Seifert and Threlfall: A Textbook of Topology

461 Pages·1980·7.061 MB·English

by  H. Seifert, W. Threlfall, Joan S. Birman, Julian Eisner, Michael A. Goodman, Wolfgang Heil

SEIFERT AND THRELFALL:
A TEXTBOOK OF TOPOLOGY
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SEI FER T:
7'0P O 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: SAMUEELI LENBEARNGD HYMANBA SS
A list of recent titles in this series appears at the end of this volunie.
SEIFERT A N D 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
COPYRIGH@T 1980, BY ACADEMIPCR ESS,I NC.
ALL RIGHTS RESERVED.
NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR
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ACADEMIC PRESS, INC.
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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
Preface to English Edition ix
...
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