Vector Bundles & K-Theory

The plan is for this to be a fairly short book focusing on topological K-theory and containing also the necessary background material on vector bundles and characteristic classes. Here is a provisional Table of Contents. At present only about half of the book is in good enough shape to be posted online, approximately 120 pages. This is available as a pdf file here. (I have reformatted this with narrower margins for a better reading experience on devices like an iPad, but for a paper copy with more standard size margins try printing at 85-90 per cent of full size.)

What is included in this installment is:

• 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.

Much of this material is already well covered in other sources, notably the classic books of Atiyah (K-theory) and Milnor & Stasheff (Characteristic Classes). These books are still in print, although they have become somewhat expensive.

Eventually I intend for the book to include a few more things that aren't readily accessible elsewhere, such as the full story on the stable J homomorphism. What is posted now is Version 2.2, dated November 2017. This is a minor revision of Version 2.0 from January 2003, with the addition (in 2017) of a section 3.3 giving the obstruction theory definition of Stiefel-Whitney classes.