Fibre BundlesSpringer 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
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 |
339 | |
348 | |
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Uobičajeni izrazi i fraze
a₁ algebra B-morphism B₁ B₂ bijection c₁ calculate characteristic classes Chern class clutching map cofunctor cohomology commutative diagram compact complex vector bundle Corollary cross section CW-complex defined Definition denote E₁ elements exact sequence exists f₁ fibre bundle fibre homotopy following commutative diagram following diagram functor G-module G-space G₁(F H-space homeomorphism homotopy equivalence Hopf invariant inclusion induced isomorphism K(II KO(X Let f line bundle linear locally trivial manifold map f map g module monomorphism Moreover multiplication n-dimensional notations paracompact paracompact space phism polynomial principal G-bundle product bundle Proof properties proves the proposition proves the theorem quotient real vector bundle relation restriction ring SO(n Spin(n statement Stiefel-Whitney Stiefel-Whitney class structure subgroup subspace topological group topology total space transition functions trivial bundle u₁ unique V₁ Vect vector bundle morphism vector fields vector space x₁ Z₂