## Topology, Geometry, and Gauge Fields: FoundationsMathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the clas sical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics (TAM). The development ofnewcourses is a natural consequence of a high levelof excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied mathe matical Sciences (AMS) series, which will focus on advanced textbooks and research level monographs. Preface In Egypt, geometry was created to measure the land. Similar motivations, on a somewhat larger scale, led Gauss to the intrinsic differential geometry of surfaces in space. Newton created the calculus to study the motion of physical objects (apples, planets, etc.) and Poincare was similarly impelled toward his deep and far-reaching topological view of dynamical systems. |

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### Contents

Physical and Geometrical Motivation | 1 |

03 The Hopf Bundle | 11 |

04 Connections on Principal Bundles | 20 |

05 NonAbelian Gauge Fields and Topology | 23 |

Topological Spaces | 27 |

12 Quotient Topologies and Projective Spaces | 48 |

13 Products and Local Products | 57 |

14 Compactness Conditions | 70 |

43 Smooth Maps on Manifolds | 193 |

44 Tangent Vectors and Derivatives | 198 |

45 Submanifolds | 207 |

46 Vector Fields and 1Forms | 215 |

47 Matrix Lie Groups | 229 |

48 VectorValued 1Forms | 245 |

49 Orientability | 263 |

410 2Forms and Riemannian Metrics | 267 |

15 Connectivity and Covering Spaces | 76 |

16 Topological Groups and Group Actions | 87 |

Homotopy Groups | 101 |

22 Path Homotopy and the Fundamental Group | 102 |

23 Contractible and Simply Connected Spaces | 112 |

24 The Covering Homotopy Theorem | 127 |

25 Higher Homotopy Groups | 142 |

Principal Bundles | 165 |

32 Transition Functions | 167 |

33 Bundle Maps and Equivalence | 169 |

34 Principal GBundles Over Spheres | 177 |

Differentiable Manifolds and Matrix Lie Groups | 185 |

42 Differentiable Manifolds | 189 |

Gauge Fields and Instantons | 289 |

52 Curvature and Gauge Fields | 307 |

53 The YangMills Functional | 314 |

54 The Hodge Dual for 2Forms in Dimension Four | 324 |

55 The Moduli Space | 333 |

Motivation | 345 |

57 Associated Fiber Bundles | 348 |

58 Matter Fields and Their Covariant Derivatives | 353 |

SU2 and SO3 | 365 |

References | 377 |

Symbols | 381 |

387 | |

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