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El. knyga: Fiber Bundles And Homotopy

  • Formatas: 336 pages
  • Išleidimo metai: 17-May-2021
  • Leidėjas: World Scientific Publishing Co Pte Ltd
  • Kalba: eng
  • ISBN-13: 9789811238109
Kitos knygos pagal šią temą:
Fiber Bundles And HomotopyDidesnis paveikslėlis
  • Formatas: 336 pages
  • Išleidimo metai: 17-May-2021
  • Leidėjas: World Scientific Publishing Co Pte Ltd
  • Kalba: eng
  • ISBN-13: 9789811238109
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This book is an introduction to fiber bundles and fibrations. But the ultimate goal is to make the reader feel comfortable with basic ideas in homotopy theory. The author found that the classification of principal fiber bundles is an ideal motivation for this purpose. The notion of homotopy appears naturally in the classification. Basic tools in homotopy theory such as homotopy groups and their long exact sequence need to be introduced. Furthermore, the notion of fibrations, which is one of three important classes of maps in homotopy theory, can be obtained by extracting the most essential properties of fiber bundles. The book begins with elementary examples and then gradually introduces abstract definitions when necessary. The reader is assumed to be familiar with point-set topology, but it is the only requirement for this book.
Preface v
Acknowledgments vii
List of Figures
xi
1 How to Bundle Fibers
1(6)
1.1 Simple Examples
1(3)
1.2 Tangent Bundles
4(3)
2 Covering Spaces as a Toy Model
7(26)
2.1 Covering Spaces
7(3)
2.2 Paths and Their Lifts
10(8)
2.3 The Fundamental Group
18(9)
2.4 Universal Covering
27(6)
3 Basic Properties of Fiber Bundles
33(66)
3.1 Defining Fiber Bundles
33(5)
3.2 Fiber Bundles with Structure Groups
38(6)
3.3 Topological Groups
44(6)
3.4 Compact-Open Topology
50(7)
3.5 Fiber Bundles and Group Action
57(10)
3.6 Quotient Spaces by Group Actions
67(19)
3.7 Principal Bundles
86(13)
4 Classification of Fiber Bundles
99(100)
4.1 Maps between Fiber Bundles
99(9)
4.2 Pullbacks
108(6)
4.3 Fiber Bundles and Homotopy
114(12)
4.4 Classification of Fiber Bundles: Simple Cases
126(6)
4.5 Classifying Fiber Bundles over CW Complexes
132(9)
4.6 CW Complexes and Homotopy
141(14)
4.7 The First Half of the Proof of the Classification Theorem
155(5)
4.8 Fiber Bundles and Homotopy Groups
160(8)
4.9 Construction of Universal Bundles: Steenrod's Approach
168(9)
4.10 Construction of Universal Bundles: The Bar Construction
177(14)
4.11 Covering Spaces Revisited
191(8)
5 Fibrations
199(82)
5.1 Why Further Generalizations of Fiber Bundles?
199(1)
5.2 Serre Fibrations and Hurewicz Fibrations
200(5)
5.3 Loop Spaces
205(8)
5.4 Comparing Fiber Bundles and Fibrations
213(11)
5.5 Deforming Continuous Maps into Fibrations
224(6)
5.6 Homotopy Fiber Sequences
230(6)
5.7 Iterated Loop Spaces
236(7)
5.8 Fibrations and Homotopy Groups
243(12)
5.9 Cofibrations
255(4)
5.10 Duality between Fibrations and Cofibrations
259(14)
5.11 Quasifibrations
273(8)
6 Postscript
281(10)
6.1 What is Homotopy Theory?
281(3)
6.2 Many Kinds of Homotopies
284(1)
6.3 Framework of Homotopy Theory
285(6)
Appendix A Related Topics
291(22)
A.1 The Meaning of Compact-Open Topology
291(4)
A.2 Vector Bundles
295(9)
A.3 Simplicial Techniques
304(9)
Bibliography 313(8)
Index 321
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