Fibre Bundles

Naslovnica
Springer Science & Business Media, 9. ožu 2013. - Broj stranica: 356
Fibre bundles play an important role in just about every aspect of modern geometry and topology. Basic properties, homotopy classification, and characteristic classes of fibre bundles have become an essential part of graduate mathematical education for students in geometry and mathematical physics. In this third edition two new chapters on the gauge group of a bundle and on the differential forms representing characteristic classes of complex vector bundles on manifolds have been added. These chapters result from the important role of the gauge group in mathematical physics and the continual usefulness of characteristic classes defined with connections on vector bundles.
 

Sadržaj

CHAPTER
1
CHAPTER
11
CHAPTER 3
24
Morphisms of Vector Bundles
26
Induced Vector Bundles
27
Homotopy Properties of Vector Bundles
28
Construction of Gauss Maps
31
Homotopies of Gauss Maps
33
Maximal Tori
182
The Representation Ring of a Torus
185
The Operations on KX and KOX
186
The Operations on S
187
CHAPTER 14
189
Maximal Tori in SUn and Un
191
The Representation Rings of SUn and Un
192
Maximal Tori in Spn
193

Functorial Description of the Homotopy Classification of Vector Bundles
34
Kernel Image and Cokernel of Morphisms with Constant Rank
35
Riemannian and Hermitian Metrics on Vector Bundles
37
Exercises
39
CHAPTER 4
40
Definition and Examples of Principal Bundles
42
Categories of Principal Bundles
43
Induced Bundles of Principal Bundles
44
Definition of Fibre Bundles
45
Functorial Properties of Fibre Bundles
46
Trivial and Locally Trivial Fibre Bundles
47
Description of Cross Sections of a Fibre Bundle
48
Numerable Principal Bundles over B x 0 1
49
The Cofunctor ko
52
The Milnor Construction
54
Homotopy Classification of Numerable Principal GBundles
56
Homotopy Classification of Principal GBundles over CWComplexes 5589
58
Exercises
59
CHAPTER 5
61
Charts and Transition Functions
62
Construction of Bundles with Given Transition Functions
64
Transition Functions and Induced Bundles
65
Local Representation of Vector Bundle Morphisms
66
Operations on Vector Bundles
67
Transition Functions for Bundles with Metrics
69
Exercises
71
CHAPTER 6
73
Local Coordinate Description of Change of Structure Group
76
CHAPTER 8
87
Homotopy Properties of Characteristic Maps
101
CHAPTER 9
111
CHAPTER 10
122
Exact Sequences in Relative KTheory
124
Products in KTheory
128
The Cofunctor LX A
129
The Difference Morphism
131
Products in LXA
133
The Clutching Construction
134
The Cofunctor LX A n
136
HalfExact Cofunctors
138
Exercises
139
CHAPTER 11
140
Complex Vector Bundles over X S²
141
Analysis of Polynomial Clutching Maps
143
Analysis of Linear Clutching Maps
145
The Inverse to the Periodicity Isomorphism
148
CHAPTER 12
151
Orthogonal Multiplications
152
Generalities on Quadratic Forms
154
Clifford Algebra of a Quadratic Form
156
Calculations of Clifford Algebras
158
Clifford Modules
161
Tensor Products of Clifford Modules
166
II
168
The Group Spink
169
Exercises
170
CHAPTER 13
171
The Adams Operations in 2Ring
172
The y Operations
175
Generalities on GModules
176
The Representation Ring of a Group G and Vector Bundles
177
Semisimplicity of GModules over Compact Groups
179
Characters and the Structure of the Group RFG
180
Formal Identities in Polynomial Rings
194
The Representation Ring of Spn
195
Maximal Tori and the Weyl Group of Spinn
196
Special Representations of SOn and Spinn
198
Calculation of RSOn and R Spinn
200
Relation Between Real and Complex Representation Rings
203
Examples of Real and Quaternionic Representations
206
Spinor Representations and the KGroups of Spheres
208
CHAPTER 15
210
Algebraic Properties of the Hopf Invariant
211
Hopf Invariant and Bidegree
213
Nonexistence of Elements of Hopf Invariant 1
215
CHAPTER 16
217
SCategory
219
SDuality and the Atiyah Duality Theorem
221
Fibre Homotopy Type
223
Stable Fibre Homotopy Equivalence
224
The Groups JS and KTopS
225
Thom Spaces and Fibre Homotopy Type
227
SDuality and SReducibility
229
Nonexistence of Vector Fields and Reducibility
230
Nonexistence of Vector Fields and Coreducibility
232
Nonexistence of Vector Fields and JRP
233
Real KGroups of Real Projective Spaces
235
Relation Between KORP and JRP
237
Remarks on the Adams Conjecture
240
PART III
243
CHAPTER 17
244
Definition of the StiefelWhitney Classes and Chern Classes
247
Axiomatic Properties of the Characteristic Classes
248
Stability Properties and Examples of Characteristic Classes
250
Splitting Maps and Uniqueness of Characteristic Classes
251
Existence of the Characteristic Classes
252
Fundamental Class of Sphere Bundles Gysin Sequence
253
Multiplicative Property of the Euler Class
256
Definition of StiefelWhitney Classes Using the Squaring Operations of Steenrod
257
The Thom Isomorphism
258
Relations Between Real and Complex Vector Bundles
259
Orientability and StiefelWhitney Classes
260
Exercises
261
CHAPTER 18
262
The Tangent Bundle to a Manifold
264
Orientation in Euclidean Spaces
266
Orientation of Manifolds
267
Duality in Manifolds
269
Thom Class of the Tangent Bundle
272
Euler Characteristic and Class of a Manifold
274
Wus Formula for the StiefelWhitney Class of a Manifold
275
StiefelWhitney Numbers and Cobordism
276
Immersions and Embeddings of Manifolds
278
Exercises
279
CHAPTER 19
280
Connections on a Vector Bundle
283
Invariant Polynomials in the Curvature of a Connection
285
Homotopy Properties of Connections and Curvature
288
Homotopy to the Trivial Connection and the ChernSimons Form
290
The LeviCivita or Riemannian Connection
291
CHAPTER 20
294
Comparison of KTheory and Cohomology Definitions
309
Bibliography
339
Index
348
Autorska prava

Ostala izdanja - Prikaži sve

Uobičajeni izrazi i fraze

Bibliografski podaci