Class of a vector bundle

Example 10.2.20 (The class of a vector bundle). Let \(X\) be a paracompact Hausdorff space and let \(E \to X\) be a rank \(n\) vector bundle, classified by a map \(f\colon X \to \Gr _n(\R ^{\infty })\). Combining the identification \(\Pi _{\infty }(\Gr _n(\R ^{\infty })) \iso \bB O(n)\) of Proposition 9.2.11 with the preceding construction, we obtain a virtual vector bundle \[ [E]\colon \Pi _{\infty }(X) \xrightarrow {\Pi _{\infty }(f)} \bB O(n) \subseteq \Vect _{\R }(\pt )^{\simeq } \longrightarrow \BOP \] of rank \(n\). For the trivial bundle \(\ul {\R ^N}\) this is the constant map with value \(\R ^N\).

This construction is compatible with direct sums. Indeed, under the identifications \(\Pi _{\infty }(\Gr _n(\R ^{\infty }))\cong \bB O(n)\), the Whitney-sum map corresponds to the map induced by the block-sum homomorphism \(O(n)\times O(m)\to O(n+m)\). To see this, observe that the orthonormal frame bundle of \(E\oplus E'\) is obtained from the product over \(X\) of the orthonormal frame bundles of \(E\) and \(E'\) by extension of structure group along this homomorphism. Since the addition on \(\Vect _{\R }(\pt )^{\simeq }\) is induced by block sum, it follows that \[ [E\oplus E']\cong [E]+[E']. \]