Introduction to Compact Transformation Groups

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