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The Wild World of 4-Manifolds

Alexandru Scorpan
American Mathematical Society
Publication Date: 
Number of Pages: 
[Reviewed by
Stephen Ahearn
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What a wonderful book! Alexandru Scorpan's The Wild World of 4-Manifolds is a lively introduction to 4-manifolds written as introductions should be: it focuses on the big picture and allows readers to delve into details when they are ready.

The book is divided into four parts. To indicate what sets 4-manifolds apart, Scorpan begins with a discussion the h-cobordism theorem and why it fails in dimension 4. The second part focuses on the main invariant of 4-manifolds, the intersection form, discusses Freeman's classification of topological 4-manifolds, and ends with exotic R4s, manifolds which are homeomorphic, but not diffeomorhpic, to R4 (in contrast to Rn, n ≠ 4, which admits a unique smooth structure). Part three is devoted to complex surfaces as examples of 4-manifolds. The fourth and largest part gives a plethora of results coming from applications of gauge theory to 4-manifolds. Both the third and fourth parts end by constructing infinite families of homeomorphic but non-diffeomorphic 4-manifolds.

As background the reader needs a reasonable understanding of manifolds and differential topology, as well as some algebraic topology (Poincaré duality, characteristic classes). Scorpan provides a very quick listing of background definitions and facts.

A key to the book's success is its multiple layers. In addition to the main text there are footnotes, inserted notes (indented text with smaller type), proofs (also indented), and end-notes. The main text tells the story and is a very reasonable 200 or so pages. The remainder of the 600 pages gives supporting details which can be read or ignored at the reader's pleasure. A very detailed index makes going back easier. Should you need more detail, Scorpan provides an extensive bibliography.

You might wonder if all of these layers are distracting; the answer is no. The main text flows very well and is enjoyable reading. I strongly recommend this book to anyone, especially graduate students, interested in getting a sense of 4-manifolds.

Stephen T. Ahearn ( teaches mathematics at Macalester College in St. Paul, MN. His primary research interests are in algebraic topology and computational topology/geometry but allows himself to be distracted by other interesting topics as in his article "Tolstoy's Integration Metaphor from War and Peace.'' He also enjoys hiking, swimming, baking bread, and reading.

* Introduction
* Front matter

Part I: Background scenery

* Introduction
* Contents of part I
* Chapter 1: Higher dimensions and the $h$-cobordism theorem
* Chapter 2: Topological 4-manifolds and $h$-cobordisms

Part II: Smooth 4-manifolds and intersection forms

* Introduction
* Contents of part II
* Chapter 3: Getting acquainted with intersection forms
* Chapter 4: Intersection forms and topology
* Chapter 5: Classifications and counterclassifications

Part III: A survey of complex surfaces

* Introduction
* Contents of part III
* Chapter 6: Running through complex geometry
* Chapter 7: The Enriques-Kodaira classification
* Chapter 8: Elliptic surfaces

Part IV: Gauge theory on 4-manifolds

* Introduction
* Contents of part IV
* Chapter 9: Prelude, and the Donaldson invariants
* Chapter 10: The Seiberg-Witten invariants
* Chapter 11: The minimum genus of embedded surfaces
* Chapter 12: Wildness unleashed: The Fintushel-Stern surgery
* Epilogue
* List of figures and tables
* List of figures graphic
* Bibliography
* Index