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SEIFERT AND THRELFALL: A TEXTBOOK OF TOPOLOGY lltld SEI FER T: 7'0PO 1.OG 1' 0 I.' 3- Dl M E N SI 0 N A I. FIRERED SPACES This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: SAMUELEILENBERG AND HYMANBASS A list of recent titles in this series appears at the end of this volunie. SEIFERT AND THRELFALL: A TEXTBOOK OF TOPOLOGY H. SEIFERT and W. THRELFALL Translated by Michael A. Goldman und S E I FE R T: TOPOLOGY OF 3-DIMENSIONAL FIBERED SPACES H. SEIFERT Translated by Wolfgang Heil Edited by Joan S. Birman and Julian Eisner @ 1980 ACADEMIC PRESS A Subsidiary of Harcourr Brace Jovanovich, Publishers NEW YORK LONDON TORONTO SYDNEY SAN FRANCISCO COPYRIGHT@ 1980, BY ACADEMICPRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. 11 1 Fifth Avenue, New York. New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NWI 7DX Mit Genehmigung des Verlager B. G. Teubner, Stuttgart, veranstaltete, akin autorisierte englische Ubersetzung, der deutschen Originalausgdbe. Library of Congress Cataloging in Publication Data Seifert, Herbert, 1897- Seifert and Threlfall: A textbook of topology. Seifert: Topology of 3-dimensional fibered spaces. (Pure and applied mathematics, a series of mono- graphs and textbooks ; ) Translation of Lehrbuch der Topologic. Bibliography: p. Includes index. 1. Topology. 1. Threlfall, William, 1888- joint author. 11. Birman, Joan S., 1927- 111. Eisner, Julian. IV. Title. V. Title: Text- book of topology. VI. Series. QA3.P8 [QA611] 510s [514] 79-28163 ISBN 0-12-634850-2 PRINTED IN THE UNITED STATES OF AMERICA 80 81 82 83 9 8 7 6 5 4 3 2 1 CON TENTS ix Preface to English Edition ... Acknowledgments Xlll Preface to German Edition xv SEIFERT AND THRELFALL: A TEXTBOOK OF TOPOLOGY CHAITER ONE ILLUSTRATIVE MATERIAL 1. The Principal Problem of Topology i 2. Closed Surfaces 5 3. Isotopy, Homotopy, Homology 14 4. Higher Dimensional Manifolds 16 CHAF'TER TWO SlMPLIClAL COMPLEXES 5. Neighborhood Spaces 22 6. Mappings 25 7. Point Sets in Euclidean Spaces 30 8. Identification Spaces 33 9. n-Simplexes 31 10. Simplicia1 Complexes 43 11. The Schema of a Simplicial Complex 45 12. Finite, Pure, Homogeneous Complexes 48 13. Normal Subdivision 50 14. Examples of Complexes 52 V vi CON TENTS CHAPTER THREE HOMOLOGY GROUPS 15. Chains 60 16. Boundary, Closed Chains 61 17. Homologous Chains 63 18. Homology Groups 66 19. Computation of the Homology Groups in Simple Cases 68 20. Homologies with Division 71 21. Computation of Homology Groups from the Incidence Matrices 73 22. Block Chains 80 23. Chains mod 2, Connectivity Numbers, Euler’s Formula 83 24. Pseudomanifolds and Orientability 90 CHAPTER FOUR SIMPLICIAL APPROXIMATIONS 25. Singular Simplexes 95 26. Singular Chains 97 27. Singular Homology Groups 98 28. The Approximation Theorem, Invariance of Simplicia1 Homology Groups 102 29. Prisms in Euclidean Spaces 103 30. Proof of the Approximation Theorem 107 3 I. Deformation and Simplicial Approximation of Mappings I I5 CHAPTER FIVE LOCAL PROPERTIES 32. Homology Groups of a Complex at a Point 123 33. Invariance of Dimension 129 34. Invariance of the Purity of a Complex 129 35. Invariance of Boundary I3 1 36. Invariance of Pseudomanifolds and of Orientability 132 CHAPTER SIX SURFACE TOPOLOGY 37. Closed Surfaces 134 38. Transformation to Normal Form 139 39. Types of Normal Form: The Principal Theorem 145 40. Surfaces with Boundary 146 41. Homology Groups of Surfaces 149 CHAPTER SEVEN THE FUNDAMENTAL GROUP 42. The Fundamental Group 154 43. Examples 161 C'ONT'ENTS vii 44. The Edge Path Group of a Simplicia1 Complex i63 45. The Edge Path Group of a Surface Complex 167 46. Generators and Relations 171 47. Edge Complexes and Closed Surfaces 174 48. The Fundamental and Homology Groups 117 49. Free Deformation of Closed Paths 180 50. Fundamental Group and Deformation of Mappings 182 51. The Fundamental Group at a Point 182 52. The Fundamental Group of a Composite Complex 183 CHAITER EIGHT COVERING COMPLEXES 53. Unbranched Covering Complexes 188 54. Base Path and Covering Path 191 55. Coverings and Subgroups of the Fundamental Group 195 56. Universal Coverings 200 57. Regular Coverings 20 1 58. The Monodromy Group 205 CHAITER NINE 3-DIMENSIONAL MANIFOLDS 59. General Principles 21 I 60. Representation by a Polyhedron 213 61. Homology Groups 218 62. The Fundamental Group 22 1 63. The Heegaard Diagram 226 64. 3-Dimensional Manifolds with Boundary 229 65. Construction of 3-Dimensional Manifolds out of Knots 23 1 CHAITER TEN n-DIMENSIONAL MANIFOLDS 66. Star Complexes 235 67. Cell Complexes 24 1 68. Manifolds 244 69. The Poincare Duality Theorem 250 70. Intersection Numbers of Cell Chains 255 71. Dual Bases 258 72. Cellular Approximations 263 73. Intersection Numbers of Singular Chains 267 74. lnvariance of Intersection Numbers 269 75. Examples 279 76. Orientability and Two-Sidedness 283 77. Linking Numbers 288 viii CON TENTS CHAPTER ELEVEN CONTINUOUS MAPPINGS 78. The Degree of a Mappings 294 79. A Trace Formula 296 80. A Fixed Point Formula 300 8 1. Applications 301 CHAPTER TWELVE AUXILIARY THEOREMS FROM THE THEORY OF GROUPS 82. Generators and Relations 305 83. Homomorphic Mappings and Factor Groups 309 84. Abelianization of Groups 312 85. Free and Direct Products 313 86. Abelian Groups 316 87. The Normal Form of Integer Matrices 323 COMMENTS 328 BIBLIOGRAPHY 341 SEIFERT: TOPOLOGY OF 3-DIMENSIONAL FIBERED SPACES I. Fibered Spaces 360 2. Orbit Surface 364 3. Fiberings of S3 368 4. Triangulations of Fibered Spaces 370 5. Drilling and Filling (Surgery) 372 6. Classes of Fibered Spaces 378 7. The Orientable Fibered Spaces 383 8. The Nonorientable Fibered Spaces 389 9. Covering Spaces 395 10. Fundamental Groups of Fibered Spaces 398 I I. Fiberings of the 3-Sphere (Complete List) 40 1 12. The Fibered Poincare Spaces 402 13. Constructing Poincar; Spaces from Torus Knots 405 14. Translation Groups of Fibered Spaces 406 15. Spaces Which Cannot Be Fibered 413 Appendix. Branched Coverings 418 Index to "A Textbook of Topology" 423 PREFACE TO ENGLISH EDITION The first German edition of Seifert and Threlfall’s “Lehrbuch der Topologie” was published in 1934. The book very quickly became the leading introductory textbook for students of geometric-algebraic topology (as distinguished from point set or “general” topology), a position which it held for possibly 30 to 35 years, during which time it was translated into Russian, Chinese, and Spanish. An English language edition is, then, long overdue. The translation presented here is due to Michael A. Goldman. In spite of the fact that with the passage of time our understanding of the subject matter has changed enormously (this is particularly true with regard to homology theory) the book continues to be of interest for its geometric insight and leisurely, careful presentation with its many beautiful examples which convey so well to the student the flavor of the subject. In fact, a quick perusal of the more successful modern textbooks aimed at advanced under- graduates or beginning graduate students reveals, inevitably, large blocks of material which appear to have been inspired by if not directly modeled on sections of this book. For example: the introductory pages on the problems of topology, the classification of surfaces, the discussion of incidence matrices and of methods for bringing them to normal form, the chapter on 3-dimensional manifolds (in particular the discussion of lens spaces), the section on intersection theory, and especially the notes at the end of the text have withstood the test of time and are as useful and readable today as they no doubt were in 1934. This volume contains, in addition to Seifert and Threlfall’s book, a translation into English, by Wolfgang Heil, of Seifert’s foundational research paper “The topology of 3-dimensional fibered spaces” [“Topologie dreidi- mensionales gefaserter Raum,” Acta Mathematics 60, 147-288 (1933)l. The manuscript treats a simple and beautiful question: what kinds of 3-dimensional manifolds can be made up as unions of disjoint circles, put together nicely? Using practically no technical machinery, Seifert takes hold ix X PREFACE TO ENGLISH EDITION of the question, lays down a set of ground rules, and with exceptional clarity of thought and particular attention to detail weaves a theory which culminates in the complete classification of his “fibered spaces” up to fiber-preserving homeomorphism. The manuscript is a model of mathematical exposition, and on these grounds alone it is worthy of translation, preservation, and study by new generations of mathematicians. But the primary reason for publishing this translation is, in our minds, because the ideas in the paper have proved to be at the origins of so many different flourishing areas of current research. We give a brief review. The question that Seifert asks and answers is not an idle question. The first paragraph of the paper shows that Seifert understood well the crucial role that the manifolds which have since come to be known as Seifert fibered spaces were likely to play in the classification problem for compact 3- manifolds. In fact it could be said that Seifert anticipated the theory of geometric structures on 3-manifolds, as it is evolving at this writing. Although Seifert only classified his fibered spaces up to fiber-preserving homeomorphism, the classification has turned out (with some exceptions) to be topological (Orlik, Vogt, and Zieschang [lo]). This classification was generalized by Waldhausen [ 1 I] to a related classification of “graph manifolds” which are Seifert fiber spaces (now with nonempty torus boundaries) pasted together along their boundaries, much as one pastes together 3-manifolds along 2-spheres in the boundary to obtain connected sums.
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