## 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. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

Topology of Curves | 1 |

12 What Is Topology Concerned With? | 4 |

13 The Simplest Topological Invariants | 8 |

14 The Euler Characteristic of a Graph | 11 |

15 Intersection Index | 15 |

16 The Jordan Curve Theorem | 19 |

17 What Is a Curve? | 22 |

18 Peano Curves | 28 |

Homotopy and Homology | 81 |

32 The Fundamental Group | 83 |

33 Cell Decompositions and Polyhedra | 87 |

34 Coverings | 91 |

35 The Degree of a Mapping and the Fundamental Theorem of Algebra | 95 |

36 Knot Groups | 99 |

37 Cycles and Homology | 104 |

38 Topological Products | 114 |

Topology of Surfaces | 31 |

22 Surfaces | 33 |

23 The Euler Characteristic of a Surface | 38 |

24 Classification of Closed Orientable Surfaces | 42 |

25 Classification of Closed Nonorientable Surfaces | 48 |

26 Vector Fields on Surfaces | 55 |

27 The Four Color Problem | 60 |

28 Coloring Maps on Surfaces | 62 |

29 Wild Spheres | 66 |

210 Knots | 70 |

211 Linking Numbers | 76 |

39 Fiber Bundles | 117 |

310 Morse Theory | 121 |

Topological Objects in Nematic Liquid Crystals VP Mineev | 127 |

AI Nematics | 128 |

A 3 Disclination and Topology | 131 |

A4 Singular Points | 134 |

A5 What Else Is There? | 136 |

137 | |

139 | |

### Other editions - View all

### Common terms and phrases

1-cycles abelian Betti numbers called cell decomposition chromatic number circle closed surface complement space compute connected graph Consider continuous deformation contours of type curvilinear image defined degree diametrically opposite points direction disclination disk edge path embedded endpoints equation Euler characteristic Example exterior fiber bundle finite number follows function fundamental group glue gluing graph G handles holes homologous to zero homology groups homotopic integral cycle interior intersection index isotopic Jordan curve theorem knot Let Q linking number midline minimum point Mobius strip modulo multivalued function number of edges number of points number of vertices obtain a surface one-dimensional parallel Peano curve point XQ polyhedra polyhedron Problems projective plane prove regions saddle point segment self-intersections Show shown in Figure simple closed curve singular points spanning tree sphere square stationary points surface homeomorphic surface Q surface with boundary tangent three-space topological invariant topological product torus traversed vector field vertex yields zero-dimensional