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4-Manifolds and Kirby Calculus (Graduate Studies in Mathematics) Hardcover – August 1, 1999
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This book is important and valuable in that both gives a comprehensive and accessible picture of an area which has developed rapidly in the past 20 years and also provides readers with techniques to begin research in the field. The book is pedagogically very strong, with many examples and exercises. The material will not go out of date, and however the field may develop in the future, this will be an important reference for many years to come. --Bulletin of the London Mathematical Society
This book gives an excellent introduction into the theory of -manifolds and can be strongly recommended to beginners in this field ... carefully and clearly written; the authors have evidently paid great attention to the presentation of the material ... contains many really pretty and interesting examples and a great number of exercises; the final chapter is then devoted to solutions of some of these ... this type of presentation makes the subject more attractive and its study easier. --European Mathematical Society Newsletter
A complete record of the folklore related to handle calculus ... All of the mathematical statements are given in absolutely precise language, and the notation and terminology used are well chosen ... a very comprehensive book ... Most low-dimensional topologists will want to have access to this as a reference book ... any student ... will be rewarded with a thorough understanding of this fascinating field. --Bulletin of the AMS
From the Publisher
I greatly recommend this wonderful book to any researcher in 4-manifold topology for the novel ideas, techniques, constructions, and computations on the topic, presented in a very fascinating way. I think really that every student, mathematician, and researcher interested in 4-manifold topology, should own a copy of this beautiful book.
Zentralblatt f\"ur Mathematik
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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.
is very expensive.
For me this doesn't deliver Kirby calculus as
claimed. It does give a vague impressing of what Kirby calculus might
be if presented as an axiomatic approach.
What is needed is a simple approach to very simple
totally defined manifolds.
A very old book gives a better starting point:
Topology of 3-Manifolds and Related Topics (Dover Books on Mathematics)
The books isn't clear about gluing, surgery and other
manipulations of manifolds in a way that can be picked up
in a two week loan period. Since I'm not really very new
at this, when I say you can't get it "easy" here,
I mean that if you buy the book
and spend full time for a long period reading and rereading
it might actually teach you some Kirby calculus.
The book isn't organized as a teacher,
but is very good at showing off the author's
knowledge which seems to be the main purpose?
Now I have to look for something better on Kirby calculus: maybe?
The Topology of 4-Manifolds (Lecture Notes in Mathematics / Nankai Institute of Mathematics, Tianjin, P.R. China (closed))
(In fact I'm interested in exotic forcing-generic R4s and their import, if any, in General Relativity. Truly wild beasts...)
Francisco Antonio Doria