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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"This book gives an excellent introduction into the theory of $4$-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
"The book under review introduces the current state of 4-manifold topology; it is almost unique in that it does so from the point of view of differential topology. Part I of the book ... would be priceless for algebraic geometers and gauge theorists who want to learn the topological aspects of the theory. Part II ... is essentially independent of Part I and would make for an excellent graduate text on its own." ---- Mathematical Reviews
"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
"The book under review introduces the current state of 4-manifold topology; it is almost unique in that it does so from the point of view of differential topology. Part I of the book ... would be priceless for algebraic geometers and gauge theorists who want to learn the topological aspects of the theory. Part II ... is essentially independent of Part I and would make for an excellent graduate text on its own." ---- Mathematical Reviews
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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6 of 6 people found the following review helpful
Format:Hardcover
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.
0 of 3 people found the following review helpful
Format:Hardcover
I got this book on inter-library loan as it
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))
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))
2 of 9 people found the following review helpful
Format:Hardcover
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
(In fact I'm interested in exotic forcing-generic R4s and their import, if any, in General Relativity. Truly wild beasts...)
Francisco Antonio Doria
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