Representability of complex K-theory

Proof. We first construct the map. Let \(f\colon X\to \Z \times BU\) be continuous. Since \(X\) is compact and \(\Z \) is discrete, the composite \(X\to \Z \) has finite image. Write \(X_m\subseteq X\) for the clopen subset on which this composite has value \(m\). By Lemma 9.2.18, the restriction \(f_m\colon X_m\to BU\) factors through some stage \(\Gr _n(\C ^\infty )\). Let \(E_m\to X_m\) be the pullback of the universal rank \(n\) bundle along this factorization. We assign to \(f_m\) the class \[ [E_m]-n[\ul {\C }]+m[\ul {\C }] \in K^0(X_m). \] These classes glue over the finite clopen decomposition \(X=\bigsqcup _m X_m\) to a class in \(K^0(X)\).

This construction is independent of all choices. Passing from \(\Gr _n(\C ^\infty )\) to \(\Gr _{n+1}(\C ^\infty )\) replaces \(E_m\) by \(E_m\oplus \ul {\C }\), and hence leaves \([E_m]-n[\ul {\C }]+m[\ul {\C }]\) unchanged. If two maps are homotopic, the homotopy has constant \(\Z \)-coordinate on each \(X_m\times [0,1]\) and, again by compactness, factors through a common Grassmannian stage after stabilization. Pulling back the universal bundle along the two ends gives isomorphic bundles by Theorem 9.1.4. Thus we get a well-defined natural map \[ \Phi _X\colon [X,\Z \times BU] \to K^0(X). \]

We prove that \(\Phi _X\) is surjective. Let \(\alpha \in K^0(X)\), and let \(X=\bigsqcup _m X_m\) be the finite clopen decomposition according to the virtual rank of \(\alpha \). It suffices to work on one \(X_m\), so assume \(\alpha \) has constant rank \(m\). Choose a presentation \(\alpha =[E]-[F]\). By Theorem 9.3.6, choose \(F'\) and \(N\) such that \(F\oplus F'\cong \ul {\C }^N\). Then \[ \alpha =[E\oplus F']-N[\ul {\C }]. \] The bundle \(E\oplus F'\) has rank \(N+m\), so Theorem 9.2.4 classifies it by a map \(g\colon X_m\to \Gr _{N+m}(\C ^\infty )\subseteq BU\). Viewed as a map \(X_m\to \{m\}\times BU\), this maps under \(\Phi _{X_m}\) to \[ [E\oplus F']-(N+m)[\ul {\C }]+m[\ul {\C }]=[E]-[F]=\alpha . \] Gluing over the finitely many \(m\) proves surjectivity.

For injectivity, suppose \(f_0,f_1\colon X\to \Z \times BU\) have the same image under \(\Phi _X\). Their images in \(K^0(X)\) have the same rank function, so the \(\Z \)-coordinates of \(f_0\) and \(f_1\) agree. We again restrict to a clopen piece \(X_m\) and choose factorizations \(f_i\colon X_m\to \Gr _{n_i}(\C ^\infty )\) classifying bundles \(E_i\). The equality \(\Phi _X(f_0)=\Phi _X(f_1)\) gives \[ [E_0]-n_0[\ul {\C }] = [E_1]-n_1[\ul {\C }] \] in \(K^0(X_m)\). By the definition of the Grothendieck group, there exists a vector bundle \(F\) and an isomorphism \[ E_0\oplus \ul {\C }^{n_1}\oplus F \cong E_1\oplus \ul {\C }^{n_0}\oplus F. \] Choose \(F'\) with \(F\oplus F'\cong \ul {\C }^q\) by Theorem 9.3.6. After stabilizing \(f_0\) by \(n_1+q\) steps and \(f_1\) by \(n_0+q\) steps, the two maps land in the same Grassmannian \(\Gr _{n_0+n_1+q}(\C ^\infty )\) and classify isomorphic vector bundles. By Theorem 9.2.4, they are homotopic. These homotopies glue over the finite clopen decomposition of \(X\), proving injectivity.

The construction is compatible with addition. If maps through \(\{m\}\times \Gr _n(\C ^\infty )\) and \(\{m'\}\times \Gr _{n'}(\C ^\infty )\) classify bundles \(E\) and \(E'\), then their sum classifies \(E\oplus E'\) at stage \(n+n'\) and has \(\Z \)-coordinate \(m+m'\). Therefore \[ [E\oplus E']-(n+n')[\ul {\C }]+(m+m')[\ul {\C }] \] equals the sum of the two associated \(K^0\)-classes. Thus \(\Phi _X\) is an isomorphism of abelian groups. □