Hatcher's KTVB - Proposed Isomorphism

Question

On Page 54 of Hatcher's K-theory and vector bundles text he outlines the reduced external product. I have understood the argument, including the fact that it is a restriction of the unreduced external product. What I don't understand is how he has obtained the commutative diagram in the middle of the page. In particular, why $K(X) \otimes K(Y)$ is isomorphic to $(\widetilde{K}(X) \otimes \widetilde{K}(Y)) \oplus \widetilde{K}(X) \oplus \widetilde{K}(Y) \oplus \mathbb{Z}$.

Answer 1

Since $K(X) \cong \tilde K(X) \oplus \def\Z{\mathbf Z}\Z$, we have $$ K(X) \otimes K(Y) \cong \bigl(\tilde K(X) \otimes \Z\bigr) \oplus \bigl(\tilde K(Y) \otimes \Z\bigr)$$ Taking termwise products, and using that $R \otimes \Z \cong R$, for each $R$, we have $$ K(X) \otimes K(Y) \cong \tilde K(X)\otimes \tilde K(Y) \oplus \tilde K(X) \oplus \tilde K(Y) \oplus \Z$$