In Hatcher's Vector Bundles and K-thery
1: http://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf a reduced version of the external product $\mu:K(X)\otimes K(Y)\to K(X\times Y)$ is used to prove the Bott periodicity theorem.
I will paste some portions from the book:
He defines $\mu(\alpha\otimes \beta)$ as $p_X^*(\alpha)p_Y^*(\beta)$ where $p_X,p_Y$ are the projections from $X\times Y$ to X and $Y$ respectively.
Then, substituting $(X,A)=(X\times Y,X\lor Y)$ in the following exact sequence
$\tilde{K}(SX)\to \tilde {K}(SA)\to \tilde{K}(X/A)\to \tilde{K}(X)\to \tilde{K}(A)$ one gets:
Then he says:
So far so good, but then he claims the induced reduced product, say $\tilde{\mu}$, induces the following commutative diagram and then $\tilde{\mu}$ is an isomorphism when $Y=S^2$.
I mean, is there an obvious reason why $\mu$ induces equality in the last on the components $\tilde{K}(X)\oplus \tilde{K}(Y) \oplus \mathbb{Z}$? Or one should go through the isomorphism given by the exact sequence of the pair $(X\times Y,X\lor Y)$ ?