## A Combinatorial Introduction to TopologyThe creation of algebraic topology is a major accomplishment of 20th-century mathematics. The goal of this book is to show how geometric and algebraic ideas met and grew together into an important branch of mathematics in the recent past. The book also conveys the fun and adventure that can be part of a mathematical investigation. |

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### Contents

2 Continuous Transformations in the Plane | 11 |

Abstract Point Set Topology | 28 |

6 Sperners Lemma and the Brouwer Fixed Point Theorem | 36 |

7 Phase Portraits and the lndex Lemma | 43 |

8 Winding Numbers | 50 |

9 lsolated Critical Points | 54 |

10 The Poincare lndex Theorem | 60 |

11 Closed lntegral Paths | 67 |

Homology of Complexes 23 Complexes | 132 |

24 Homology Groups of a Complex | 143 |

25 lnvariance | 153 |

26 Betti Numbers and the Euler Characteristic | 159 |

27 Map Coloring and Regular Complexes | 169 |

28 Gradient Vector Fields | 177 |

29 lntegral Homology | 185 |

30 Torsion and Orientability | 192 |

12 Further Results and Applications | 73 |

Chapter Three Plane Homology and the Jordan Curve Theorem 13 Polygonal Chains | 79 |

14 The Algebra of Chains on a Grating | 84 |

15 The Boundary Operator | 88 |

16 The Fundamental Lemma | 91 |

17 Alexanders Lemma | 97 |

18 Proof of the Jordan Curve Theorem | 100 |

Chapter Five | 103 |

Chapter Four Surfaces 19 Examples of Surfaces | 104 |

20 The Combinatorial Definition of a Surface | 116 |

21 The Classification Theorem | 122 |

22 Surfaces with Boundary | 129 |

31 The Poincare lndex Theorem Again | 200 |

Chapter Six Continuous Transformations 32 Covering Spaces | 209 |

33 Simplicial Transformations | 221 |

34 lnvariance Again | 228 |

35 Matrixes | 234 |

36 The Lefschetz Fixed Point Theorem | 242 |

37 Homotopy | 251 |

38 Other Homologies | 259 |

Topics in Point Set Topology | 265 |

41 Compactness Again | 279 |

Suggestions for Further Reading | 302 |