Classical Topology and Combinatorial Group Theory / Edition 2

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This is a well-balanced introduction to topology that stresses geometric aspects. Focusing on historical background and visual interpretation of results, it emphasizes spaces with few dimensions, where visualization is possible, and interaction with combinatorial group theory via the fundamental group. It also present algorithms for topological problems. Most of the results and proofs are known, but some have been simplified or placed in a new perspective. Over 300 illustrations, many interesting exercises, and challenging open problems are included. New in this edition is a chapter on unsolvable problems, which includes the first textbook proof that the main problem of topology, the homeomorphism problem, is unsolvable.
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Editorial Reviews

Focusing on historical background and visual interpretation of results, the author emphasizes spaces with few dimensions, where visualization is possible, and interaction with combinatorial group theory via the fundamental group. Algorithms for topological problems are also presented. Most of the results and proofs are known, but some have been simplified or placed in a new perspective. Exercises and many illustrations are included. This second edition contains a new chapter on unsolvable problems. Annotation c. Book News, Inc., Portland, OR (
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Product Details

  • ISBN-13: 9780387979700
  • Publisher: Springer New York
  • Publication date: 3/25/1993
  • Series: Graduate Texts in Mathematics Series , #72
  • Edition description: 2nd ed. 1993. Corr. 2nd printing 1995
  • Edition number: 2
  • Pages: 334
  • Product dimensions: 0.88 (w) x 6.14 (h) x 9.21 (d)

Table of Contents

Preface to the First Edition
Preface to the Second Edition
Ch. 0 Introduction and Foundations 1
0.1 The Fundamental Concepts and Problems of Topology 2
0.2 Simplicial Complexes 19
0.3 The Jordan Curve Theorem 26
0.4 Algorithms 36
0.5 Combinatorial Group Theory 40
Ch. 1 Complex Analysis and Surface Topology 53
1.1 Riemann Surfaces 54
1.2 Nonorientable Surfaces 62
1.3 The Classification Theorem for Surfaces 69
1.4 Covering Surfaces 80
Ch. 2 Graphs and Free Groups 89
2.1 Realization of Free Groups by Graphs 90
2.2 Realization of Subgroups 99
Ch. 3 Foundations for the Fundamental Group 109
3.1 The Fundamental Group 110
3.2 The Fundamental Group of the Circle 116
3.3 Deformation Retracts 121
3.4 The Seifert-Van Kampen Theorem 124
3.5 Direct Products 132
Ch. 4 Fundamental Groups of Complexes 135
4.1 Poincare's Method for Computing Presentations 136
4.2 Examples 141
4.3 Surface Complexes and Subgroup Theorems 156
Ch. 5 Homology Theory and Abelianization 169
5.1 Homology Theory 170
5.2 The Structure Theorem for Finitely Generated Abelian Groups 175
5.3 Abelianization 181
Ch. 6 Curves on Surfaces 185
6.1 Dehn's Algorithm 186
6.2 Simple Curves on Surfaces 190
6.3 Simplification of Simple Curves by Homeomorphisms 196
6.4 The Mapping Class Group of the Torus 206
Ch. 7 Knots and Braids 217
7.1 Dehn and Schreier's Analysis of the Torus Knot Groups 218
7.2 Cyclic Coverings 225
7.3 Braids 233
Ch. 8 Three-Dimensional Manifolds 241
8.1 Open Problems in Three-Dimensional Topology 242
8.2 Polyhedral Schemata 248
8.3 Heegaard Splittings 252
8.4 Surgery 263
8.5 Branched Coverings 270
Ch. 9 Unsolvable Problems 275
9.1 Computation 276
9.2 HNN Extensions 285
9.3 Unsolvable Problems in Group Theory 290
9.4 The Homeomorphism Problem 298
Bibliography and Chronology 307
Index 319
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