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Seifert Fiberings Mathematical Surveys and Monographs Volume 166 Seifert Fiberings Kyung Bai Lee Frank Raymond American Mathematical Society surv-166-lee3-cov.indd 1 10/21/10 11:26 AM http://dx.doi.org/10.1090/surv/166 Seifert Fiberings Seifert Fiberings Kyung Bai Lee Frank Raymond Mathematical Surveys and Monographs Volume 166 Seifert Fiberings Kyung Bai Lee Frank Raymond American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair MichaelA.Singer Eric M. Friedlander Benjamin Sudakov MichaelI.Weinstein 2000 Mathematics Subject Classification. Primary 55R55, 57S30, 57–XX; Secondary 53C30, 55R91, 58E40, 58D19, 57N16. For additional information and updates on this book, visit www.ams.org/bookpages/surv-166 Library of Congress Cataloging-in-Publication Data Lee, Kyung Bai, 1943– Seifert fiberings / Kyung Bai Lee, Frank Raymond. p. cm. — (Mathematical surveys and monographs : v. 166) Includes bibliographical references and index. ISBN 978-0-8218-5231-6 (alk. paper) 1. Fiberings (Mathematics). 2. Complex manifolds. I. Raymond, Frank, 1932– II. Title. QA612.6 .L44 2010 514.2—22 2010022528 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2010 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 151413121110 To Hyang, Myung, Kwang, Chung, Natasha, Nyra and Rob Acknowledgments We wish to acknowledge our debt to our many collaborators, teachers, students, and colleagues who have, over the years, contributed so much to our work and enjoyment of mathematics. We are also grateful for past financial support from University of Oklahoma and University of Michigan as well as from the National Science Foundation and RIP of the Mathematisches Institut Oberwolfach. Contents Preface xiii Chapter 1. Transformation Groups 1 1.1. Introduction 1 1.2. (Locally) Proper G-spaces 4 1.3. Fiber bundles 6 1.4. Classifying spaces 9 1.5. Borel spaces and classifying spaces 10 1.6. Tubular neighborhoods and slices 11 1.7. Existence of slices 15 1.8. Cohomology manifolds and the Smith theorems 18 1.9. Actions of G Π (G Lie group, Π discrete) 20 · Chapter 2. Group Actions and the Fundamental Group 23 2.1. Covering spaces 23 2.2. Lifting group actions to covering spaces 26 2.3. Lifting an action of G when G has a fixed point 28 2.4. Evaluation homomorphism 31 2.5. Lifting connected group actions 32 2.6. Example (Semi-free S1-actions on 3-manifolds) 35 2.7. Lifting the slice representation 39 2.8. Locally injective actions 42 Chapter 3. Actions of Compact Lie Groups on Manifolds 47 3.1. Actions of compact Lie groups on aspherical manifolds 47 3.2. Actions of compact Lie groups on admissible manifolds 54 3.3. Compact Lie group actions on spaces which map into K(Γ, 1) 59 3.4. Manifolds with few or no periodic homeomorphisms 65 3.5. Injective torus actions 66 Chapter 4. Definition of Seifert Fibering 69 4.1. Examples 69 4.2. TOPG(P ), the group of weak G-equivalences 73 4.3. Seifert fiberings modeled on a principal G-bundle 78 4.4. The topology and geometry of the fibers 81 4.5. Examples with Π discrete 83 4.6. The Seifert Construction 91 Chapter 5. Group Cohomology 95 5.1. Introduction 95 5.2. Group extensions 95 ix x CONTENTS 5.3. Pullback and pushout of short exact sequences 97 5.4. Extensions with Abelian kernel A and H2(Q; A) 100 5.5. Central extensions 102 5.6. Extensions with non-Abelian kernel G and H2(Q; (G)) 104 5.7. H1(Q; C) with a non-Abelian C Z 105 Chapter 6. Lie Groups 109 6.1. Introduction 109 6.2. Nilpotent Lie groups 111 6.3. Solvable Lie groups 114 6.4. Semisimple Lie groups 115 Chapter 7. Seifert Fiber Space Construction for G W 119 7.1. Introduction × 119 7.2. Cohomological criteria 120 7.3. Main Construction Theorem for TOPG(G W ) 122 7.4. The meaning of existence, uniqueness and× rigidity 123 7.5. Hp(Q; M(W, Rk)) = 0, p > 0 127 7.6. Proof of the Construction Theorem 131 7.7. When is θ injective? 135 7.8. Smooth case 137 Chapter 8. Generalization of Bieberbach’s Theorems 139 8.1. Bieberbach’s theorems 139 8.2. Proof of Bieberbach’s theorems 139 8.3. The First Bieberbach Theorem 143 8.4. The Second Bieberbach Theorem 151 8.5. The Third Bieberbach Theorem 156 Chapter 9. Seifert Manifolds with Γ G/K-Fiber 159 9.1. Introduction \ 159 9.2. The group TOPG,K(G/K W ) 160 ×0 9.3. When NG(K) = K and Aut (G, K) = 1 164 9.4. Symmetric spaces of noncompact type 166 9.5. Solvmanifolds 172 Chapter 10. Locally Injective Seifert Fiberings with Torus Fibers 179 10.1. Introduction 179 10.2. When does E(P, Q) split? 187 10.3. From local to global: H∗(Q; k) and H∗(Q; k) 190 10.4. The product case, Rk W T Z 198 10.5. Some aspects of the ′E×p,q-spectral sequence 201 Chapter 11. Applications 205 11.1. Existence of closed K(Π, 1)-manifolds 206 11.2. Rigidity of Seifert fibering 210 11.3. Lifting problem for homotopy classes 216 11.4. Polynomial structures for solvmanifolds 227 11.5. Applications to fixed-point theory 231 CONTENTS xi 11.6. Homologically injective torus operations 236 11.7. Maximal torus actions 248 11.8. Toral rank of spherical space forms 265 Chapter 12. Seifert Fiberings with Compact Connected Q 269 12.1. Introduction 269 12.2. Lifting Q-actions 269 12.3. Lifting Q-actions (for connected Q) 271 12.4. Examples 278 Chapter 13. Deformation Spaces 283 13.1. Uniformizing groups 283 13.2. PSL(2, R)-geometry 284 13.3. Lorentz structures and PSL(2, R)-geometry 288 13.4. Deformationg spaces for PSL(2, R)-geometry 290 13.5. Deformation spaces for Nil-geometryg 296 g Chapter 14. S1-actions on 3-dimensional Manifolds 299 14.1. Introduction 299 14.2. S1-actions on 3-manifolds 302 14.3. S1-actions on 3-manifolds as Seifert fiberings 304 14.4. The classification of S1-actions on closed aspherical 3-manifolds 309 14.5. The classification of S1-actions with fixed points on 3-manifolds 312 14.6. A complete set of invariants 314 14.7. The Euler number 318 14.8. Γ G as 3-dimensional Seifert manifolds 322 \ 14.9. H1(M; Z) 329 14.10. Injective holomorphic Seifert fiberings 331 14.11. Brieskorn complete intersections 335 14.12. Generalized Seifert 3-manifolds as local SO(2)-actions 338 14.13. A complete set of invariants for the 3-dimensional Seifert fiberings 344 14.14. Historical remarks 348 Chapter 15. Classification of Seifert 3-manifolds via equivariant cohomology 353 15.1. H2(Q; Zk) and codimension-2 injective actions 353 15.2. A presentation for H2(Q; ) 356 15.3. 3-dimensional spherical spaceZ forms 366 15.4. Seifert fiberings with Q Euclidean crystallographic 374 15.5. Seifert fiberings with Q hyperbolic 376 15.6. Equivariant classification 377 15.7. An Illustration 377 Bibliography 383 Index 393 Preface This book is an exploration of Seifert fiberings. These are mappings which extend the notion of fiber bundle mappings by allowing some of the fibers to be singular. Seifert fiberings are mappings whose typical fibers are homeomorphic to a fixed homogeneous space. The singular fibers are quotients of the homogeneous space by distinguished groups of homeomorphisms. In a remarkable paper, in 1933, Herbert Seifert, introduced a class of 3-manifolds which became known as Seifert manifolds. They play a very significant role in low dimensional topology and remain under intense scrutiny today. A Seifert manifold maps onto a 2-dimensional surface such that the inverse image of each point on the surface is homeomorphic to a circle. The set of singular fibers are isolated from each other and the typical fibers wind nontrivially around the singular fibers. We will describe in detail the Seifert 3-manifolds as a special case of the general con- struction of Seifert fiberings. Our major focus, however, is on higher dimensional phenomenon where the typical fiber is a homogeneous space. A major inspiration for a generalization to higher dimensions comes from trans- formation groups. Let (G, X) be a proper action of the connected Lie group G on x a path-connected X and examine the homomorphism ev∗ : π1(G, e) π1(X, x) induced by the evaluation map evx : g gx, g G. The image H →of evx is a 7→ ∈ ∗ central subgroup of π1(X, x) independent of the base point x. The G-action on X can be lifted to the covering space XH of X associated with the subgroup H of π1(X, x) so that π1(XH , x) = H.
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