Complex Bott periodicity

The wedge inclusion $X\vee S^2\to X\times S^2$ has restriction map split by the two projections. Hence reduced exactness [F4] identifies ${\tilde{K}}^0(X\wedge S^2)$ with the subgroup of classes on the product restricting to zero on both axes. In the normal form $a+b\beta$ from [F1], restriction to $X\times \{\ast \}$ is $a$, while restriction to $\{x_0\}\times S^2$ is $a(x_0)+b(x_0)\beta$. Both vanish exactly when $a=0$ and $b\in {\tilde{K}}^0(X)$.

By [F2], the class corresponding to $b$ in step 1.1 is exactly the reduced external product $b⊠\beta$. Therefore $b\mapsto b⊠\beta$ is a natural isomorphism ${\tilde{K}}^0(X)\to {\tilde{K}}^0(X\wedge S^2)={\tilde{K}}^0({\mathrm{\Sigma }}^{2}X)$. Taking $X=S^0$ with one nonbasepoint sends its rank-difference generator to $\beta$ on $S^0\wedge S^2\cong S^2$. Since the unbased point has ${\ast }_{+}=S^0$, [F3] identifies this target with the coefficient group $K^{-2}(\ast )$.

Apply step 2.1 to ${\mathrm{\Sigma }}^{n}X$. A suspension of a finite CW complex is again a compact based finite CW complex, and $S^2\wedge {\mathrm{\Sigma }}^{n}X\cong {\mathrm{\Sigma }}^{n+2}X$. Using [F3] gives the displayed isomorphism ${\tilde{K}}^{-n}(X)\cong {\tilde{K}}^{-n-2}(X)$ for every $n\ge 0$. Replacing $X$ by the compact quotient $X\mathrm{/}A$ gives the relative statement; naturality follows from naturality of external product and quotient maps.

Define positive degrees by transporting the already defined nonpositive groups along the inverse of step 3.1: choose $r$ with $q-2r\le 0$ and set $K^q(X)=K^{q-2r}(X)$ using the canonical composite of inverse Bott maps. If a larger $r$ is used, the two composites differ by a Bott isomorphism followed by its inverse, so the identification is independent of $r$. This is the unique extension for which multiplication by $\beta$ gives $K^q\cong K^{q-2}$ in every degree.

The maps in [F4] commute with external product by naturality, so the two-periodic identifications respect absolute, reduced, and relative maps and their exact sequences. This proves the stated natural periodic theory, including the zero group and the one-point boundary cases.