The Wild World of 4-Manifolds by Alexandru Scorpan (PDF)

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.

User’s Reviews

Editorial Reviews: Review “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”What a wonderful book! I strongly recommend this book to anyone, especially graduate students, interested in getting a sense of 4-manifolds.” —- MAA Online”The author records many spectacular results in the subject … (the author) gives the reader a taste of the techniques involved in the proofs, geometric topology, gauge theory and complex and symplectic structures. “The book has a large and up-to-date collection of references for the reader wishing to get a more detailed or rigorous knowledge of a specific topic. The exposition is user-friendly, with a large numer of illustrations and examples.” —- Mathematical Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐A fascinating book about 4-manifolds designed for professional mathematicians and graduate students who have a very solid background in topology and algebraic topology. The book is a bit casual about fine points, and contains a number of minor errors, but nothing that really bothered me. I hope it comes out in a corrected edition . . . but since it’s been out now for over 10 years, I’m not going to hold my breath.Despite the casualness, the author goes to great lengths to be clear — with many illustrations, footnotes, and elaborations on parts of proofs that are supplied. So it is a great way to get one’s feet wet very quickly in this field.And many of the results, as you may have heard, are quite astonishing, like there being uncountably many distinct differential structures on Euclidean 4-space, despite there being only one differential structure on Euclidean space of every dimension unequal to 4.

⭐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

⭐or Kirby’s

⭐(but see also

⭐, 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

⭐or Kosinski’s

⭐for the h-cobordism theorem; Gompf & Stipcisz’s

⭐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

⭐, Morgan’s

⭐, or Moore’s

⭐) 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.

⭐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.

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