Let $(E,\pi,X)$ a complex vector bundle over X (which we assume to be Compact-Hausdorff) Let $$f \colon E \times S^1 \to E \times S^1$$ an automorphism of the product bundle $E \times S^1$. We define:
$[E,f] := E\times D^2 \sqcup E \times D^2$ with the identification $(v,x) \sim f(x,v)$
Let's call $f$ a clutching function for the bundle $[E,f]$.
The author asks to prove that $[E,z^n] \approx \mu(E \otimes H^n)$ where, $z^n$ means the scalar multiplication by $z^n \in S^1$, $H^n$ the tensor product of $n$ copies of the Tautological Line Bundle and $\mu $ is the exterior product from $K(X) \otimes K(S^2) \to K(X \times S^2)$
My question: I don't know where to start to prove the example n. $3$, I've worked out the example 1 and 2,but for 3 I don't have a clear strategy.
Just for clarity I've uploaded an extract of the book by Hatcher - K theory
