20 - Lectures on K-theory

Summary

In these two lectures on K-theory I shall:

1. (i) Give the elementary proof (due to Atiyah and Bott) of the Bott periodicity theorem.

2. (ii) Develop the basic machinery of K-theory (using Dold's lectures on half-exact functors) and show how Adams and Dyer have applied it to obtain Adams' result on the nonexistence of elements of Hopf invariant one.

I shall under (i) only give the surjectivity of the Bott homomorphism whereas the injectivity is obtained under (ii) from ‘general nonsense’. This shortens the exposition. For the full elementary proof see the notes of Atiyah and Bott distributed during this Summer Institute.

Vector bundles on X and vector bundles on X × S²

Let X be a compact topological space. We shall prove the Bott periodicity theorem by examining the relationship between complex vector bundles on X and complex vector bundles on X × s². A complex vector bundle may have different fibre dimensions over the various connectedness components of its base space. We consider S² as C ∪ ∞ and S¹ as {z ∥ z ∣ = 1}.

If E is a complex vector bundle on X, E_x denotes the fibre of E at x ∈ X. A clutching function for E is a function p which continuously assigns to each (x, z) ∈ X × S¹ an automorphism p(x, z) of E_x. An endomorphism of E is a function a which continuously assigns to each x ∈ X an endomorphism a(x) of E_x. a is an automorphism of E if a(x) is non-singular for x all x ∈ X.