Fibre BundlesSpringer Science & Business Media, 29. lip 2013. - Broj stranica: 327 The notion of a fibre bundle first arose out of questions posed in the 1930s on the topology and geometry of manifolds. By the year 1950 the defini tion of fibre bundle had been clearly formulated, the homotopy classifica tion of fibre bundles achieved, and the theory of characteristic classes of fibre bundles developed by several mathematicians, Chern, Pontrjagin, Stiefel, and Whitney. Steenrod's book, which appeared in 1950, gave a coherent treatment of the subject up to that time. About 1955 Milnor gave a construction of a universal fibre bundle for any topological group. This construction is also included in Part I along with an elementary proof that the bundle is universal. During the five years from 1950 to 1955, Hirzebruch clarified the notion of characteristic class and used it to prove a general Riemann-Roch theorem for algebraic varieties. This was published in his Ergebnisse Monograph. A systematic development of characteristic classes and their applications to manifolds is given in Part III and is based on the approach of Hirze bruch as modified by Grothendieck. |
Sadržaj
7 | |
Morphisms of bundles | 14 |
10 | 19 |
Local properties of bundles | 20 |
Induced vector bundles | 26 |
Functorial description of the homotopy classification of vector bundles | 32 |
GENERAL FIBRE BUNDLES | 39 |
Functorial properties of fibre bundles | 45 |
Nonexistence of elements of Hopf invariant 1 | 201 |
VECTOR FIELDS ON THE SPHERE AND STABLE HOMOTOPY 203 1 Thom spaces of vector bundles | 203 |
Scategory | 205 |
Sduality and the Atiyah duality theorem | 207 |
Fibre homotopy type | 208 |
Stable fibre homotopy equivalence | 210 |
The groups JS and Top Sk | 211 |
Thom spaces and fibre homotopy type | 213 |
The cofunctor ko | 51 |
Homotopy classification of principal G bundles over CW complexes | 57 |
Local representation of vector bundle morphisms | 64 |
CHANGE OF STRUCTURE GROUP IN FIBRE BUNDLES | 70 |
Some of the homotopy groups of the classical groups | 92 |
Corepresentations of Kr | 104 |
9 | 110 |
Products in Ktheory | 116 |
The clutching construction | 122 |
BOTT PERIODICITY IN THE COMPLEX CASE | 128 |
The inverse to the periodicity isomorphism | 136 |
Calculations of Clifford algebras | 146 |
Tensor products of Clifford modules | 154 |
The Adams 4operations in Xring | 160 |
Semisimplicity of Gmodules over compact groups | 166 |
The representation ring of a torus | 172 |
Maximal tori in SUn and Un | 178 |
Maximal tori and the Weyl group of SOn | 182 |
Maximal tori and the Weyl group of Spinn | 183 |
Special representations of SOn and Spinn | 184 |
Calculation of RSOn and RSpinn | 187 |
Relation between real and complex representation rings | 190 |
Examples of real and quaternionic representations | 193 |
Spinor representations and the Kgroups of spheres | 195 |
THE HOPF INVARIANT | 196 |
Algebraic properties of the Hopf invariant | 197 |
Hopf invariant and bidegree | 199 |
Sduality and Sreducibility | 215 |
Nonexistence of vector fields and reducibility | 216 |
Nonexistence of vector fields and coreducibility | 218 |
Nonexistence of vector fields and JRP | 219 |
Real Kgroups of real projective spaces | 222 |
Relation between KORP and JRP | 223 |
Remarks on the Adams conjecture | 226 |
PART III | 230 |
CHERN CLASSES AND STIEFELWHITNEY CLASSES | 231 |
Definition of the StiefelWhitney classes and Chern classes | 232 |
Axiomatic properties of the characteristic classes | 234 |
Stability properties and examples of characteristic classes | 236 |
Splitting maps and uniqueness of characteristic classes | 237 |
Existence of the characteristic classes | 238 |
Fundamental class of sphere bundles Gysin sequence | 239 |
Orientability and StiefelWhitney classes | 246 |
6 | 257 |
Immersions and embeddings of manifolds | 263 |
Complex characteristic classes | 269 |
Examples and applications | 278 |
Dolds theory of local properties of bundles | 285 |
Connectivity of the pair 2²S²n+1 S²n1 localized at p | 291 |
Spaces where the pth power is zero | 302 |
N³S²np+1 NS2np1 | 309 |
323 | |
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a₁ algebra B-isomorphic B-morphism B₁ bijection characteristic classes clutching map cofunctor cohomology coker commutative diagram compact Corollary cross section CW-complex defined Definition denote dimension E₁ elements epimorphism exact sequence exists fibre bundle fibre F fibre map finite CW-complexes following diagram functor G-module G-space Gauss map homeomorphism homotopy classes homotopy equivalence Hopf invariant induced isomorphism classes Let f line bundle linear locally trivial manifold map f map g mod G mod H monomorphism Moreover notations open covering open neighborhood open set orthogonal paracompact space phism polynomial principal G-bundle product bundle Proof properties proves the proposition proves the theorem quotient relation restriction ring semigroup SO(n Spin statement Stiefel-Whitney Stiefel-Whitney class structure subgroup subspace surjective topology torus total space transition functions trivial bundle unique Ur(n V₁ vector bundle morphism vector fields vector space X₁