Ordinary cohomology orientations

Example 10.4.8. Since \(\Line _{H\F _2}\) is contractible, every virtual vector bundle of fixed rank has a unique \(H\F _2\)-orientation. For \(R=H\Z \) and an honest real vector bundle \(E\to X\), the local system \(\xi _{[E],H\Z }\) is classified by \[ X \longrightarrow \bB O(n) \xrightarrow {\det } \bB C_2. \] Thus an \(H\Z \)-orientation is a classical orientation, equivalently a nullhomotopy of \(w_1(E)\).