K-theory

by Michael Francis Atiyah

These notes are based on the course of lectures I gave at Harvard in the fall of 1964. They constitute a self-contained account of vector bundles and K-theory assuming only the rudiments of point-set topology and linear algebra. One of the features of the treatment is that no use is made of ordinary homology or cohomology theory. In fact, rational cohomology is defined in terms of K-theory.The theory is taken as far as the solution of the Hopf invariant problem and a start is mode on the J-homomorphism. In addition to the lecture notes proper, two papers of mine published since 1964 have been reproduced at the end. The first, dealing with operations, is a natural supplement to the material in Chapter III. It provides an alternative approach to operations which is less slick but more fundamental than the Grothendieck method of Chapter III, and it relates operations and filtration. Actually, the lectures deal with compact spaces, not cell-complexes, and so the skeleton-filtration does not figure in the notes. The second paper provides a new approach to K-theory and so fills an obvious gap in the lecture notes.

Genres: Mathematics

240 pages, Kindle Edition

First published July 1, 1989

About the author

Sir Michael Francis was a British-Lebanese mathematician specialising in geometry.

Reviews

[Dow · Rating 4 out of 5]

Pretty poorly written, and the comment in the introduction that this only relies on linear algebra is laughable (unless that's where you learned homotopy theory...). With that aside, everything is pretty elementary, which makes this The Book on K-theory. Anyone who wants to learn the K-Theory proof of complex Bott Periodicity ought to read this book first (one should also read Milnor's Morse Theory, which contains Bott's original, beautiful, proof as well as an extension to the real case due to Atiyah, Bott, and Shapiro) to avoid ultra-modern treatments using Fredholm operators and all of that. That stuff is meant to prepare the reader for the K-Theoretic proof of the Atiyah-Singer index theorem; this is a proof which is great to know, but certainly would make a first pass at K-theory far harder than it need be.