph is suitable for researchers and graduate students in differential geometry, complex analysis, and PDEs.

Product Description
In the millennium's last great sf novel, Stephen Baxter takes us a short step byond Y2K. The year is 2010. We have survived ! so far.Cornelius Taine of Eschatology, Inc., mathematical genius, predicts that in just 200 years our species will be wiped out. Even evacuation from Earth will not save us from extinction.Reid Malenfant, entrepreneur, has Big Dumb Boosters ready to fly from the California desert, to be piloted by an enhanced squid named Sheena 5. When Taine offers Malenfant the ultimate dream of saving the species, Sheena's mission is diverted to investigate Earth's recently discovered -- and very remote -- second moon.What Sheena 5 discovers there is nothing less than a revelation: the secret reason for our existence visible at last beneath the rippled surface of Time's river. Malenfant and Taine must follow Sheena! but they are pursued by an enraged US Air and Space Force, and a mighty battle in space may cut short their hopes for the ultimate transformation of mankind. ... Read more

Customer Reviews (1)

Excellent read
Baxter has written a very clever book, very enjoyable. I really liked the way he uses things like Feynman's radio in the story. ... Read more

Product Description

Customer Reviews (3)

Captain Obvious Handles Another Review
This book contains a lot of information about manifolds, particularly those with differentiable structures. It used to only be available with a boring green cover and it was expensive. Now, its cover is colorful and has a wacky picture on it. I thought that this would surely make the price go up but it got cheaper! The picture on the front cover concerns operations on manofolds, particularly differentiable manifolds.

Almost everything about higher-dim smooth manfolds in 220 pgs
Don't be deceived by the title of Kosinski's "Differential Manifolds," which sounds like a book covering differential forms, such as Lee's Introduction to Smooth Manifolds, or by claims that it is self-contained or for beginning graduate students. In fact, the purpose of this book is to lay out the theory of (higher-dimensional, i.e., >= 5) smooth manifolds as it was known in the '60s, namely, the techniques of handle decompositions, framed cobordism including the Thom-Pontrjagin construction, and surgery (sometimes called spherical modification). Offhand, I can't think of another book that covers all these topics as thoroughly and concisely, and does so in a way that is readily comprehensible.

The first 4 chapters are an overview of the basic background of differential topology - differential manifolds, diffeomorphism, imbeddings and immersions, isotopy, normal bundles, tubular neighborhoods, Morse functions, intersection numbers, transversality - as one would find in, e.g., Guillemin and Pollack's Differential Topology, Milnor's Topology from the Differentiable Viewpoint, or Hirsch's Differential Topology, albeit at a higher level and with much less explanation. As the author himself states, with some understatement, "The presentation is complete, but it is assumed, implicitly, that the subject is not totally unfamiliar to the reader." Although I would dispute somewhat the notion that it is complete, as several very important results on immersions and isotopies of Whitney and Haefliger are cited and used repeatedly, but not proved, since, as the author explains, it would have taken the reader too far a field. The reader should also have a good knowledge of algebraic topology (Dold and Spanier are frequently used as references), as well as the classification of bundles over spheres as found in Steenrod.

Since the purpose of the first 4 chapters (about 75 pp) is to develop the machinery of differential topology to the point where the results on handles, cobordism, and surgery can be proved, several topics are briefly touched upon that are generally not encountered in introductory diff top books, such as the group Gamma of differential structures on the m-sphere mod those that extend over the m-disk or the bidegree of a map from a product of spheres to a sphere, in addition to the aforementioned results of Whitney and Haefliger, but just enough is given so that they may be used in later proofs. Most perplexing is Chapter V, on foliations, which has only a tenuous connection to the preceding material and absolutely none to the following. It seems that the author just included it because he felt that knowledge of the subject was essential for a topologist, not because it was necessary for the purposes of this book; it certainly could be skipped, but is worth reading as a brief introduction to foliations.

The heart of the book is Chapter VI, where the concept of gluing manifolds together is explored. Normally, connected sums are defined by removing imbedded balls in 2 closed manifolds and gluing them along the spherical boundaries, but Kosinski instead constructs, explicitly in local coordinates, an orientation-reversing diffeomorphism of a punctured ball and then uses that to identify punctured balls in each manifold. Similarly, handle attachment is defined, rather than by just attaching a handle to an imbedded sphere in the boundary, but instead by again explicitly constructing an orientation reversing diffeomorphism of a (in the 0-dim case) punctured hemisphere and then identifying it with the normal bundle of a point in the boundary of the manifold. In this way, one automatically constructs smooth manifolds without having to resort to "vigorous hand waving" to smooth corners. The downside to this method (which is likely to be unfamiliar to modern readers) is that much time is spent constructing explicit formulas for handle attachments, e.g., in local coordinates, but after Chapter VI the details of these maps are no longer needed.

The last 4 chapters are the most interesting, as all the tools developed in the first 6 chapters are used to prove results such as the existence of handle decompositions for manifolds; the classification of handlebodies; the h-cobordism theorem, proved much easier than in Milnor's Lectures on the h-Cobordism Theorem; the Poincare conjecture for dimensions > 4; Poincare duality (for smooth manifolds only); the Morse inequalities; the existence of Heegard diagrams; the equivalence of the aforementioned group Gamma with the group of differential structures on the sphere and with h-cobordism classes of homotopy spheres (Theta); the Pontrjagin-Thom isomorphism; results on stably parallelizable and almost parallelizable manifolds; conditions under which surgery can eliminate homology in the middle dimension of a framed manifold that is closed or has a boundary that is a homotopy sphere, thus leading to corollaries about when a manifold is cobordant to a highly-connected manifold (such as a sphere); and the computation of some of the aforementioned Theta groups. As you can see, a lot of important results are derived, whose proofs are complete except for a few technical lemmas that are cited.

Most chapters conclude with a section titled "Historical Remarks" or just "Remarks," that explains the history of the development of the subject, including many references. The author himself, now almost 80, had in hand in some of these developments and was personally well-familiar with the giants of 20-c. mathematics who discovered them, such as Thom, Bott, Milnor, Smale, Whitney, Wall, Browder, Morse, etc. The text is also interlaced with exercises, most of which are relatively straightforward.

The book concludes with a new appendix, written last year by John Morgan (my former thesis adviser), on Perelman's proof of the Poincare conjecture. It's just an overview of the proof and feels really out of place, the only connections being that it concerns the Poincare conjecture in dim 3, whose proof for dimensions higher than 4 is one of the highlights of this book, and also that Perelman's proof involves a kind of surgery. This appendix does little to enhance the value of the book.

The book is not without it faults, however. In addition to the above observations about it being too advanced for an introductory text and the incongruity of Chapter V, there are the usual batch of typos: an arrow pointing the wrong way in a diagram on pg 231; a wrong sign in the second displayed equation on pg 102; the switching between indices 0,1 and 1,2 on pg 93; the reversal of the equations for the equator and meridian, as well as the words themselves, on pg. 212; 1/2 in place of epsilon 3 lines above eqn (2.2.6) on pg 128; missing bars over the h in many places in pp. 110-11, as well as omitting the -1 exponent for g in one place; etc. There are also errors of exposition, such as reversing the order of the i and j terms in the definition of M1 and M2 on pg. 211, which leads to factors of +/- missing from subsequent formulas, that fortunately do not impact the results, but do waste the reader's time; this category would also include Case 2 on pg. 214, whose proof is identical to that for Case 1 after a framed surgery and thus unnecessary, or even the 2 possibilities for m, listed in the first sentence for both Case 1 and Case 2 on that page, that are in fact identical, as well as an extraneous condition on n on pg 171. A more serious omission is Theorem X,5.1 (in the notation of the book), which should have been stated in 2 ways, one of which being analogous to Theorems X,4.5 and X,3.4 for use in proving the corollaries 5.2 and 5.3.

Probably the worst mistake is when the term "framed manifold" is introduced and defined to mean exactly the same thing as "pi-manifold," without ever acknowledging this fact, and then the terms are used interchangeably afterward, with theorems about framed manifolds being proved by reference to results about pi-manifolds, and even with the redundant expression "framed pi-manifold" being used in a few places. Moreover, "framed cobordant" is then defined in Chapter X to mean something different than it meant in Chapter IX.

Another group of complaints that I have is with the system of references. First of all, the chapter numbers do not appear in either the running heads or the theorem numbers, so when a result is cited in a previous chapter, the reader must flip back and forth through the book to find it, remembering the chapter numbers for each chapter, or must go back to the table of contents to locate it. Moreover, many theorems from earlier chapters are used without comment, or a reference is made to a theorem when in fact a corollary is being used (or vice versa!). Sometimes a theorem from another source is cited as the justification for a statement, when in fact the author is directly applying a theorem from his own book that just happened to use that other author's result in its proof - citing his own theorem, by number, would save the reader a lot of effort. And then there's the important imbedding theorem of Haefliger that he frequently cites, even though he never actually states what the theorem says! (I had to read Haefliger's paper to verify that it actually could be used to produce the results that Kosinski wanted.)

Rigorous but not inaccessible.
This book treats differential topology from scratch to the works by Pontryagin, Thom, Milnor, and Smale. The best thing with this book is that you don't too often have to read between the lines. That is, the exposition is detailed and user-friendly, so I highly recommend this book. If something is to be blamed, the price is horrendous. ... Read more

Product Description

This heavily class-tested book is an exposition of the theoretical foundations of hyperbolic manifolds. It is a both a textbook and a reference. A basic knowledge of algebra and topology at the first year graduate level of an American university is assumed. The first part is concerned with hyperbolic geometry and discrete groups. The second part is devoted to the theory of hyperbolic manifolds. The third part integrates the first two parts in a development of the theory of hyperbolic orbifolds. Each chapter contains exercises and a section of historical remarks. A solutions manual is available separately.

Customer Reviews (3)

Great reference
The book is nothing if not comprehensive, and if you work in the field, it is a useful reference to have close at hand. However, I would not recommend it to a student, since there is a good chance a student would be bored to death by the time he slogged his way through this. Thurston's notes (NOT his book), available for free from MSRI are available for free from MSRI, and are vastly superior as an introduction. Various survey papers by Vinberg (including the one in the Russian Encyclopedia of Mathematics: Geometry: Volume 2: Spaces of Constant Curvature (v. 2) are very lucid, and bring out the beauty of the subject much better than the book under review.

Best on the market
This is a wonderful book on both hyperbolic geometry **and** spherical geometry--non-Euclidean geometry in general.It's more comprehensive than all of the others.The prerequisites for this book vary greatly from chapter to chapter.If you want to read, and understand, all of the material right away, the prerequisites are somewhat steep.I would study smooth and riemannian manifolds first (I heavily recommend John Lee's two books).I would also get some basic algebraic topology (Hatcher's is a classic).If you have these, it's smooth sailing ahead.

An excellent overview for mathematicians and physicists
The advent of non-Euclidean geometry resulted in many different areas of mathematics, some being specifically related to geometry, others being more general, such as proof theory and model theory. This book is an excellent overview of a particular branch of non-Euclidean geometry called hyperbolic geometry. There are good exercises in the book, and the author gives a detailed history of the subjects after the end of each chapter. After a brief review of Euclidean geometry in chapter 1, emphasizing the metric properties of Euclidean space, orthogonal transformations, and isometries, the author discusses spherical geometry in chapter 2. Spherical and hyperbolic geometries are dual to each other, in the sense that in spherical geometry, a line through a point outside a given line is never parallel to the given line; but in hyperbolic geometry there are infinitely many such lines. Also, the sum of the angles of a spherical triangle is always greater than 180 degrees ; but in hyperbolic geometry less than 180 degrees. Hyperbolic geometry is of crucial importance in physics, particularly in the theory of relativity, and the author begins a discussion of this kind of geometry in chapter 3. Hyperbolic n-space is viewed more as dual to elliptic geometry in the sense that it is modeled as a unit sphere of imaginary radius with only the positive sheet of this (disconnected) set retained. The author outlines in detail the important properties of hyperbolic geometry along with its trigonometry. This is followed in the next chapters by a model of hyperbolic n-space as a conformal ball and an upper half-space, and a consideration of the isometries of hyperbolic space. The Mobius transformations are given detailed treatment. The famous classical discrete groups are introduced, along with the crystallographic groups. The discussion gets more abstract in some parts here, for the author introduces some algebraic notions such as valuation rings, in order to prove Selberg's lemma. The author finally lays the groundwork for a theory of hyperbolic manifolds in chapter 8, by first introducing geometric spaces. These are defined by four axioms, which are generalizations of Euclid's first four axioms, and two of these axioms imply that any geometric manifold is an n-manifold. The discussion is specialized in the next chapter to geometric surfaces, where the famous Gauss-Bonnet theorem, relating the area of a surface to its Euler characteristic, is proved for spherical, Euclidean, or hyperbolic surfaces. The author studies the collection of similarity equivalence classes of complete structures for a geometric surface, namely the moduli space of such structures. Physicists, particularly string theorists, will appreciate the resulting discussion on Teichmuller space and the Dehn-Nielsen theorem. Considerations of a nature more familiar to geometric topologists follows in the next chapter, where it is shown how to explicitly construct hyperbolic 3-manifolds. Dehn surgery is employed to study the complement of the figure 8 knot. The discussion is very interesting, for it employs explicit detailed constructions that would take many hours to dig out of the literature. The general case of n-dimensional hyperbolic manifolds is the subject of chapter 11, with the constructions in chapter 10 generalized to deal with high dimensions. The author considers also the two closed, orientable, hyperbolic manifolds of the same homotopy type have the same volume by using the Gromov invariant, a quantity defined in terms of the singular homology on the manifold. The reader will get a taste of the Haar measure in the proof of the result, and later an overview of measure homology. The later is very interesting, as it brings in techniques from differential topology and the de Rham complex, and it also defines a notion of a "straightening" and smearing of a singular complex. Mostow rigidity, which says that for any two closed, connected, orientable, hyperbolic n-manifolds, with n greater than 2, a homotopy between these will also be an isometry, is also proven here. The next chapter is more involved than the rest, and deals with the case of geometrically finite n-manifolds. Dealing with cusps and "sharp corners" from the actions of discrete groups is given detailed and rigorous discussion here. The discussion leads naturally to a treatment of orbifolds in the next chapter. These objects have been extremely important in string theories in high energy physics, and the author does an excellent job of detailing their properties. ... Read more

Product Description
For many years, John Hempel's book has been a standard text on the topology of 3-manifolds. Even though the field has grown tremendously during that time, the book remains one of the best and most popular introductions to the subject.

The theme of this book is the role of the fundamental group in determining the topology of a given 3-manifold. The essential ideas and techniques are covered in the first part of the book: Heegaard splittings, connected sums, the loop and sphere theorems, incompressible surfaces, free groups, and so on. Along the way, many useful and insightful results are proved, usually in full detail. Later chapters address more advanced topics, including Waldhausen's theorem on a class of 3-manifolds that is completely determined by its fundamental group. The book concludes with a list of problems that were unsolved at the time of publication.

Hempel's book remains an ideal text to learn about the world of 3-manifolds. The prerequisites are few and are typical of a beginning graduate student. Exercises occur throughout the text.

Other key books on low-dimensional topology available from the AMS are Knots and Links, Lectures on Three-Manifold Topology, and The Knot Book. ... Read more

Customer Reviews (1)

An indispensable classic
It seems odd that nobody has reviewed this book until now, but undoubtedly this is due to the fame of the book and the fact that the majority of those looking for it have no need for a review; however, a minority may find a review of what this book is and isn't useful in their decision to buy or not.

What this book isn't:
1) An introduction to topology, or even to low-dimensional topology.Someone who has heard of 3-manifolds and gotten excited would do better to get a taste of the subject elsewhere first, e.g. in Rolfsen's _Knots and Links_.
2) A research monograph designed to bring the reader up to speed on current research on 3-manifolds.This book is about 30 years old and doesn't even mention the Geometrization Conjecture of Thurston.
3) A book on the role of knot theory in 3-manifolds.Knots play an important role in the theory, not only theoretically, but as a rich source of examples to sharpen the intuition and test conjectures (through Dehn surgeries on knots and links).This role is not discussed in this book.

What this book is:
1) A primer for topologists seeking to become specialists in 3-manifolds.The basic theorems regarding prime decomposition, loop and sphere theorems, Haken hierarchy, and Waldhausen's theorems on Haken manifolds are explained in detail.These can be considered some of the highlights although much relevant material is necessarily also explained.As perhaps befitting a primer, the JSJ decomposition and characteristic submanifold theory is not included.Jaco's book complements Hempel by covering this material.
2) A reference for those already familiar with the material.The writing style is very concise and to the point.This makes it simple to look up a theorem to refresh one's memory on a sticky detail in a proof.As an introduction to the material, some passages may be terse, but inevitably after some effort, they can be "decoded" completely, unlike some texts that may be more verbose but can never be entirely deciphered.I think there could be a lot more pictures; there aren't very many, to say the least.But if the reader draws his/her own pictures, this shouldn't be too much of a problem.

Some final remarks:
This book serves its dual role as a primer and reference admirably, but the reader may get lost in the details and lose the forest for the trees.Unfortunately, the only way to rectify this seems to be to read various papers on the subject to get a good feel of the various threads that motivate current research.But with Hempel's _3-manifolds_ in hand, this task is much easier and enjoyable. ... Read more

Product Description

By bringing together various ideas and methods for extracting the slow manifolds, the authors show that it is possible to establish a more macroscopic description in nonequilibrium systems. The book treats slowness as stability. A unifying geometrical viewpoint of the thermodynamics of slow and fast motion enables the development of reduction techniques, both analytical and numerical. Examples considered in the book range from the Boltzmann kinetic equation and hydrodynamics to the Fokker-Planck equations of polymer dynamics and models of chemical kinetics describing oxidation reactions. Special chapters are devoted to model reduction in classical statistical dynamics, natural selection, and exact solutions for slow hydrodynamic manifolds. The book will be a major reference source for both theoretical and applied model reduction. Intended primarily as a postgraduate-level text in nonequilibrium kinetics and model reduction, it will also be valuable to PhD students and researchers in applied mathematics, physics and various fields of engineering.

Product Description

Customer Reviews (1)

Well done Pierre & Co
This is a very useful book, especially for beginners in the subject. The essentials from differential geometry and topology are carefully collected and illustrated with the most popular matrix manifolds. The optimization algorithms are considered in great details usually omitted in the research papers. The list of references shows the great variety of areas where optimization on matrix manifolds would be appropriate. ... Read more

Product Description
General relativity is now essential to the understanding of modern physics, but the power of the theory cannot be exploited fully without a detailed knowledge of its mathematical structure.This book aims to implement this structure, and then to develop those applications that have been central to the growth of the theory. ... Read more

Product Description

Differential geometry techniques have very useful and important applications in partial differential equations and quantum mechanics. This work presents a purely geometric treatment of problems in physics involving quantum harmonic oscillators, quartic oscillators, minimal surfaces, Schrödinger's, Einstein's and Newton's equations, and others. Historically, problems in these areas were approached using the Fourier transform or path integrals, although in some cases, e.g., the case of quartic oscillators, these methods do not work. New geometric methods, which have the advantage of providing quantitative or at least qualitative descriptions of operators, many of which cannot be treated by other methods, are introduced. And, conservation laws of the Euler--Lagrange equations are employed to solve the equations of motion qualitatively when quantitative analysis is not possible.

... Read more

Product Description
This is the third version of a book on Differential Manifolds; in this latest expansion three chapters have been added on Riemannian and pseudo-Riemannian geometry, and the section on sprays and Stokes' theorem have been rewritten. This text provides an introduction to basic concepts in differential topology, differential geometry and differential equations. In differential topology one studies classes of maps and the possibility of finding differentiable maps in them, and one uses differentiable structures on manifolds to determine their topological structure. In differential geometry one adds structures to the manifold (vector fields, sprays, a metric, and so forth) and studies their properties. In differential equations one studies vector fields and their integral curves, singular points, stable and unstable manifolds, and the like. ... Read more

Customer Reviews (3)

Modern, but....
This book is a proper subset of Lang's later book "Fundamentals of Differential Geometry (Graduate Texts in Mathematics, 191)".

Not a "first book", ok as reference
Lang's book is definitely not useful as textbook for classes or for self-guided study (learnt this the hard way). He is rather abstract and provides zero motivation for the theory. The book is obviously made for people who learnt diff. geometry elsewhere but want to read a cleaner and more modern treatment. To this end, Lang's book is useful. The best part is that manifolds are infinite-dimensional right away. This is probably the only reason for buying Lang instead of/in addition to Dieudonne as a reference. Otherwise, the book is a little too terse; fiber bundles are merely hinted at. Moreover, I think some of the proofs are unnecessarily complicated, such as the one for Frobenius theorem.

Maybe Lang's best book.
Well, we have here another book on differential manifolds, and another book by Serge Lang. Lang is well-known by writing (lots of) books on different topics in analysis and algebra, all of them in a quite "Bourbaki-like" style: attaining maximum generality, with less motivation than most students would like. This is no surprise, because Lang himself is a Bourbakist.

So, what's interesting about D&RM? It's a book very much like Lang's other books, only that here the Bourbakist's approach is quite happy: it's one of the very few books on his subject to present most of his results in infinite-dimensional(Banach) version, a must if you are interested in nonlinear functional analysis or dynamical systems. The exposition is very clean and clear: Lang uses categories all the way to estabilish the main relations between the different differential-topological structures and tools, and he does not hesitate in stating and using tools from analysis, such as Lebesgue measure and functional analysis' main theorems. The proofs are very polished and, in a certain sense, beautiful, a philosophy thatpermeates most of the book. As if it weren't enough, the book still contains an appendix with a Von Neumann's seminar about the spectral theorem.

All things considered, it's a quite "state-of-the-art" book about the basics of differential manifolds, from an analyst's perspective. This perspective provides differential topology with a lot of additional clarity and power. I don't know if most physicists would like this book, because its motivations, if any, are sparse and sometimes quite obscure, as long as physical applications are concerned. For a mathematician, however, this book is a gem: it's Lang at its best, and the perfect opening door to global analysis (the nonlinear analysis on infinite dimensional manifolds, a vast field of mathematics that encompasses dynamical systems and nonlinear functional analysis). Despite all that, I would also recommend to physicists to at least tackle this book, as an antidote to all the crap that the so-called "differential topology for physicists" books put on their heads, because I don't know a cleaner and more precise presentation of differential manifolds so far. ... Read more

Product Description
Recently there has been considerable interest in developing techniques based on number theory to attack problems of 3-manifolds; Contains many examples and lots of problems; Brings together much of the existing literature of Kleinian groups in a clear and concise way; At present no such text exists ... Read more

Customer Reviews (1)

Sweet book on a really interesting topic
One of the things that attracted me to geometry/topology was the tremendous interconnection between this field and other fields of math.You never feel like you're missing out on algebra, analysis, physics, and, apparently number theory as well.Walk in knowing algebraic number theory and some hyperbolic geometry (I recommend Foundations of Hyperbolic Manifolds by Ratcliffe).This is great fun for anyone interested in both geometry and number theory--even if it's just a passing interest. ... 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