The Wild World of 4-Manifolds

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American Mathematical Soc., May 10, 2005 - Mathematics - 609 pages
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What a wonderful book! I strongly recommend this book to anyone, especially graduate students, interested in getting a sense of 4-manifolds. --MAA Reviews 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 book offers 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 small enough so that there is no room to undo the wildness. For example, only manifolds of dimension four can exhibit infinitely many distinct smooth structures. Indeed, their topology remains the least understood today. To put things in context, the book starts with a survey of higher dimensions and of topological 4-manifolds. In the second part, the main invariant of a 4-manifold--the intersection form--and its interaction with the topology of the manifold are investigated. In the third part, as an important source of examples, complex surfaces are reviewed. In the final fourth part of the book, gauge theory is presented; this differential-geometric method has brought to light how unwieldy smooth 4-manifolds truly are, and while bringing new insights, has raised more questions than answers. The structure of the book is modular, organized into a main track of about two hundred pages, augmented by extensive notes at the end of each chapter, where many extra details, proofs and developments are presented. To help the reader, the text is peppered with over 250 illustrations and has an extensive index.

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Serious mathematics cannot get more entertaining than in this book.
Alexandru Scorpan's "The Willd World of 4-Manifolds" is one of the best pieces of mathematical writing that I have come across
. The author manages not only to teach you a variety of (difficult) concepts reaching from geometric topology over complex geometry to gauge theory but also makes it fun to read. I have not yet found a single boring page. There is always something that puts a smile on my face or even makes me laugh.*
Mathematically speaking, the best parts are probably the whole chapter on intersections forms (Part II) and the discussion of the Seiberg-Witten invariants (Chapter IV.9). Scorpan has found the perfect balance between technical detail and insightfulness. You really get a feel for the theory and you feel well prepared to read up all the details in the references (which are given extensively).
Another nice feature that makes the book easy to read are the well placed cross references and the extensive index. You never feel lost in this book and whenever a complicated theorem or definition is appealed to there is a footnote telling you where at was stated in the book.
Everyone with an interest in geometric topology should give this book a try.
* I like to believe that this is because the book is seriously funny and not because I'm dense enough to laugh about bad mathematical jokes. ;-)

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