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3-Manifolds

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It seems odd that nobody has reviewed this book until now, but undoubtedly this is due to the fame of the book and the fact that the majority of those looking for it have no need for a review; however, a minority may find a review of what this book is and isn't useful in their decision to buy or not.

What this book isn't: 1) An introduction to topology, or even to low-dimensional topology. Someone who has heard of 3-manifolds and gotten excited would do better to get a taste of the subject elsewhere first, e.g. in Rolfsen's _Knots and Links_. 2) A research monograph designed to bring the reader up to speed on current research on 3-manifolds. This book is about 30 years old and doesn't even mention the Geometrization Conjecture of Thurston. 3) A book on the role of knot theory in 3-manifolds. Knots play an important role in the theory, not only theoretically, but as a rich source of examples to sharpen the intuition and test conjectures (through Dehn surgeries on knots and links). This role is not discussed in this book.

What this book is: 1) A primer for topologists seeking to become specialists in 3-manifolds. The basic theorems regarding prime decomposition, loop and sphere theorems, Haken hierarchy, and Waldhausen's theorems on Haken manifolds are explained in detail. These can be considered some of the highlights although much relevant material is necessarily also explained. As perhaps befitting a primer, the JSJ decomposition and characteristic submanifold theory is not included. Jaco's book complements Hempel by covering this material. 2) A reference for those already familiar with the material. The writing style is very concise and to the point. This makes it simple to look up a theorem to refresh one's memory on a sticky detail in a proof. As an introduction to the material, some passages may be terse, but inevitably after some effort, they can be "decoded" completely, unlike some texts that may be more verbose but can never be entirely deciphered. I think there could be a lot more pictures; there aren't very many, to say the least. But if the reader draws his/her own pictures, this shouldn't be too much of a problem.

Some final remarks: This book serves its dual role as a primer and reference admirably, but the reader may get lost in the details and lose the forest for the trees. Unfortunately, the only way to rectify this seems to be to read various papers on the subject to get a good feel of the various threads that motivate current research. But with Hempel's _3-manifolds_ in hand, this task is much easier and enjoyable.

Author(s): John Hempel
Series: Annals of Mathematics Studies
Publisher: Princeton University Press
Year: 1976

Language: English
Commentary: +OCR
Pages: 184

NEW RESEARCH ON THREE-MANIFOLDS AND MATHEMATICS......Page 5
NOTICE TO THE READER......Page 6
CONTENTS......Page 7
PREFACE......Page 9
1 Introduction......Page 13
2 Notation......Page 16
3 Proof......Page 21
4 Remark to the Poincare Conjecture......Page 24
References......Page 25
Abstract......Page 29
2 A Quantum Gauge Model......Page 30
3 Classical Wilson Loop......Page 32
4 A Gauge Fixing Condition and Affine Kac-Moody Algebra......Page 34
5 Quantum Knizhnik-Zamolodchikov Equation in Dual Form......Page 39
6 Solving Quantum KZ Equation in Dual Form......Page 43
7 Computation of Quantum Wilson Lines......Page 44
8 Representing Braiding of Curves by Quantum Wilson Lines......Page 45
10 Defining Quantum Knots and Knot Invariant......Page 47
11 Examples of Quantum Knots......Page 48
12 Generalized Wilson Loops as Quantum Knots......Page 50
13 More Examples of Quantum Knots and Knot Invariant......Page 56
14 A Classification Table of Knots......Page 58
15 Examples of Quantum Links and Link Invariant......Page 74
16 Classification of Links......Page 76
17 Quantum Invariants of 3-manifolds and Classification......Page 80
18 An Investigation of the Proof of Poincare Conjecture......Page 90
19 Counterexamples of the Geometrization Conjecture......Page 93
References......Page 96
1 Introduction......Page 99
2 Proof of Theorem 1.1......Page 101
3 Additive Map Preserving Ranks and Geometry of Matrices......Page 105
References......Page 106
1 Preliminaries......Page 109
2 Motivation......Page 110
3 Exactness of the Expected Value......Page 111
4 Exactness of the Variance......Page 113
5 Error Bounds......Page 114
References......Page 117
1 Introduction......Page 119
2 Problem Formulation and the Scalings......Page 125
3 The Basic and Interaction Equations......Page 127
4 Ill-Posed System......Page 131
5 A Uniformly Valid Composite Approach......Page 132
6 Conclusion......Page 137
References......Page 139
1 Introduction......Page 143
2 Initial Layer Thickness......Page 145
3 Initial Layer Correction Equations......Page 148
4 Initial Layer Correction Functions......Page 149
5 Transcendental Smallness and Initial Layer Functions......Page 153
References......Page 155
1 Introduction......Page 157
2 Generalities......Page 158
3 Arithmetic Functions, Families of Rational Functions and Cardinality......Page 159
4 The Action of the Projective Line Homographies......Page 162
5 Constructions of Optical Orthogonal Codes (OOC)......Page 163
References......Page 167
1 Introduction......Page 169
2 L2-eigenforms of the Pauli-Dirac Operator......Page 171
3 L2-eigenprojector Kernels for the Pauli-Dirac Operator......Page 175
4 Resolvent Kernel for the Pauli-Dirac Operator......Page 176
5 Concluding Remarks......Page 177
References......Page 178
INDEX......Page 181