## History of TopologyTopology, for many years, has been one of the most exciting and influential fields of research in modern mathematics. Although its origins may be traced back several hundred years, it was Poincaré who "gave topology wings" in a classic series of articles published around the turn of the century. While the earlier history, sometimes called the prehistory, is also considered, this volume is mainly concerned with the more recent history of topology, from Poincaré onwards. As will be seen from the list of contents the articles cover a wide range of topics. Some are more technical than others, but the reader without a great deal of technical knowledge should still find most of the articles accessible. Some are written by professional historians of mathematics, others by historically-minded mathematicians, who tend to have a different viewpoint. |

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### Contents

1 | |

25 | |

65 | |

103 | |

Chapter 5 Differential forms | 111 |

Chapter 6 The topological work of Henri Poincaré | 123 |

Chapter 7 Weyl and the topology of continuous groups | 169 |

Some remarks on the interaction of general Topology With Other Areas of Mathematics | 199 |

Origins to 1953 | 631 |

Chapter 24 Stable algebraic topology 19451966 | 665 |

Chapter 25 A history of duality in algebraic topology | 725 |

Chapter 26 A short history of Hspaces | 747 |

Chapter 27 A history of rational homotopy theory | 757 |

Chapter 28 History of homological algebra | 797 |

Chapter 29 Topologists at conferences | 837 |

Chapter 30 Topologists in Hitlers Germany | 849 |

Chapter 9 Absolute neighborhood retracts and shape theory | 241 |

Chapter 10 Fixed point theory | 271 |

Chapter 11 Geometric aspects in the development of knot theory | 301 |

Chapter 12 Topology and physics a historical essay | 359 |

Chapter 13 Singularities | 417 |

Chapter 14 One hundred years of manifold topology | 435 |

Chapter 15 3dimensional topology up to 1960 | 449 |

Chapter 16 A short history of triangulation and related matters | 491 |

Chapter 17 Graph theory | 503 |

Chapter 18 The early development of algebraic topology | 531 |

Chapter 19 From combinatorial topology to algebraic topology | 561 |

Chapter 20 p3S2 H Hopf WK Clifford F Klein | 575 |

Chapter 21 A history of cohomology theory | 579 |

Chapter 22 Fibre bundles fibre maps | 605 |

Chapter 31 The Japanese school of topology | 863 |

Chapter 32 Some topologists | 883 |

Chapter 33 Johann Benedikt Listing | 909 |

Chapter 34 Poul Heegaard | 925 |

Chapter 35 Luitzen Egbertus Jan Brouwer | 947 |

Chapter 36 Max Dehn | 965 |

Chapter 37 Jakob Nielsen and his contributions to topology | 979 |

Chapter 38 Heinz Hopf | 991 |

Chapter 39 Hans Freudenthal | 1009 |

Chapter 40 Herbert Seifert 19071996 | 1021 |

Appendix Some dates | 1029 |

1033 | |

### Common terms and phrases

3-manifolds Abelian Alexander algebraic topology Amer Analysis Situs Atiyah axioms Betti numbers boundary Brouwer bundle C. R. Acad Cartan closed curve cobordism coefficients cohomology theory combinatorial compact complex compute concept conjecture construction cup products CW-complexes defined definition Dehn developed differential dimension dimensional duality Eilenberg equations espaces example fibration fibre finite fixed point function fundamental group geometry given graph H-space Halperin Heegaard Hilbert homeomorphic homology homotopy equivalence homotopy groups homotopy type Hopf Hurewicz integral intersection invariant isomorphism J.H.C. Whitehead K-theory knot theory Lefschetz Lie algebra Lie groups linear loop manifolds map f Math mathematical mathematicians metric Milnor n-dimensional notion obtained orientable paper Poincaré Poincaré duality polyhedron polynomial problem Proc proof proved rational homotopy theory relation result Seifert Serre showed simplicial simply connected singular space spectral sequence sphere Steenrod structure theorem Thom torsion transformations triangulation Weyl Whitehead

### Popular passages

Page 13 - — if to divide a continuum it suffices to consider as cuts a certain number of elements all distinguishable from one another, we say that this continuum is of one dimension ; if, on the contrary, to divide a continuum it is necessary to consider as cuts a system of elements themselves forming one or several continua, we shall say that this continuum is of several dimensions. "If to divide a continuum C, cuts which form one or several continua of one dimension suffice, we shall say that C is a...