Introduction to Compact Transformation Groups (eBook)

Front Cover 1
Introduction to Compact Transformation Groups 4
Copyright Page 5
CONTENTS 6
Preface 10
Acknowledgments 14
Chapter 0. Background on Topological Groups and Lie Groups 16
1. Elementary Properties of Topological Groups 16
2. The Classical Groups 20
3. Integration on Compact Groups 26
4. Characteristic Functions on Compact Groups 30
5. Lie Groups 36
6. The Structure of Compact Lie Groups 41
Chapter I. Transformation Groups 47
1. Group Actions 47
2. Equivariant Maps and Isotropy Groups 50
3. Orbits and Orbit Spaces 52
4. Homogeneous Spaces and Orbit Types 55
5. Fixed Points 59
6. Elementary Constructions 61
7. Some Examples of O(n)-Spaces 64
8. Two Further Examples 70
9. Covering Actions 77
Exercises for Chapter I 82
Chapter II. General Theory of G-Spaces 85
1. Fiber Bundles 85
2. Twisted Products and Associated Bundles 87
3. Twisted Products with a Compact Group 94
4. Tubes and Slices 97
5. Existence of Tubes 99
6. Path Lifting 105
7. The Covering Homotopy Theorem 107
8. Conical Orbit Structures 113
9. Classification of G-Spaces 119
10. Linear Embedding of G-Spaces 125
Exercises for Chapter II 127
Chapter III. Homological Theory of Finite Group Actions 129
1. Simplicial Actions 129
2. The Transfer 133
3. Transformations of Prime Period 137
4. Euler Characteristics and Ranks 141
5. Homology Spheres and Disks 144
6. G-Coverings and Cech Theory 147
7. Finite Group Actions on General Spaces 156
8. Groups Acting Freely on Spheres 163
9. Newman's Theorem 169
10. Toral Actions 173
Exercises for Chapter III 181
Chapter IV. Locally Smooth Actions on Manifolds 185
1. Locally Smooth Actions 185
2. Fixed Point Sets of Maps of Prime Period 190
3. Principal Orbits 194
4. The Manifold Part of M* 201
5. Reduction to Finite Principal Isotropy Groups 205
6 . Actions on Sn with One Orbit Type 211
7. Components of B U E 215
8. Actions with Orbits of Codimension 1 or 2 220
9. Actions on Tori 229
10. Finiteness of Number of Orbit Types 233
Exercises for Chapter IV 237
Chapter V. Actions with Few Orbit Types 239
1. The Equivariant Collaring Theorem 239
2. The Complementary Dimension Theorem 245
3. Reduction of Structure Groups 248
4. The Straightening Lemma and the Tube Theorem 253
5. Classification of Actions with Two Orbit Types 261
6. The Second Classification Theorem 268
7. Classification of Self-Equivalences 276
8. Equivariant Plumbing 282
9. Actions on Brieskorn Varieties 287
10. Actions with Three Orbit Types 295
11. Knot Manifolds 302
Exercises for Chapter V 308
Chapter VI. Smooth Actions 311
1. Functional Structures and Smooth Actions 311
2. Tubular Neighborhoods 318
3. Integration of Isotopies 327
4. Equivariant Smooth Embeddings and Approximations 329
5. Functional Structures on Certain Orbit Spaces 334
6. Special G-Manifolds 341
7. Smooth Knot Manifolds 348
8. Groups of Involutions 352
9. Semifree Circle Group Actions 362
10. Representations at Fixed Points 367
11. Refinements Using Real K-Theory 374
Exercises for Chapter VI 381
Chapter VII. Cohomology Structure of Fixed Point Sets 384
1. Preliminaries 384
2. Some Inequalities 390
3. Zp- Actions on Projective Spaces 393
4. Some Examples 403
5. Circle Actions on Projective Spaces 408
6. Actions on Poincaré Duality Spaces 415
7. A Theorem on Involutions 420
8. Involutions on Sn × Sm 425
9. Zp- Actions on Sn × Sm 431
10. Circle Actions on a Product of Odd-Dimensional Spheres 437
11. An Application to Equivariant Maps 440
Exercises for Chapter VII 443
References 447
Author Index 469
Subject Index 472
Pure and Applied Mathematics 475