Customer Reviews

Topology of 4-Manifolds. (PMS-39)

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It is too bad this book is out of print, for it introduces the reader to a fascinating branch of topology and has the clearest proof of the 4-dimensional Poincare conjecture. In addition, the authors do not hesitate to employ diagrams as needed to illustrate the main points and to assist the reader in visualizing 4-dimensional objects. The authors give a fine discussion as to the reasons why four dimensions is harder to deal with topologically than dimensions five or greater, this being essentially due to the behavior of 2-dimensional disks: mapping 2-disks into 3-manifolds results (generically) with 1-dimensional self-intersections; in 4-dimensions the intersections are isolated points, and in 5 dimensions or more the 2-disks can be embedded.

Interestingly, the authors choose not to employ the famous "Kirby calculus" in the proofs of the main results, despite the fact that it was used extensively in their earlier works. They break the book into two parts, the first one emphasizing embedding theorems and the second one the structure of manifolds. Those readers interested in the proof of the 4-dimensional Poincare conjecture will find it in chapter 7, as a consequence of the authors proof of the h-cobordism theorem, the latter being nontrivial. It is the absence of a smooth structure on the h-cobordism that makes it so difficult in dimension four.

The existence of exotic structures on 4-manifolds is discussed in detail in chapter 8 and the authors endeavor to show why dimension 4 is unique compared to higher dimensions. The existence of exotic structures on 4-manifolds is definitely interesting, and has recently been shown to have importance in physics. But physicists who need an explicit example of one of these structures will not find one here, and I know of no such examples in the literature. Such an example would be interesting from the standpoint of the behavior of quantum field theories on such 4-manifolds, as one would like to know if this behavior would indeed be different than that on the manifold with the "standard structure".

Lee Carlson's review can hardly be improved upon, but I have one addendum, responding to his lament at the end of the review:

A concrete construction of a fake 4-space can be found in the last section of Chapter 1 of Dan Freed and Karen Uhlenbeck's (1984, revised 1991) book, "Instantons and Four-Manifolds", a great compliment to Freedman and Quinn's book. Unfortunately, it's just as hard to find and about three times as costly as the reviewed book. Unless you really can't live without seeing how this is done, take it on faith (not much consolation to a physicist who wants to know if black holes behave differently in fake spacetime). A cheaper alternative: If you visit a major university math library and take a bunch of quarters for their copier, the relevant Ch.1, "Fake R^4", is fourteen pages long.

Do you know what a Capped grope is? If so, this is the book for you! This is a really readable, though understandably advanced, book. It seems to contain everything a person could possibly want to know about embedding a disk in a 4-manifold. I'm far from an expert on 4-manifolds, but I haven't seen much of this material presented anywhere else. If you want to become a master of 4-manifolds, you need this book like Luke Skywalker needed Yoda.

My copy doesn't have the cool orange cover so I feel a little sad. Hopefully you won't get the brown one like me.