3-manifolds

John Hempel0691081832, 9780691081830, 0691081786, 0691081832-pbk

For many years, John Hempel’s book has been a standard text on the topology of 3-manifolds. Even though the field has grown tremendously during that time, the book remains one of the best and most popular introductions to the subject.
The theme of this book is the role of the fundamental group in determining the topology of a given 3-manifold. The essential ideas and techniques are covered in the first part of the book: Heegaard splittings, connected sums, the loop and sphere theorems, incompressible surfaces, free groups, and so on. Along the way, many useful and insightful results are proved, usually in full detail. Later chapters address more advanced topics, including Waldhausen’s theorem on a class of 3-manifolds that is completely determined by its fundamental group. The book concludes with a list of problems that were unsolved at the time of publication.
Hempel’s book remains an ideal text to learn about the world of 3-manifolds. The prerequisites are few and are typical of a beginning graduate student. Exercises occur throughout the text.
Other key books on low-dimensional topology available from the AMS are Knots and Links, Lectures on Three-Manifold Topology, and The Knot Book.

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Table of contents :
Contents……Page 6
Preface……Page 3
1 Preliminaries……Page 9
Definitions……Page 10
Basic Theorems……Page 12
Regular Neighborhoods……Page 13
General Position……Page 14
2 Heegaard Splittings……Page 20
Cubes with Handles……Page 21
Splittings and Diagrams……Page 23
Genus One Splittings……Page 26
3 Connected Sums……Page 30
Primes……Page 33
Existence of Factorizations……Page 35
Uniqueness of Factorizations……Page 38
4 The loop and sphere theorems……Page 45
Double Curve Surgery……Page 47
Proof of the Loop Theorem……Page 53
Proof of the Sphere Theorem……Page 56
The Projective Plane Theorem……Page 60
5 Free groups……Page 62
6 Incompressible Surfaces……Page 64
7 Kneser’s Conjecture on Free Products……Page 72
8 Finitely Generated Subgroups……Page 75
Group Homology……Page 81
Finite Groups: The Nonorientable Case……Page 82
Subgroups with Higher Homology……Page 86
Abelian Groups……Page 90
10 I-Bundles……Page 94
Products……Page 95
Twisted Bundles……Page 97
Surface Subgroups of Finite Index……Page 104
11 Group Extensions and Fibrations……Page 106
Algebraic Preliminaries……Page 107
Bundles……Page 109
Proof of Theorem 11.1……Page 116
12 Seifert fibered spaces……Page 121
Fuchsian Groups……Page 124
Bundles with Period Structure Groups……Page 127
Cyclic Normal Subgroups……Page 131
Centers……Page 135
Cyclic Actions on S^1 x S^1 x S^1……Page 136
13 Classification of P^2-irreducible, Sufficiently Large 3-Manifolds……Page 140
The Analogue for Surfaces……Page 141
Hierarchies……Page 144
Classification Theorems……Page 147
Peripheral Systems……Page 153
Remarks and Examples……Page 154
14 Some Approaches to the Poincaré Conjecture……Page 158
Contractible Open 3-Manifolds……Page 159
A Characterization of S^3……Page 161
Splitting Homomorphisms……Page 162
The Mapping Class Group……Page 166
Involutions on Homotopy 3-Spheres……Page 168
The Fundamental Groups……Page 173
Peripheral Systems……Page 176
Hopficity……Page 179
Residual Finiteness……Page 180
References……Page 189
Index……Page 196
Symbols and notation……Page 198