## Topology of SurfacesSpringer Science & Business Media, 26. sep. 1997 - 281 sider " . . . that famous pedagogical method whereby one begins with the general and proceeds to the particular only after the student is too confused to understand even that anymore. " Michael Spivak This text was written as an antidote to topology courses such as Spivak It is meant to provide the student with an experience in geomet describes. ric topology. Traditionally, the only topology an undergraduate might see is point-set topology at a fairly abstract level. The next course the average stu dent would take would be a graduate course in algebraic topology, and such courses are commonly very homological in nature, providing quick access to current research, but not developing any intuition or geometric sense. I have tried in this text to provide the undergraduate with a pragmatic introduction to the field, including a sampling from point-set, geometric, and algebraic topology, and trying not to include anything that the student cannot immediately experience. The exercises are to be considered as an in tegral part of the text and, ideally, should be addressed when they are met, rather than at the end of a block of material. Many of them are quite easy and are intended to give the student practice working with the definitions and digesting the current topic before proceeding. The appendix provides a brief survey of the group theory needed. |

### Indhold

Introduction to topology | 1 |

Pointset topology in Rⁿ | 7 |

22 Relative neighborhoods | 16 |

23 Continuity | 19 |

24 Compact sets | 26 |

25 Connected sets | 29 |

26 Applications | 31 |

Pointset topology | 37 |

61 The algebra of chains | 126 |

62 Simplicial Complexes | 135 |

63 Homology | 139 |

64 More computations | 151 |

65 Betti numbers and the euler characteristic | 154 |

Cellular functions | 157 |

72 Homology and cellular functions | 165 |

73 Examples | 170 |

32 Continuity connectedness and compactness | 43 |

33 Separation axioms | 47 |

34 Product spaces | 48 |

35 Quotient spaces | 52 |

Surfaces | 56 |

42 Cell complexes | 60 |

43 Surfaces | 67 |

44 Triangulations | 75 |

45 Classification of surfaces | 81 |

46 Surfaces with boundaryn | 91 |

The euler characteristic | 94 |

52 Graphs and trees | 95 |

53 The euler characteristic and the sphere | 101 |

54 The euler characteristic and surfaces | 108 |

55 Mapcoloring problems | 113 |

56 Graphs revisited | 119 |

Homology | 125 |

74 Covering spaces | 175 |

Invariance of homology | 183 |

82 The Simplicial Approximation Theorem | 187 |

Homotopy | 197 |

92 The fundamental group | 203 |

Miscellany | 212 |

102 The Jordan Curve Theorem | 221 |

103 3Manifolds | 226 |

Topology and calculus | 235 |

112 Differentiable manifolds | 248 |

113 Vector fields on manifolds | 250 |

114 Integration on manifolds | 257 |

Groups | 263 |

References | 272 |

274 | |

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### Almindelige termer og sætninger

1-cycle 2-complex 3-manifold abelian group Algebra antipodal points barycentric subdivision C₁ Calculus cell complex cellular function chain Chapter Ck(K Cl(A closed sets colors compact connected sum considered continuous function Corollary cube curve cylinder defined Definition deformation retract denote differential dimensions disc elements endpoints equivalence relation euler characteristic example Exercise faces finite number fixed point function f geometric glued graph H₁(K Ho(K homeomorphic homology groups homotopy identified illustrated in Figure induces integer interior interval inverse k-cell k-chain Klein bottle Lemma limit point loop manifolds Mathematics Möbius band neighborhood Note number of edges number of vertices open sets orientable paths pictured in Figure planar diagram polygon projective plane Proof Prove Theorem rectangle regular complex sequence simplex simplicial approximation simplicial complex simplicial function subset topological space topologically equivalent torus triangles v₁ vector field vertex მყ