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 about half of the book is in good enough shape to be posted online, approximately 110 pages. 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, but unfortunately they have become rather expensive, especially since they aren't even nicely typeset.

Eventually I intend for the book to include things that aren't readily accessible elsewhere, such as the full story on the stable J homomorphism.

The part of the book currently available, about 110 pages, can be downloaded in the following formats:

• PDF file, singlepage (700 KB)
• PDF file, doublepage (700 KB)
• Postscript file, singlepage (1.8 MB, zipped)
• Postscript file, doublepage (1.8 MB, zipped)

The doublepage versions have two pages side-by-side, reduced in size to fit onto a single sheet of paper.

These files contain Version 2.0, dated January 2003. Unfortunately I haven't had time to work on the book since then, apart from correcting a few minor typos. I plan to get back to serious work on it in the spring of 2009. First on the slate of things to do is to extend the proof of Bott periodicity from the complex case to the real case, following the argument in Atiyah's book, using Clifford algebras.