Lecture Notes in Mathematics

By: Klaus Johannson

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Paperback | 21 April 1995

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This book is a study of combinatorial structures of 3-mani- folds, especially Haken 3-manifolds. Specifically, it is concerned with Heegard graphs in Haken 3-manifolds, i.e., with graphs whose complements have a free fundamental group. These graphs always exist. They fix not only a combinatorial stucture but also a presentation for the fundamental group of the underlying 3-manifold. The starting point of the book is the result that the intersection of Heegard graphs with incompressible surfaces, or hierarchies of such surfaces, is very rigid. A number of finiteness results lead up to a ri- gidity theorem for Heegard graphs. The book is intended for graduate students and researchers in low-dimensional topolo- gy as well as combinatorial theory. It is self-contained and requires only a basic knowledge of the theory of 3-manifolds

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Introduction
Standard Graphs and Meridian-Systems	p. 1
The Disc-Complex	p. 10
Homeomorphisms	p. 14
Pseudo-Anosov Homeomorphisms	p. 23
Branching Complexes	p. 24
Heegaard-Diagrams	p. 37
Surfaces	p. 42
Homeomorphisms	p. 59
Duality	p. 64
Presentations for Surfaces	p. 84
The Seifert Characteristic	p. 94
Algorithms for Surfaces	p. 110
The Haken Spectrum	p. 130
Good Pencils	p. 148
The Handle-Addition Lemma (Absolute Form)	p. 152
The Handle-Addition Lemma (Relative Form)	p. 162
Homotopy and Isotopy	p. 184
The Mapping Class Group	p. 212
Incompressibility	p. 217
The Compression-Algorithm	p. 221
The Haken Spectrum	p. 231
An Example	p. 239
The Handle-Addition Lemma (General Case)	p. 247
The Unknotting Theorem	p. 269
Distances for Heegaard Graphs	p. 276
Constructing Heegaard Graphs	p. 289
Haken Complexes and Heegaard Complexes	p. 310
Short Companions for Haken Graphs	p. 333
Invariant Regions for Long Arc-Sequences	p. 359
The Universal Limit for Induced Haken Graphs	p. 376
Finiteness for Heegaard Graphs	p. 389
Rigidity for Heegaard Graphs	p. 401
Reidemeister-Zieschang Problem	p. 423
Appendix. Computing Surfaces	p. 427
References	p. 436
Index	p. 442
Table of Contents provided by Blackwell. All Rights Reserved.

Lecture Notes in Mathematics

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Paperback

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