# Classical Topology and Combinatorial Group Theory

Springer Science & Business Media, Mar 25, 1993 - Mathematics - 334 pages
In recent years, many students have been introduced to topology in high school mathematics. Having met the Mobius band, the seven bridges of Konigsberg, Euler's polyhedron formula, and knots, the student is led to expect that these picturesque ideas will come to full flower in university topology courses. What a disappointment "undergraduate topology" proves to be! In most institutions it is either a service course for analysts, on abstract spaces, or else an introduction to homological algebra in which the only geometric activity is the completion of commutative diagrams. Pictures are kept to a minimum, and at the end the student still does nr~ understand the simplest topological facts, such as the rcason why knots exist. In my opinion, a well-balanced introduction to topology should stress its intuitive geometric aspect, while admitting the legitimate interest that analysts and algebraists have in the subject. At any rate, this is the aim of the present book. In support of this view, I have followed the historical development where practicable, since it clearly shows the influence of geometric thought at all stages. This is not to claim that topology received its main impetus from geometric recreations like the seven bridges; rather, it resulted from the l'isualization of problems from other parts of mathematics-complex analysis (Riemann), mechanics (Poincare), and group theory (Dehn). It is these connec tions to other parts of mathematics which make topology an important as well as a beautiful subject.

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### Contents

 01 The Fundamental Concepts and Problems of Topology 2 02 Simplicial Complexes 19 03 The Jordan Curve Theorem 26 04 Algorithms 36 05 Combinatorial Group Theory 40 Complex Analysis and Surface Topology 53 11 Riemann Surfaces 54 12 Nonorientable Surfaces 62
 52 The Structure Theorem for Finitely Generated Abelian Groups 175 53 Abelianization 181 Curves on Surfaces 185 61 Dehns Algorithm 186 62 Simple Curves on Surfaces 190 63 Simplification of Simple Curves by Homeomorphisms 196 64 The Mapping Class Group of the Torus 206 Knots and Braids 217

 13 The Classification Theorem for Surfaces 69 14 Covering Surfaces 80 Graphs and Free Groups 89 21 Realization of Free Groups by Graphs 90 22 Realization of Subgroups 99 Foundations for the Fundamental Group 109 31 The Fundamental Group 110 32 The Fundamental Group of the Circle 116 33 Deformation Retracts 121 34 The SeifertVan Kampen Theorem 124 35 Direct Products 132 Fundamental Groups of Complexes 135 41 Poincares Method for Computing Presentations 136 42 Examples 141 43 Surface Complexes and Subgroup Theorems 156 Homology Theory and Abelianization 169 51 Homology Theory 170
 71 Dehn and Schreiers Analysis of the Torus Knot Groups 218 72 Cyclic Coverings 225 73 Braids 233 ThreeDimensional Manifolds 241 81 Open Problems in ThreeDimensional Topology 242 82 Polyhedral Schemata 248 83 Heegaard Splittings 252 84 Surgery 263 85 Branched Coverings 270 Unsolvable Problems 275 91 Computation 276 92 HNN Extensions 285 93 Unsolvable Problems in Group Theory 290 94 The Homeomorphism Problem 298 Bibliography and Chronology 307 Index 319 Copyright

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Page iii - I have followed the historical development where practicable, since it clearly shows the influence of geometric thought at all stages. This is not to claim that topology received its main impetus from geometric recreations like the seven bridges; rather, it resulted from the visualization of problems from other parts of mathematics -complex analysis (Riemann). mechanics (Poincare), and group theory (Dehn).
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