How to get isomorphism $K(X)\cong \mathbb{Z}\times \underset{\to n}{\lim} {\rm Vect}_n(X)$

Question

I'm reading Atiyah's K-theory, on page 44, the Lemma 2.1.1 claims that

$$ K(X)\cong \mathbb{Z}\times \underset{\to n}{\lim} {\rm Vect}_n(X) $$

I'm confused about how to get this isomorphism. Please help me, thanks!

Answer 1

The isomorphism is defined using the material above. We want to define a map $\phi:K(X)\rightarrow \mathbb Z\times \lim_{n\rightarrow\infty} \mathrm{Vect}_n(X)$. How to do that?

Well, by the material above the lemma, there exists a bundle $G$ such that $F\oplus G\cong \underline n$, where $\underline n$ denotes the trivial bundle of rank $n$.

Then define $\phi([E]-[F])=(\dim E-\dim F,\overline{E\oplus G})$, where $\overline{E\oplus G}$ denotes the class of the bundle $E\oplus G$ in $\lim_{n\rightarrow\infty} \mathrm{Vect}_n(X)$.

We check that this is well defined: If $G'$ is another bundle such that $F\oplus G'=\underline n'$, then $E\oplus G'\oplus \underline n\cong E\oplus G'\oplus F\oplus G\cong E\oplus \underline n\oplus G\cong E\oplus G\oplus n$. Hence $\overline{E\oplus G}$ and $\overline{E\oplus G'}$ are the same element in $\lim_{n\rightarrow\infty} \mathrm{Vect}_n(X)$.

We should also check that it does not depend on the representatives $E$ and $F$. So if $H$ is a third bundle we want to check that $[E\oplus H]-[F\oplus H]$ is send to the same element. Clearly $\dim(E\oplus H)-\dim(F\oplus H)=\dim E-\dim F$. Let $G''$ be a bundle such that $H\oplus G''=\underline n''$. Then $F\oplus H\oplus (G\oplus G'')\cong \underline {n+n''}$ and $E\oplus H\oplus (G\oplus G'')\cong E\oplus G\oplus n''$. We conclude that $\overline{E\oplus H\oplus (G\oplus G'')}=\overline{E\oplus G}$.

I invite you to write down the inverse of $\phi$ explicitly. If you need help, let me know!