Section 17.3 Stable Bundles and the Spaces \(BU\) and \(BO\)

By Proposition 16.22, \(\widetilde{K}(X)\) only sees a bundle up to adding trivial summands. To classify \(\widetilde{K}\) by a single space, we glue the spaces \(BU(k)\) together along a map that realises \(E \mapsto E \oplus \varepsilon^1\) on classifying maps.

Let \(e_1, e_2, \dots\) be the standard basis of \(\mathbb{C}^\infty\text{,}\) and let \(\sigma \colon \mathbb{C}^\infty \to \mathbb{C}^\infty\) be the shift \(e_i \mapsto e_{i+1}\text{.}\) Since \(\sigma\) is linear and injective with image orthogonal to \(e_1\text{,}\) we obtain a stabilisation map

\begin{equation*} i_k \colon BU(k) \;\longrightarrow\; BU(k+1), \qquad W \longmapsto \sigma(W) \oplus \mathbb{C}e_1. \end{equation*}

By construction \(i_k^*\gamma_{k+1} \cong \gamma_k \oplus \varepsilon^1\text{.}\) So if \(f \colon X \to BU(k)\) classifies \(E\text{,}\) then \(i_k \circ f\) classifies \(E \oplus \varepsilon^1\text{.}\)

Definition 17.10.

The infinite unitary classifying space is

\begin{equation*} BU \;:=\; \mathop{\mathrm{colim}}_{k \to \infty} BU(k), \end{equation*}

with structure maps \(i_k\text{.}\) The infinite orthogonal classifying space \(BO\) is defined analogously.

Now let \(X\) be a finite CW complex. A compact subset of a CW complex lies in a finite subcomplex, so every map \(X \to BU\) factors through some \(BU(k)\text{,}\) and any homotopy between two such maps factors through some \(BU(k')\text{.}\) Together with Theorem 17.2, this shows that for connected \(X\) the set \([X, BU]\) is the set of stable equivalence classes of complex vector bundles on \(X\text{,}\) in the sense of Proposition 16.22. Combining this with that proposition and Remark 16.17 gives the fundamental identification

\begin{equation*} K(X) \;\cong\; [X,\, \mathbb{Z} \times BU], \qquad \widetilde{K}(X) \;\cong\; [X,\, \mathbb{Z} \times BU]_*, \end{equation*}

where the second statement is for based \(X\text{.}\) The \(\mathbb{Z}\)-coordinate records the rank on each component. If \(X\) is connected, a based map lands in the basepoint component \(\{0\} \times BU\text{,}\) and since \(BU\) is simply connected we get \(\widetilde{K}(X) \cong [X, BU]_* \cong [X, BU]\text{.}\)

The first identification must use unbased homotopy classes, since for connected \(X\) the based classes only compute \(\widetilde{K}(X)\text{.}\) Equivalently, \(K(X) \cong [X_+, \mathbb{Z} \times BU]_*\text{.}\)

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