Seifert and Threlfall, A Textbook of Topology (eBook)

Front Cover 1
Seifert and Threlfall: A Textbook of Topology and Seifert: Topology of 3-Dimensional Fibered Spaces 4
Copyright Page 5
CONTENTS 6
Preface to English Edition 10
Acknowledgments 14
Preface to German Edition 16
PART I: SEIFERT AND THRELFALL: A TEXTBOOK OF TOPOLOGY 18
CHAPTER ONE. ILLUSTRATIVE MATERIAL 20
1. The Principal Problem of Topology 20
2. Closed Surfaces 24
3. Isotopy, Homotopy, Homology 33
4. Higher Dimensional Manifolds 35
CHAPTER TWO. SIMPLICIAL COMPLEXES 41
5. Neighborhood Spaces 41
6. Mappings 44
7. Point Sets in Euclidean Spaces 49
8. Identification Spaces 52
9. n-Simplexes 56
10. Simplicial Complexes 62
11. The Schema of a Simplicial Complex 64
12. Finite, Pure, Homogeneous Complexes 67
13. Normal Subdivision 69
14. Examples of Complexes 71
CHAPTER THREE. HOMOLOGY GROUPS 79
15. Chains 79
16. Boundary, Closed Chains 80
17. Homologous Chains 82
18. Homology Groups 85
19. Computation of the Homology Groups in Simple Cases 87
20. Homologies with Division 90
21. Computation of Homology Groups from the Incidence Matrices 92
22. Block Chains 99
23. Chains mod 2, Connectivity Numbers, Euler’s Formula 102
24. Pseudomanifolds and Orientability 109
CHAPTER FOUR. SIMPLICIAL APPROXIMATIONS 114
25. Singular Simplexes 114
26. Singular Chains 116
27. Singular Homology Groups 117
28. The Approximation Theorem, Invariance of Simplicial Homology Groups 121
29. Prisms in Euclidean Spaces 122
30. Proof of the Approximation Theorem 126
3I. Deformation and Simplicial Approximation of Mappings 134
CHAPTER FIVE. LOCAL PROPERTIES 142
32. Homology Groups of a Complex at a Point 142
33. Invariance of Dimension 148
34. Invariance of the Purity of a Complex 148
35. Invariance of Boundary 150
36. Invariance of Pseudomanifolds and of Orientability 151
CHAPTER SIX. SURFACE TOPOLOGY 153
37. Closed Surfaces 153
38. Transformation to Normal Form 158
39. Types of Normal Form: The Principal Theorem 164
40. Surfaces with Boundary 165
41. Homology Groups of Surfaces 168
CHAPTER SEVEN. THE FUNDAMENTAL GROUP 173
42. The Fundamental Group 173
43. Examples 180
44. The Edge Path Group of a Simplicial Complex 182
45. The Edge Path Group of a Surface Complex 186
46. Generators and Relations 190
47. Edge Complexes and Closed Surfaces 193
48. The Fundamental and Homology Groups 196
49. Free Deformation of Closed Paths 199
50. Fundamental Group and Deformation of Mappings 201
51. The Fundamental Group at a Point 201
52. The Fundamental Group of a Composite Complex 202
CHAPTER EIGHT. COVERING COMPLEXES 207
53. Unbranched Covering Complexes 207
54. Base Path and Covering Path 210
55. Coverings and Subgroups of the Fundamental Group 214
56. Universal Coverings 219
57. Regular Coverings 220
58. The Monodromy Group 224
CHAPTER NINE. 3-DIMENSIONAL MANIFOLDS 230
59. General Principles 230
60. Representation by a Polyhedron 232
61. Homology Groups 237
62. The Fundamental Group 240
63. The Heegaard Diagram 245
64. 3-Dimensional Manifolds with Boundary 248
65. Construction of 3-Dimensional Manifolds out of Knots 250
CHAPTER TEN. n-DIMENSIONAL MANIFOLDS 254
66. Star Complexes 254
67. Cell Complexes 260
68. Manifolds 263
69. The Poincaré Duality Theorem 269
70. Intersection Numbers of Cell Chains 274
71. Dual Bases 277
72. Cellular Approximations 282
73. Intersection Numbers of Singular Chains 286
74. lnvariance of Intersection Numbers 288
75. Examples 298
76. Orientability and Two-Sidedness 302
77. Linking Numbers 307
CHAPTER ELEVEN. CONTINUOUS MAPPINGS 313
78. The Degree of a Mappings 313
79. A Trace Formula 313
80. A Fixed Point Formula 319
81. Applications 320
CHAPTER TWELVE. AUXILIARY THEOREMS FROM THE THEORY OF GROUPS 324
82. Generators and Relations 324
83. Homomorphic Mappings and Factor Groups 328
84. Abelianization of Groups 331
85. Free and Direct Products 332
86. Abelian Groups 335
87. The Normal Form of Integer Matrices 342
COMMENTS 347
BIBLIOGRAPHY 360
PART II: SEIFERT: TOPOLOGY OF 3-DIMENSIONAL FIBERED SPACES 378
1. Fibered Spaces 381
2. Orbit Surface 385
3. Fiberings of S3 389
4. Triangulations of Fibered Spaces 391
5. Drilling and Filling (Surgery) 393
6. Classes of Fibered Spaces 399
7. The Orientable Fibered Spaces 404
8. The Nonorientable Fibered Spaces 410
9. Covering Spaces 416
10. Fundamental Groups of Fibered Spaces 419
11. Fiberings of the 3-Sphere (Complete List) 422
12. The Fibered Poincaré Spaces 423
13. Constructing Poincaré Spaces from Torus Knots 426
14. Translation Groups of Fibered Spaces 427
15. Spaces Which Cannot Be Fibered 434
Appendix. Branched Coverings 439
Index to "A Textbook of Topology" 444