Fibre Bundles

Naslovnica
Springer 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

Fibre maps
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
index
323
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