INTRODUCTION TO COMPACT TRANSFORMATION GROUPS This is Volume 46 in PURE AND APPLIED MATHEMATICS A series of Monographs and Textbooks Editors: PAULA. SMITHAND SAMUELEILENBERG A complete list of titles in this series appears at the end of this volume INTRODUCTION TO COMPACT TRANSFORMATION GROUPS GLEN E. BREDON Department of Mathematics Rutgers University New Brunswick, New Jersey @ 1972 ACADEMIC PRESS New York and London COPYRIGHT 0 1972, BY ACADEMICPRESS, INC. ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRIlTEN PERMISSION FROM THE PUBLISHERS. ACADEMIC PRESS INC. 111 Fifth Avenue, New qork, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NW1 LIBRARYOF CONGRESSCATALOG CARD NUMBER: 75-180794 AMS(M0S) 1970 Subject Classification: 57E10 PRINTED IN THE UNITED STATES OF AMERICA CONTENTS PREFACE ................................ ix ACKNOWLEDGMENTS........................... xiii Chapter 0 Background on Topological Groups and Lie Groups 1. Elementary Properties of Topological Groups ........... 1 2 . The Classical Groups ...................... 5 3 . Integration on Compact Groups ................. 11 4 . Characteristic Functions on Compact Groups ........... 15 5. Lie Groups ........................... 21 6 . The Structure of Compact Lie Groups .............. 26 Chapter I Transformation Groups 1 . Group Actions ......................... 32 2 . Equivariant Maps and Isotropy Groups .............. 35 3 . Orbits and Orbit Spaces ..................... 37 4 . Homogeneous Spaces and Orbit Types .............. 40 5 . Fixed Points .......................... 44 6. Elementary Constructions ..................... 46 7. Some Examples of O(n)-Spaces .................. 49 8 . Two Further Examples ..................... 55 9 . Covering Actions ........................ 62 Exercises for Chapter I ..................... 67 Chapter II General Theory of G-Spaces 1. Fiber Bundles .......................... 70 2 . Twisted Products and Associated Bundles ............. 72 3 . Twisted Products with a Compact Group ............. 79 4. Tubes and Slices ........................ 82 5 . Existence of Tubes ....................... 84 6 . Path Lifting .......................... 90 V vi CONTENTS 7. The Covering Homotopy Theorem ................ 92 8 . Conical Orbit Structures ..................... 98 9. Classification of G-Spaces .................... 104 10. Linear Embedding of G-Spaces .................. 110 Exercises for Chapter I1 ..................... 112 Chapter 111 Homological Theory of Finite Group Actions 1. Simplicia1 Actions ........................ 114 2. The Transfer .......................... 118 3. Transformations of Prime Period ................. 122 4. Euler Characteristics and Ranks ................. 126 5. Homology Spheres and Disks ................... 129 6. G-Coverings and cech Theory .................. 132 7. Finite Group Actions on General Spaces ............. 141 8. Groups Acting Freely on Spheres ................ 148 9. Newman's Theorem ....................... 154 10. Toral Actions .......................... 158 Exercises for Chapter I11 .................... 166 Chapter IV Locally Smooth Actions on Manifolds 1. Locally Smooth Actions ..................... 170 2. Fixed Point Sets of Maps of Prime Period ............ 175 3. Principal Orbits ......................... 179 4. The Manifold Part of M* .................... 186 5 . Reduction to Finite Principal Isotropy Groups .......... 190 6. Actions on S" with One Orbit Type ............... 196 7 . Components of B v E ...................... 200 8. Actions with Orbits of Codimension 1 or 2 ............ 205 9. Actions on Tori ........................ 214 10. Finiteness of Number of Orbit Types ............... 218 Exercises for Chapter IV .................... 222 Chapter V Actions with Few Orbit Types 1. The Equivariant Collaring Theorem ................ 224 2. The Complementary Dimension Theorem ............. 230 3. Reduction of Structure Groups .................. 233 4 . The Straightening Lemma and the Tube Theorem ......... 238 5 . Classification of Actions with Two Orbit Types .......... 246 6. The Second Classification Theorem ................ 253 7. Classification of Self-Equivalences ................. 261 8. Equivariant Plumbing ...................... 267 CONTENTS vii 9. Actions on Brieskorn Varieties .................. 272 10. Actions with Three Orbit Types ................. 280 11. Knot Manifolds ......................... 287 Exercises for Chapter V ..................... 293 Chapter VI Smooth Actions 1. Functional Structures and Smooth Actions ............. 296 2 . Tubular Neighborhoods ..................... 303 3. Integration of Isotopies ..................... 312 4 . Equivariant Smooth Embeddings and Approximations ....... 314 5 . Functional Structures on Certain Orbit Spaces ........... 319 6. Special G-Manifolds ....................... 326 7. Smooth Knot Manifolds ..................... 333 8. Groups of Involutions ...................... 337 9 . Semifree Circle Group Actions .................. 347 10. Representations at Fixed Points ................. 352 11. Refinements Using Real K-Theory ................ 359 Exercises for Chapter VI .................... 366 Chapter VII Cohomology Structure of Fixed Point Sets 1. Preliminaries .................... ...... 369 2 . Some Inequalities ........................ 375 3. Z,. Actions on Projective Spaces ........... ...... 378 4 . Some Examples ................... ...... 388 5 . Circle Actions on Projective Spaces .......... ...... 393 6. Actions on Poincar6 Duality Spaces .......... ...... 400 7. A Theorem on Involutions .................... 405 8. Involutions on Sn x Sm ..................... 410 9 . Z,. Actions on Sn x Sm ..................... 416 10. Circle Actions on a Product of Odd-Dimensional Spheres ...... 422 11. An Application to Equivariant Maps ......... ...... 425 Exercises for Chapter VII .............. ...... 428 References 432 AUTHORINDEX ............................. 454 SUBJECTINDEX ............................. 457 This Page Intentionally Left Blank P R E FAC E In topology, one studies such objects as topological spaces, topological manifolds, differentiable manifolds, polyhedra, and so on. In the theory of transformation groups, one studies the symmetries of such objects, or generally subgroups of the full group of symmetries. Usually, the group of symmetries comes equipped with a naturally defined topology (such as the compact-open topology) and it is important to consider this topology as part of the structure studied. In some cases of importance, such as the group of isometries of a compact riemannian manifold, the group of symmetries is a compact Lie group. This should be sufficient reason for studying compact groups of transformations of a space or of a manifold. An even more com- pelling reason for singling out the case of compact groups is the fact that one can obtain many strong results and tools in this case that are not available for the case of noncompact groups. Indeed, the theory of compact trans- formation groups has a completely different flavor from that of noncompact transformation groups. There has been a good deal of research done on this subject in recent years. as a glance at the bibliography will show. This has convinced us of the need for a reasonably extensive introduction to the subject which would be comprehensible to a wide range of readers at the graduate level. The main obstacle to the writing of a successful introduction to this subject is the fact that it draws on so many disparate parts of mathematics. This makes it difficult to write such an introducton which would be readable by most second-year graduate students, which would cover a large portion of the subject, and which would also touch on a good amount of interesting nontrivial mathematics of current interest. To overcome this obstacle, we have endeavored to keep the prerequisites to a minimum, especially in early parts of the book. (This does not apply to all of Chapter 0. For a reader with minimal background, we recommend the reading of the first three sections of that chapter, then skipping to Chapter I, with a return to parts of Chapter 0 when needed. Many readers would do well to skip Chap- ter 0 altogether.) An indispensable prerequisite for reading this book is a first course in ix X PREFACE algebraic topology. The requirements in this direction are fairly minimal until the last half of Chapter 111, where some Cech theory is needed. PoincarC duality is not used until Chapter IV, and spectral sequences appear only in Chapter VII. A considerable saving in the algebraic topological demands on the reader results from the fact that we do not consider the theory of gen- eralized manifolds in this book. There is, of course, a resulting loss in the generality of some of the theorems, but we believe this is minimal. (Most current interest is in the case of smooth or locally smooth actions, and there the loss is practically nonexistent.) Although we are almost entirely concerned with actions of compact Lie groups in this book, there is really very little about Lie groups which the reader needs to know, outside of a few simple facts about maximal tori which we develop in Chapter 0, Section 6. This results from the fact that we concentrate on those theorems for which the classification theory of compact Lie groups, detailed case by case calculations in representation theory, and similar considerations