Complex K-theory is a two-periodic generalized cohomology theory

For $q\le 0$, functoriality and homotopy invariance follow by applying [F1] to the suspended maps in [F3]. For arbitrary $q$, transport these maps through the natural Bott isomorphisms [F4]. Identity and composition are preserved by conjugating with natural isomorphisms, and homotopic maps remain equal.

Apply [F2] after each suspension in [F3]. This gives the natural long exact cofiber sequence in every nonpositive degree, with the connecting map induced by the next mapping-cone arrow. Transport through [F4] gives the long exact sequence for every integer degree. Taking the cofiber of $X\to CX$ identifies its quotient with $\mathrm{\Sigma }X$ and yields the suspension isomorphism, with the reflection signs already fixed in [F2].

For a finite wedge $W=X_1\vee \cdots \vee X_r$, restriction gives ${\tilde{K}}^0(W)\to {⨁}_i{\tilde{K}}^0(X_i)$. Let $p_i:W\to X_i$ collapse the other summands. For reduced classes $a_i$, the sum ${\sum }_i{p}_i^{\ast }{a}_i$ restricts to $a_i$ on $X_i$, because every other $p_j$ is constant there and reduced classes vanish at the basepoint. This is a two-sided inverse. Suspending and then applying [F4] proves the finite-wedge axiom in every degree; $r=0$ gives the zero group and $r=1$ the identity.

For based reduced groups and $i,j\ge 0$, apply [F5] to ${\mathrm{\Sigma }}^{i}X$ and ${\mathrm{\Sigma }}^{j}Y$ and use ${\mathrm{\Sigma }}^{i}X\wedge {\mathrm{\Sigma }}^{j}Y\cong {\mathrm{\Sigma }}^{i+j}(X\wedge Y)$. For absolute groups, apply the same construction to ${\mathrm{\Sigma }}^i{X}_{+}$ and ${\mathrm{\Sigma }}^j{Y}_{+}$; the canonical homeomorphism $X_{+}\wedge Y_{+}\cong (X\times Y{)}_{+}$ gives $K^{-i}(X)\otimes K^{-j}(Y)⟶K^{-(i+j)}(X\times Y)\mathrm{.}$ Relative products are obtained by applying the reduced construction to quotient spaces. Diagonal pullback gives internal products. Tensor associativity, the trivial-line unit, and naturality hold at degree zero. The reduced products are uniquely characterized by their pullbacks to products, so these identities commute with suspension; [F4] transports them to all degrees. Thus the graded theory has natural associative unital external and internal products and is multiplicative.

By [F3], $K^q(\ast )={\tilde{K}}^q(S^0)$. The parity calculation [F6] gives $\mathbb{Z}$ for even $q$ and zero for odd $q$, and [F4] identifies all even generators with Bott translates of $1$. Together, the preceding four steps verify the homotopy, exactness, suspension, finite-wedge, and multiplicative axioms, including zero and one-point cases.