Fundamental product theorem for complex K-theory

By [F1] and [F6], $\mu$ is the natural ring map determined by $e\otimes 1\mapsto {\mathrm{pr}}_X^{\ast }e$ and $e\otimes \gamma \mapsto {\mathrm{pr}}_X^{\ast }e{\mathrm{pr}}_{S^2}^{\ast }\gamma$.

Let $V$ be a bundle on $X\times S^2$. By [F2]–[F4], after harmless stabilization and homotopy it has data $[E,z^{-m}q]$ with $m\ge 0$ and $q$ polynomial of degree at most $n$. By [F5], if $M_{+}\oplus M_{-}$ is the spectral splitting of $(n+1)E$ for $L_{n}q$, then $[E,q]=[M_{+},I]+[M_{-},z]-[nE,I]$. Multiplying by ${\gamma }^{-m}$ gives $[V]=M_{+}⊠{\gamma }^{-m}+M_{-}⊠{\gamma }^{1-m}-nE⊠{\gamma }^{-m}$, which lies in the image of $\mu$. Since bundle classes generate $K^0(X\times S^2)$, $\mu$ is surjective.

The explicit block matrices in [F5] give two stabilization identities. Padding $q$ to degree at most $n+1$ and clearing the first $-z$ block yields $[(n+2)E,L_{n+1}q]\cong [(n+1)E,L_{n}q]\oplus [E,I]$. Applying the same matrix to $zq$ and clearing the final $-z$ block yields $[(n+2)E,L_{n+1}(zq)]\cong [(n+1)E,L_{n}q]\oplus [E,z]$; the possible sign $-z$ is absorbed by the constant gauge $-I$.

Under the spectral procedure of [F5], $[E,I]$ has minus bundle $0$ and $[E,z]$ has minus bundle $E$: for $z$ the monic endomorphism is $A=0$, while the Möbius reduction of the constant $I$ produces $z+t_0^{-1}I$, whose associated $A=-t_0^{-1}I$ has all eigenvalues outside $S^1$. Direct-sum compatibility in [F5] therefore turns the first identity of step 3.1 into $M_{-}(n+1,q)\cong M_{-}(n,q)$ and the second into $M_{-}(n+1,zq)\cong M_{-}(n,q)\oplus E$.

Define on a bundle represented by $[E,z^{-m}q]$ the element $\nu ([E,z^{-m}q])=[M_{-}(n,q)]\otimes \beta +[E]\otimes {\gamma }^{-m}$. The first identity in step 4.1 shows independence of the chosen degree bound $n\ge \mathrm{deg}q$.

Replacing $(m,q)$ by $(m+1,zq)$ changes the formula of step 5.1 to $([M_{-}]+[E])\otimes \beta +[E]\otimes {\gamma }^{-m-1}$. Since [F6] gives ${\gamma }^k=1+k\beta$, one has $\beta ={\gamma }^{-m}-{\gamma }^{-m-1}$; the new expression is therefore $[M_{-}]\otimes \beta +[E]\otimes {\gamma }^{-m}$. Thus $\nu$ is independent of the Laurent shift.

The remaining choices also do not change $\nu$. Varying the Möbius parameter $t_0$ through values sufficiently close to $1$ gives the spectral endomorphism over $X\times I$, whose inside subbundle has isomorphic endpoint restrictions by [F7]. By [F2] any two normalized presentations of the same bundle are homotopic, and [F3] joins their Laurent approximations by a Laurent homotopy. Applying the finite block formula and spectral splitting over $X\times I$ again identifies the endpoint minus bundles by [F7]. Isomorphic initial bundles transport all data along their restriction over $X\times \{1\}$. Hence the formula depends only on the isomorphism class of $V$.

Block linearization and spectral splitting preserve direct sums by [F5], so the formula in step 5.1 takes Whitney sums to sums. It therefore extends uniquely from bundle classes to a homomorphism $\nu :K^0(X\times S^2)\to K^0(X)\otimes \mathbb{Z}[\beta ]\mathrm{/}({\beta }^2)$.

It remains to compute $\nu \mu$. The domain is additively generated by $[E]\otimes {\gamma }^{-m}$ with $m\ge 0$: $m=0,1$ already give the basis $1,\beta$ because $\beta =1-{\gamma }^{-1}$. Now $\mu ([E]\otimes {\gamma }^{-m})=[E,z^{-m}]$, so take $q=I$ and $n=0$. Step 4.1 gives $M_{-}=0$, and step 5.1 yields $\nu \mu ([E]\otimes {\gamma }^{-m})=[E]\otimes {\gamma }^{-m}$. Additivity proves $\nu \mu =I$.

Step 9.1 makes $\mu$ injective, while step 2.1 makes it surjective; by step 1.1 it is a natural ring isomorphism. Finally [F6] identifies its domain additively with $K^0(X)\oplus K^0(X)\beta$, so bijectivity gives existence and uniqueness of the displayed normal form, including $X=\mathrm{\varnothing }$ and the zero class.