## Classical Topology and Combinatorial Group TheoryIn 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 |

307 | |

319 | |

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### Common terms and phrases

3-manifolds abelian group algebraic algorithm automorphism boundary paths bouquet of circles braid called canonical curves Cayley diagram closed path combinatorial computation construction corresponding covering surface crosscaps define deformation retraction Dehn denote disc disjoint element embedding endpoints equivalence class example EXERCISE fact finitely presented follows free group function fundamental group genus geometric given graph halting problem Heegaard diagram hence homeomorphism problem homotopy identified infinite cyclic integer intersection isomorphism isotopy knot groups lens space manifold meridian Mobius band neighbourhood nonorientable normal form obtained open set orientable surface oriented edges pair permutation plane Poincare polygon proof prove realized relations result schema Schreier Seifert-Van Kampen theorem Show shown in Figure simple closed curve simple curve simplexes simplicial complex simplicial decomposition solid torus spanning tree sphere surface complex tessellation Tietze topological transformations trefoil knot triangulation Turing machine twist universal covering unsolvable vertex vertices word problem

### Popular passages

Page 310 - A practical method for enumerating cosets of a finite abstract group, Proc.

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).

Page 314 - Fomenko. AT : The problem of the algorithmic discrimination of the standard three-dimensional sphere. Russian Math.