4-Manifolds and Kirby Calculus (Graduate Studies in Mathematics) (Graduate Studies in Mathematics) | 
 enlarge | Author: Andras I. Stipsicz Robert E. Gompf
Publisher: American Mathematical Society
Category: Book
List Price: $68.00
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Rating:  3 reviews
Sales Rank: 643227
Media: Hardcover
Pages: 558
Number Of Items: 1
Shipping Weight (lbs): 2.7
Dimensions (in): 10.1 x 7.1 x 1.3
ISBN: 0821809946
Dewey Decimal Number: 514.3
EAN: 9780821809945
Publication Date: August 1, 1999
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Product Description
The past two decades have brought explosive growth in 4-manifold theory. Many books are currently appearing that approach the topic from viewpoints such as gauge theory or algebraic geometry. This volume, however, offers an exposition from a topological point of view. It bridges the gap to other disciplines and presents classical but important topological techniques that have not previously appeared in the literature. Part I of the text presents the basics of the theory at the second-year graduate level and offers an overview of current research. Part II is devoted to an exposition of Kirby calculus, or handlebody theory on 4-manifolds. It is both elementary and comprehensive. Part III offers in depth a broad range of topics from current 4-manifold research. Topics include branched coverings and the geography of complex surfaces, elliptic and Lefschetz fibrations, $h$-cobordisms, symplectic 4-manifolds, and Stein surfaces. Applications are featured, and there are over 300 illustrations and numerous exercises with solutions in the book.
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| Customer Reviews:
Extremely detailed overview of Kirby calculus October 28, 2001
Dr. Lee D. Carlson (Saint Louis, Missouri USA)
3 out of 3 found this review helpful
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.
Review September 18, 2007
Francisco A. Doria (Rio, Brazil)
1 out of 3 found this review helpful
Actually I was looking for loose ends - things that do not appear in Scorpan's _The Wild World of 4-Manifolds_, like the Buzaca construction of exotic R4s, or the construction of an ``universal'' R4, and I found it it Gompf's book.
(In fact I'm interested in exotic forcing-generic R4s and their import, if any, in General Relativity. Truly wild beasts...)
Francisco Antonio Doria
Complexity is the name... June 13, 2000
Darunee Suwannakoon (Palo Alto, CA)
5 out of 10 found this review helpful
If you really into mathematics, this book is for you. It contains comprehensive explanation of the Kirby calculas. The complexity of this book require graduate level mathematics knowleadge as a prerequisite. It describes in the detail of a closed 4-manifold which admits a finite decomposition into geometric pieces of finite volume. It also consider the homotopy types of closed 4-manifolds which are Seifert fibred or which are the total spaces of bundles with base and fibre closed aspherical surfaces.
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