## Intuitive Combinatorial TopologyTopology is a relatively young and very important branch of mathematics. It studies properties of objects that are preserved by deformations, twistings, and stretchings, but not tearing. This book deals with the topology of curves and surfaces as well as with the fundamental concepts of homotopy and homology, and does this in a lively and well-motivated way. There is hardly an area of mathematics that does not make use of topological results and concepts. The importance of topological methods for different areas of physics is also beyond doubt. They are used in field theory and general relativity, in the physics of low temperatures, and in modern quantum theory. The book is well suited not only as preparation for students who plan to take a course in algebraic topology but also for advanced undergraduates or beginning graduates interested in finding out what topology is all about. The book has more than 200 problems, many examples, and over 200 illustrations. |

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### Contents

Topology of Curves | 3 |

12 What Is Topology Concerned With? | 6 |

13 The Simplest Topological Invariants | 10 |

14 The Euler Characteristic of a Graph | 13 |

15 Intersection Index | 17 |

16 The Jordan Curve Theorem | 21 |

17 What Is a Curve? | 24 |

18 Peano Curves | 30 |

211 Linking Numbers | 78 |

Homotopy and Homology | 83 |

32 The Fundamental Group | 85 |

33 Cell Decompositions and Polyhedra | 89 |

34 Coverings | 93 |

35 The Degree of a Mapping and the Fundamental Theorem of Algebra | 97 |

36 Knot Groups | 101 |

37 Cycles and Homology | 106 |

Topology of Surfaces | 33 |

22 Surfaces | 35 |

23 The Euler Characteristic of a Surface | 40 |

24 Classification of Closed Orientable Surfaces | 44 |

25 Classification of Closed Nonorientable Surfaces | 50 |

26 Vector Fields on Surfaces | 57 |

27 The Four Color Problem | 62 |

28 Coloring Maps on Surfaces | 64 |

29 Wild Spheres | 68 |

210 Knots | 72 |

38 Topological Products | 116 |

39 Fiber Bundles | 119 |

310 Morse Theory | 123 |

Topological Objects in Nematic Liquid Crystals VP Mineev | 129 |

AI Nematics | 130 |

A 3 Disclination and Topology | 133 |

A4 Singular Points | 136 |

139 | |

141 | |

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### Common terms and phrases

Algebra associate boundary called cell decomposition changes circle closed curve colors common complete compute connected connected graph Consider consists construct contains continuous contour contractible corresponding countries covering curve cycle defined deformation denote determined direction disclination disk divides draw edges embedded equal equation Example faces fiber Figure finite follows four function fundamental group given gluing graph handles Hence holes homeomorphic integral interior intersection index join knot least liquid mapping means membrane Mobius strip move nematic obtain opposite oriented parallel path piece plane polygons polyhedron positive possible Problems projective plane properties prove regions remains remove represented respectively resulting segment Show sides simple singular points space spanning sphere square surface Q theorem Theory topological topological product torus trail traversed tree vector field vertex vertices yields

### Popular passages

Page xi - In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain.