Vector Bundles and K-Theory

Table of Contents

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Introduction.

Chapter 1. Vector Bundles.

1. Basic Definitions and Constructions.

Sections. Direct Sums. Inner Products. Tensor Products. Associated Fiber Bundles.

2. Classifying Vector Bundles.

Pullback Bundles. Clutching Functions. The Universal Bundle. Cell Structures on Grassmannians. Appendix: Paracompactness

 

Chapter 2. K-Theory.

1. The Functor K(X).

Ring Structure. The Fundamental Product Theorem.

2. Bott Periodicity.

Exact Sequences. Deducing Periodicity from the Product Theorem. Extending to a Cohomology Theory. Elementary Applications.

3. Division Algebras and Parallelizable Spheres.

H-Spaces. Adams Operations. The Splitting Principle.

4. Bott Periodicity in the Real Case. [Not yet written]

5. Vector Fields on Spheres. [Not yet written]

Chapter 3. Characteristic Classes.

1. Stiefel-Whitney and Chern Classes.

Axioms and Constructions. Cohomology of Grassmannians.

2. Euler and Pontryagin Classes.

The Euler Class. Pontrygin Classes.

3. Characteristic Classes as Obstructions.

Obstructions to Sections. Stiefel-Whitney Classes as Obstructions. Euler Classes as Obstructions.

 

Chapter 4. The J-Homomorphism.

1. Lower Bounds on Im J.

The Chern Character. The e Invariant. Thom Spaces. Bernoulli Denominators.

2. Upper Bounds on Im J. [Not yet written]