Vector Bundles and KTheory
Table of Contents
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. KTheory.
 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.
 HSpaces. 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. StiefelWhitney 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. StiefelWhitney Classes as Obstructions. Euler Classes as Obstructions.


Chapter 4. The JHomomorphism.
 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]
