Readers familiar with the proof of Stephen Smale's proof of the high-dimensional Poincare conjecture will know that handle calculus was employed in the proof. This book is an overview of Kirby calculus, which is essentially handle calculus in dimensions less than or equal to four.
Kirby calculus can be used to describe four-dimensional manifolds such as elliptic surfaces, and gives a pictorial description of its handle decomposition. Its utility lies further than this however, as Kirby calculus has been used to answer questions that would have been very difficult otherwise.
The book begins with a very quick overview of the algebraic topology and gauge theory of four-dimensional manifolds. Readers not familiar with this material will have to consult other books or papers on the subject.
Part two takes up Kirby calculus, and handle decompositions are described with examples given for disk bundles over surfaces and tori. Handle moves are employed as processes that allow one to go from one description of a manifold to another. Handlebody descriptions are given for spin manifolds, and more exotic topics, such as Casson handles and branched covers are treated.
Part 3 of the book uses techniques from algebraic geometry to describe branched covers of algebraic surfaces. Handle decompositions of Lefschetz fibrations are given, and its is shown that a Stein structure on a manifold is completely described by a handle diagram. There is also a thorough discussion of exotic structures on Euclidean 4-space. In spite of the non-constructive nature of these results, namely that no explicit example of an exotic structure is given, the discussion is a fascinating one and has recently been shown to be important in physics.
The reader will no doubt attempt many of the exercises; the solutions of some of these given in the back of the book. The book serves well the needs of those dedicated individuals who are interested in specializing in low-dimensional topology. In addition, physicists interested in these ideas couuld benefit from its reading, although some of the results may seem a little heavy-handed and abtruse at times.
- Hardcover: 558 pages
- Publisher: American Mathematical Society; UK ed. edition (31 August 1999)
- Language: English
- ISBN-10: 0821809946
- ISBN-13: 978-0821809945
- Product Dimensions: 19 x 3.2 x 26.7 cm
- Boxed-product Weight: 1.2 Kg
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- Amazon Bestsellers Rank: 66,178 in Books (See Top 100 in Books)