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Chapter 1. Vector Bundles

1.1. Basic Definitions and Constructions . . . . . . . . . . . . 1

Sections 3. Direct Sums 5. Pullback Bundles 5. Inner Products 7.

Subbundles 8. Tensor Products 9. Associated Bundles 11.

1.2. Classifying Vector Bundles . . . . . . . . . . . . . . . . . 12

The Universal Bundle 12. Vector Bundles over Spheres 16.

Orientable Vector Bundles 21. A Cell Structure on Grassmann Manifolds 22.

Appendix: Paracompactness 24.

Chapter 2. Complex K-Theory

2.1. The Functor K(X) . . . . . . . . . . . . . . . . . . . . . . . 28

Ring Structure 31. Cohomological Properties 32.

2.2. Bott Periodicity . . . . . . . . . . . . . . . . . . . . . . . . 38

Clutching Functions 38. Linear Clutching Functions 43.

Conclusion of the Proof 45.

2.3. Adams’ Hopf Invariant One Theorem . . . . . . . . . . . 48

Adams Operations 51. The Splitting Principle 55.

2.4. Further Calculations . . . . . . . . . . . . . . . . . . . . . 61

The Thom Isomorphism 61.

Chapter 3. Characteristic Classes

3.1. Stiefel-Whitney and Chern Classes . . . . . . . . . . . . 63

Axioms and Construction 64. Cohomology of Grassmannians 69.

Applications of w1 and c1 72.

3.2. The Chern Character . . . . . . . . . . . . . . . . . . . . . 73

The J–Homomorphism 76.

3.3. Euler and Pontryagin Classes . . . . . . . . . . . . . . . . 83

The Euler Class 87. Pontryagin Classes 90.

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