文档介绍：The Topology of Fiber Bundles
Lecture Notes
Ralph L. Cohen
Dept. of Mathematics
Stanford University
Contents
Introduction v
Chapter 1. Locally Trival Fibrations 1
1. Definitions and examples 1
. Vector Bundles 3
. Lie Groups and Principal Bundles 7
. Clutching Functions and Structure Groups 15
2. Pull Backs and Bundle Algebra 21
. Pull Backs 21
. The tangent bundle of Projective Space 24
. K - theory 25
. Differential Forms 30
. Connections and Curvature 33
. The Levi - Civita Connection 39
Chapter 2. Classification of Bundles 45
1. The homotopy invariance of fiber bundles 45
2. Universal bundles and classifying spaces 50
3. Classifying Gauge Groups 60
4. Existence of universal bundles: the Milnor join construction and the simplicial classifying
space 63
. The join construction 63
. Simplicial spaces and classifying spaces 66
5. Some Applications 72
. Line bundles over projective spaces 73
. Structures on bundles and homotopy liftings 74
. Embedded bundles and K -theory 77
. Representations and flat connections 78
Chapter 3. Characteristic Classes 81
1. Preliminaries 81
2. Chern Classes and Stiefel - Whitney Classes 85
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iv CONTENTS
. The Thom Isomorphism Theorem 88
. The Gysin sequence 94
. Proof of theorem 95
3. The product formula and the splitting principle 97
4. Applications 102
. Characteristic classes of manifolds 102
. Normal bundles and immersions 105
5. Pontrjagin Classes 108
. Orientations plex Conjugates 109
. Pontrjagin classes 111
. Oriented characteristic classes 114
6. Connections, Curvature, and Characteristic Classes 115
Chapter 4. Homotopy Theory of Fibrations 125
1. Homotopy Groups 125
2. Fibrations 130
3. Obstruction Theory 135
4. Eilenberg - MacLane Spaces 140
. Obstruction theory and the existence of Eilenberg - MacLane spaces 140
. The Hopf - Whitney theorem and the