Customer Reviews

Metrics, Connections and Gluing Theorems (Cbms Regional Conference Series in Mathematics)

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First of all, don't pay too much heed to Dr. Carlson's review of this book, since he apparently didn't read more than a page or two (cf. my comment on his review). While much of the book discusses Donaldson's work on the moduli spaces of anti-self-dual connections over a closed 4-manifold, that isn't really the focus or purpose of this work. The real purpose is to (sort of) prove the following theorem on anti-self-dual (ASD) metrics (i.e., Riemannian metrics with zero self-dual Weyl curvature; W+ = 0):

Let M be a compact, oriented, 4-dimensional manifold (w/o boundary). Then there exists a positive integer N such that the connected sum of M with n copies of -CP^2 (i.e., 2-d complex projective space with the reverse orientation) has metrics with W+ = 0 for all n >= N.

So the author is interested in proving an existence theorem, one which bears much similarity to the earlier existence theorem ("Taubes's existence theorem") he provided for anti-self-dual connections on the same kinds of 4-manifolds, which was one of the key steps in Donaldson's theorem in gauge theory. Thus in this book Taubes first gives an overview of Donaldson theory (which is why Dr. Carlson was confused) and then sketches a (30-page) proof of its existence theorem. Afterward he returns to self-dual metrics, and provides another 30-page proof of the aforementioned main theorem, semi-modeled on the preceding one and involving many of the same analytic techniques. Finally he concludes with some open (at that time, and still open as far as I know) problems concerning self-dual metrics and Donaldson theory.

I say sketch because even though he devotes 30 pages to the proof, he doesn't quite succeed in proving it, having to refer to other papers at a number of points and making frequent errors. Infact, he botches the proof so badly (and shockingly, considering that it was his theorem!), I had to supply my own. He tries to make the proof easier to digest for novices by first considering several special cases and working up to the full generality, but he ends up repeating a number of statements and equations and even unnecessarily reproving things in different ways. What's worse is that some statements are in fact incorrect, such as eq'ns (4.26), (5.6), (5.4), and especially (5.27) and the condition derived for E immediately after it. I wrote a page of equations and attached it to p. 53 of the book to replace the disorganized mess that concludes the proof. Ironically, he made this proof much longer than the versions that appear in his original paper and in Freed & Uhlenbeck's Instantons and Four-Manifolds, Lawson's Theory of Gauge Fields in Four Dimensions, and Donaldson & Kronheimer's The Geometry of Four-Manifolds in an effort to appeal to students, and while this does have the advantage of making it easier to understand what he is trying to do, it also has the unfortunate result of seeing him fail to accomplish it.

The proof of the main theorem on ASD metrics suffers from many of the same deficiencies. While it is much longer, with more detail and background information, than the proof in his original papers, it also contains several serious errors and there are gaps where technical results are only stated but not proved. Most embarrassing are a statement on p. 72 where it is stated that a certain function is square integrable when it isn't and the equations (9.9) and (9.10), which are not only incorrect, but even if they weren't, still fail to provide an expression for Q-bar over the entire manifold in terms of Q. This latter point is so serious, it is unclear whether the theorem has even been proved - it will be necessary to consult the original papers.

The editing for the book is also virtually nonexistent; infact, I seriously doubt it was edited at all. Among the plethora of typos include "have" instead of "half," "a priori" spelled as one word on a number of occasions, and even more mysteriously "in fact" written as one word ("infact"), not once but a dozen times or more (I have done it twice in this review as an illustration). There are also problems with the references to theorems and equations (it seems that they were renumbered but the references weren't amended) and variables are often written in roman instead of italic font, which becomes a problem when the variable is an "a."

Despite all these demerits there are a few positives. First of all, the level is a little lower than most of his original papers, with definitions provided for such basic concepts as the connected sum, the Fredholm property, connections, and the contraction mapping theorem. Second, his explanations of how the theorems work, rather than the details, are valuable in introducing the beginner to gauge theory, which is notoriously difficult to learn. Third, the main theorem itself may be of interest to some differential geometers. Fourth, the problems in the final chapter, and even more significantly, the connections between ASD metrics and connections alluded to in the all-too-brief Chapter 6, although not relevant to the rest of the book, offer a glimpse into a potentially deeper and more fascinating theory, in which the Donaldson polynomials are encoded in twister spaces of manifolds with ASD metrics and in which knowledge of the moduli space of based ASD metrics leads to information about the group of orientation-preserving diffeomorphisms of a manifold.

Don't get me wrong - Taubes is a brilliant mathematician, one of the best in gauge theory; but this was far from his best work.

The results in this book have been superseded by the results of Nathan Seiberg and Edward Witten that appeared a short time after this book was published. The Seiberg-Witten theory caused great excitement among the mathematical community who do research in differential geometry and differential topology, particularly in the area of the differential toplogy of 4-dimensional manifolds. Proclaiming the "end of the Donaldson theory", mathematicians moved on to study the properties of the Seiberg-Witten equation, which is still nonlinear enough to be interesting.

Thus the discussions in this book are somewhat dated, but still could be of interest from a mathematical history standpoint or with the goal perhaps of motivating the Seiberg-Witten theory, although the self-dual Yang-Mills equations, a cornerstone of the Donaldson theory, are not discussed in the book. Instead the author studies the anti-self dual equations. These arise uniquely in four dimensions due to the fact that the Lie algebra so(4) of the special orthogonal group SO(4) is not simple, but instead decomposes as the direct sum of two copies of the Lie algebra so(3) of SO(3). This enables one to decompose the tangent bundle of the manifold into a direct sum of two oriented 3-plane bundles. The curvature of the Levi-Civita connection then splits with respect to this decomposition, the decomposition having entries involving the scalar curvature, the traceless Ricci tensor, and the self-dual and anti-self dual Weyl curvature tensors. The question of the existence of metrics on the manifold for which the anti-self dual part vanishes is the subject of the book, with particular attention paid to complex vector bundles over the manifold. The anti-self dual equations are consequently a set of algebraic equations for the curvature, which are equivalent, over an open set in the manifold, to a first-order differential equation involving a 1-form over this open set and taking values in the Lie algebra of complex 2-space. The main strategy the author employs for studying these equations is to show that they linearize to Fredholm equations. Thus a kind of generalization of the Fredholm property holds here in the context of (infinite-dimensional) vector bundles. A section of an infinite-dimensional vector bundle which linearizes to a Fredholm operator acts essentially like a section of a finite dimensional bundle over a finite dimensional manifold.

The book could also be used to motivate a study of the connection between the Donaldson and Seiberg-Witten invariants of smooth four-manifolds. That there is such a connection was recently conjectured and some promising work in proving this conjecture has appeared lately. There has also been recent interest in examining the anti-self dual equations on noncommutative four-dimensional Euclidean space, because of its connection with string theory.