I'm trying to understand Atiyah's proof of Bott Periodicity from his little book on K-Theory - in particular his formulation in terms of $K(P(L \oplus 1))$ where $L$ is a line bundle on a space $X$.
On page 47, I am perfectly happy with everything up to and including the sentence which reads:
"Thus, any bundle $E$ on $P(L \oplus 1)$ is isomorphic to one of the form $(\pi^\ast_0 (E^0), f, \pi^\ast_\infty(E^\ast))$ where $f$ is a clutching function."
What I cannot understand is then (in descending order of precision):
- What is meant by the "obvious [isomorphism] over the zero and infinite sections"?
- In general, how can we expect to be able to modify the clutching function $f$ at all?
- How to make sense of the latter half of page 48.
The proof of the easier statement in Hatcher, for example, is far easier to set up, and I can follow most of it - it's rather straightforward since we can talk very explicitly about spheres.
Any help will be much appreciated!