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Tables of Contents for Vector Bundles and Their Applications
Chapter/Section Title
Page #
Page Count
PREFACE
vii
1 INTRODUCTION TO THE LOCALLY TRIVIAL BUNDLES THEORY
1
72
1.1 Locally trivial bundles
1
8
1.2 The structure groups of the locally trivial bundles
9
12
1.3 Vector bundles
21
13
1.4 Linear transformations of bundles
34
11
1.5 Vector bundles related to manifolds
45
19
1.6 Linear groups and related bundles
64
9
2 HOMOTOPY INVARIANTS OF VECTOR BUNDLES
73
64
2.1 The classification theorems
73
4
2.2 Exact homotopy sequence
77
14
2.3 Constructions of the classifying spaces
91
8
2.4 Characteristic classes
99
21
2.5 Geometric interpretation of some characteristic classes
120
6
2.6 K -- theory and the Chern character
126
11
3 GEOMETRIC CONSTRUCTIONS OF BUNDLES
137
32
3.1 The difference construction
137
10
3.2 Bott periodicity
147
3
3.3 Periodic K -theory
150
3
3.4 Linear representations and bundles
153
4
3.5 Equivariant bundles
157
3
3.6 Relations between complex, symplectic and real bundles
160
9
4 CALCULATION METHODS IN K-THEORY
169
26
4.1 Spectral sequences
169
10
4.2 Operations in K-theory
179
6
4.3 The Thom isomorphism and direct image
185
4
4.4 The Riemann-Roch theorem
189
6
5 ELLIPTIC OPERATORS ON SMOOTH MANIFOLDS AND K-THEORY
195
20
5.1 Symbols of pseudodifferential operators
195
5
5.2 Fredholm operators
200
6
5.3 The Sobolev norms
206
4
5.4 The Atiyah-Singer formula for the index of an elliptic operator
210
5
6 SOME APPLICATIONS OF VECTOR BUNDLE THEORY
215
32
6.1 Signatures of manifolds
215
8
6.2 C(*) -algebras and K -theory
223
8
6.3 Families of elliptic operators
231
5
6.4 Fredholm representations and asymptotic representations of discrete groups
236
4
6.5 Conclusion
240
7
INDEX
247
4
REFERENCES
251