Vector Bundles and K-Theory

Contents: Chapter 1, containing basics about vector bundles. Part of Chapter 2, introducing K-theory, then proving Bott periodicity in the complex case and Adams\' theorem on the Hopf invariant, with its famous applications to division algebras and parallelizability of spheres. Not yet written is the proof of Bott Periodicity in the real case, with its application to vector fields on spheres. Most of Chapter 3, constructing Stiefel-Whitney, Chern, Euler, and Pontryagin classes and establishing their basic properties. Part of Chapter 4 on the stable J-homomorphism. What is written so far is just the application of complex K-theory, using the Chern character, to give a lower bound on the order of the image of the stable J-homomorphism.