Fibre bundles:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York u.a.
Springer
1994
|
| Edition: | 3. ed. |
| Series: | Graduate texts in mathematics
20 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIX, 353 S. graph. Darst. |
| ISBN: | 0387940871 3540940871 |
Staff View
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Record in the Search Index
| _version_ | 1804123208781135872 |
|---|---|
| adam_text | Contents
Preface to the Third Edition vii
Preface to the Second Edition ix
Preface to the First Edition xi
Chapter 1
Preliminaries on Homotopy Theory 1
1. Category Theory and Homotopy Theory 1
2. Complexes 2
3. The Spaces Map (A , Y) and Map0 (X,Y) 4
4. Homotopy Groups of Spaces 6
5. Fibre Maps 7
Part I
The General Theory of Fibre Bundles 9
Chapter 2
Generalities on Bundles 11
1. Definition of Bundles and Cross Sections 11
2. Examples of Bundles and Cross Sections 12
3. Morphisms of Bundles 14
4. Products and Fibre Products 15
5. Restrictions of Bundles and Induced Bundles 17
6. Local Properties of Bundles 20
7. Prolongation of Cross Sections 21
Exercises 22
xiv Contents
Chapter 3
Vector Bundles 24
1. Definition and Examples of Vector Bundles 24
2. Morphisms of Vector Bundles 26
3. Induced Vector Bundles 27
4. Homotopy Properties of Vector Bundles 28
5. Construction of Gauss Maps 31
6. Homotopies of Gauss Maps 33
7. Functorial Description of the Homotopy Classification of Vector
Bundles 34
8. Kernel, Image, and Cokernel of Morphisms with Constant Rank 35
9. Riemannian and Hermitian Metrics on Vector Bundles 37
Exercises 39
Chapter 4
General Fibre Bundles 40
1. Bundles Defined by Transformation Groups 40
2. Definition and Examples of Principal Bundles 42
3. Categories of Principal Bundles 43
4. Induced Bundles of Principal Bundles 44
5. Definition of Fibre Bundles 45
6. Functorial Properties of Fibre Bundles 46
7. Trivial and Locally Trivial Fibre Bundles 47
8. Description of Cross Sections of a Fibre Bundle 48
9. Numerable Principal Bundles over B x [0,1] 49
10. The Cofunctor kG 52
11. The Milnor Construction 54
12. Homotopy Classification of Numerable Principal G Bundles .... 56
13. Homotopy Classification of Principal G Bundles over
CW Complexes 58
Exercises 59
Chapter 5
Local Coordinate Description of Fibre Bundles 61
1. Automorphisms of Trivial Fibre Bundles 61
2. Charts and Transition Functions 62
3. Construction of Bundles with Given Transition Functions 64
4. Transition Functions and Induced Bundles 65
5. Local Representation of Vector Bundle Morphisms 66
6. Operations on Vector Bundles 67
7. Transition Functions for Bundles with Metrics 69
Exercises 71
Chapter 6
Change of Structure Group in Fibre Bundles 73
1. Fibre Bundles with Homogeneous Spaces as Fibres 73
Contents xv
2. Prolongation and Restriction of Principal Bundles 74
3. Restriction and Prolongation of Structure Group for Fibre
Bundles 75
4. Local Coordinate Description of Change of Structure Group ... 76
5. Classifying Spaces and the Reduction of Structure Group 77
Exercises 77
Chapter 7
The Gauge Group of a Principal Bundle 79
1. Definition of the Gauge Group 79
2. The Universal Standard Principal Bundle of the Gauge Group .. 81
3. The Standard Principal Bundle as a Universal Bundle 82
4. Abelian Gauge Groups and the Runneth Formula 83
Chapter 8
Calculations Involving the Classical Groups 87
1. Stiefel Varieties and the Classical Groups 87
2. Grassmann Manifolds and the Classical Groups 90
3. Local Triviality of Projections from Stiefel Varieties 91
4. Stability of the Homotopy Groups of the Classical Groups 94
5. Vanishing of Lower Homotopy Groups of Stiefel Varieties 95
6. Universal Bundles and Classifying Spaces for the Classical Groups 95
7. Universal Vector Bundles 96
8. Description of all Locally Trivial Fibre Bundles over Suspensions 97
9. Characteristic Map of the Tangent Bundle over S 98
10. Homotopy Properties of Characteristic Maps 101
11. Homotopy Groups of Stiefel Varieties 103
12. Some of the Homotopy Groups of the Classical Groups 104
Exercises 107
Part II
Elements of K Theory 109
Chapter 9
Stability Properties of Vector Bundles Ill
1. Trivial Summands of Vector Bundles Ill
2. Homotopy Classification and Whitney Sums 113
3. The K Cofunctors . . ._ 114
4. Corepresentations of KF 118
5. Homotopy Groups of Classical Groups and KF{S ) 120
Exercises 121
Chapter 10
Relative K Theory 122
1. Collapsing of Trivialized Bundles 122
xvi Contents
2. Exact Sequences in Relative K Theory 124
3. Products in K Theory 128
4. The Cofunctor L(X.A) 129
5. The Difference Morphism 131
6. Products in L(X. A) 133
7. The Clutching Construction 134
8. The Cofunctor LjX.A) 136
9. Half Exact Cofunctors 138
Exercises 139
Chapter 11
Bott Periodicity in the Complex Case 140
1. AC Theory Interpretation of the Periodicity Result 140
2. Complex Vector Bundles over X x S2 141
3. Analysis of Polynomial Clutching Maps 143
4. Analysis of Linear Clutching Maps 145
5. The Inverse to the Periodicity Isomorphism 148
Chapter 12
Clifford Algebras 151
1. Unit Tangent Vector Fields on Spheres: I 151
2. Orthogonal Multiplications 152
3. Generalities on Quadratic Forms 154
4. Clifford Algebra of a Quadratic Form 156
5. Calculations of Clifford Algebras 158
6. Clifford Modules 161
7. Tensor Products of Clifford Modules 166
8. Unit Tangent Vector Fields on Spheres: II 168
9. The Group Spin(Jt) 169
Exercises 170
Chapter 13
The Adams Operations and Representations 171
1. ;. Rings 171
2. The Adams ^ Operations in /. Ring 172
3. The / Operations 175
4. Generalities on C Modules 176
5. The Representation Ring of a Group G and Vector Bundles 177
6. Semisimplicity of G Modules over Compact Groups 179
7. Characters and the Structure of the Group RF(G) 180
8. Maximal Tori 182
9. The Representation Ring of a Torus 185
Contents xvii
10. The (// Operations on K(X) and K0(X) 186
11. The (// Operations on K(S ) 187
Chapter 14
Representation Rings of Classical Groups 189
1. Symmetric Functions 189
2. Maximal Tori in SU(n) and U{n) 191
3. The Representation Rings of SU(n) and U(n) 192
4. Maximal Tori in Sp{n) 193
5. Formal Identities in Polynomial Rings 194
6. The Representation Ring of Sp{n) 195
7. Maximal Tori and the Weyl Group of SO(n) 195
8. Maximal Tori and the Weyl Group of Spin(n) 196
9. Special Representations of SO(n) and Spin(n) 198
10. Calculation of RSO(n) and R Spin(n) 200
11. Relation Between Real and Complex Representation Rings 203
12. Examples of Real and Quaternionic Representations 206
13. Spinor Representations and the K Groups of Spheres 208
Chapter 15
The Hopf Invariant 210
1. K Theory Definition of the Hopf Invariant 210
2. Algebraic Properties of the Hopf Invariant 211
3. Hopf Invariant and Bidegree 213
4. Nonexistence of Elements of Hopf Invariant 1 215
Chapter 16
Vector Fields on the Sphere 217
1. Thorn Spaces of Vector Bundles 217
2. S Category 219
3. S Duality and the Atiyah Duality Theorem 221
4. Fibre Homotopy Type 223
5. Stable Fibre Homotopy Equivalence 224
6. The Groups J(Sk) and KTop(Sk) 225
7. Thorn Spaces and Fibre Homotopy Type 227
8. S Duality and S Reducibility 229
9. Nonexistence of Vector Fields and Reducibility 230
10. Nonexistence of Vector Fields and Coreducibility 232
11. Nonexistence of Vector Fields and J{RPk) 233
12. Real K Groups of Real Projective Spaces 235
13. Relation Between KO(RP ) and J(RPn) 237
14. Remarks on the Adams Conjecture 240
xviii Contents
Part III
Characteristic Classes 243
Chapter 17
Chern Classes and Stiefel Whitney Classes 245
1. The Leray Hirsch Theorem 245
2. Definition of the Stiefel Whitney Classes and Chern Classes 247
3. Axiomatic Properties of the Characteristic Classes 248
4. Stability Properties and Examples of Characteristic Classes 250
5. Splitting Maps and Uniqueness of Characteristic Classes 251
6. Existence of the Characteristic Classes 252
7. Fundamental Class of Sphere Bundles. Gysin Sequence 253
8. Multiplicative Property of the Euler Class 256
9. Definition of Stiefel Whitney Classes Using the Squaring
Operations of Steenrod 257
10. The Thorn Isomorphism 258
11. Relations Between Real and Complex Vector Bundles 259
12. Orientability and Stiefel Whitney Classes 260
Exercises 261
Chapter 18
Differentiable Manifolds 262
1. Generalities on Manifolds 262
2. The Tangent Bundle to a Manifold 263
3. Orientation in Euclidean Spaces 266
4. Orientation of Manifolds 267
5. Duality in Manifolds 269
6. Thorn Class of the Tangent Bundle 272
7. Euler Characteristic and Class of a Manifold 274
8. Wu s Formula for the Stiefel Whitney Class of a Manifold 275
9. Stiefel Whitney Numbers and Cobordism 276
10. Immersions and Embeddings of Manifolds 278
Exercises 279
Chapter 19
Characteristic Classes and Connections 280
1. Differential Forms and de Rham Cohomology 280
2. Connections on a Vector Bundle 283
3. Invariant Polynomials in the Curvature of a Connection 285
4. Homotopy Properties of Connections and Curvature 288
5. Homotopy to the Trivial Connection and the Chern Simons Form 290
6. The Levi Civita or Riemannian Connection 291
Contents xix
Chapter 20
General Theory of Characteristic Classes 294
1. The Yoneda Representation Theorem 294
2. Generalities on Characteristic Classes 295
3. Complex Characteristic Classes in Dimension n 296
4. Complex Characteristic Classes 298
5. Real Characteristic Classes Mod 2 300
6. 2 Divisible Real Characteristic Classes in Dimension n 301
7. Oriented Even Dimensional Real Characteristic Classes 304
8. Examples and Applications 306
9. Bott Periodicity and Integrality Theorems 307
10. Comparison of K Theory and Cohomology Definitions
of Hopf Invariant 309
11. The Borel Hirzebruch Description of Characteristic Classes 309
Appendix 1
Dold s Theory of Local Properties of Bundles 312
Appendix 2
On the Double Suspension 314
1. HJQS(X)) as an Algebraic Functor of H^X) 314
2. Connectivity of the Pair (fi2S2 + S2 1) Localized at p 318
3. Decomposition of Suspensions of Products and iXS(A ) 319
4. Single Suspension Sequences 322
5. Mod p Hopf Invariant 326
6. Spaces Where the pth Power Is Zero 329
7. Double Suspension Sequences 333
8. Study of the Boundary Map A: Q3S2 P+1 ^as2 1 337
Bibliography 339
Index 348
|
| any_adam_object | 1 |
| author | Husemöller, Dale |
| author_GND | (DE-588)117713058 |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 514 - Topology |
| dewey-raw | 514/.224 |
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| dewey-sort | 3514 3224 |
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| indexdate | 2024-07-09T17:26:07Z |
| institution | BVB |
| isbn | 0387940871 3540940871 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-005856571 |
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| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Husemöller, Dale Verfasser (DE-588)117713058 aut Fibre bundles Dale Husemoller 3. ed. New York u.a. Springer 1994 XIX, 353 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 20 Differentiaalmeetkunde gtt Faisceaux fibrés (Mathématiques) Faisceaux fibrés (Mathématiques) ram Vezels (wiskunde) gtt Fiber bundles (Mathematics) Algebraische Topologie (DE-588)4120861-4 gnd rswk-swf Faserbündel (DE-588)4135582-9 gnd rswk-swf K-Theorie (DE-588)4033335-8 gnd rswk-swf 1\p (DE-588)4151278-9 Einführung gnd-content K-Theorie (DE-588)4033335-8 s DE-604 Algebraische Topologie (DE-588)4120861-4 s Faserbündel (DE-588)4135582-9 s 2\p DE-604 Graduate texts in mathematics 20 (DE-604)BV000000067 20 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005856571&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Husemöller, Dale Fibre bundles Graduate texts in mathematics Differentiaalmeetkunde gtt Faisceaux fibrés (Mathématiques) Faisceaux fibrés (Mathématiques) ram Vezels (wiskunde) gtt Fiber bundles (Mathematics) Algebraische Topologie (DE-588)4120861-4 gnd Faserbündel (DE-588)4135582-9 gnd K-Theorie (DE-588)4033335-8 gnd |
| subject_GND | (DE-588)4120861-4 (DE-588)4135582-9 (DE-588)4033335-8 (DE-588)4151278-9 |
| title | Fibre bundles |
| title_auth | Fibre bundles |
| title_exact_search | Fibre bundles |
| title_full | Fibre bundles Dale Husemoller |
| title_fullStr | Fibre bundles Dale Husemoller |
| title_full_unstemmed | Fibre bundles Dale Husemoller |
| title_short | Fibre bundles |
| title_sort | fibre bundles |
| topic | Differentiaalmeetkunde gtt Faisceaux fibrés (Mathématiques) Faisceaux fibrés (Mathématiques) ram Vezels (wiskunde) gtt Fiber bundles (Mathematics) Algebraische Topologie (DE-588)4120861-4 gnd Faserbündel (DE-588)4135582-9 gnd K-Theorie (DE-588)4033335-8 gnd |
| topic_facet | Differentiaalmeetkunde Faisceaux fibrés (Mathématiques) Vezels (wiskunde) Fiber bundles (Mathematics) Algebraische Topologie Faserbündel K-Theorie Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005856571&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT husemollerdale fibrebundles |