Proposition 10.4.7. Let \(E \to X\) be a rank \(n\) vector bundle over a paracompact Hausdorff space of the homotopy type of a cell complex. Under the identification \(\th ([E])\cong \Sigma ^\infty \Th (E)\) of Lemma 10.2.21, an \(R\)-orientation of \([E]\) is the same as a class \(\theta \in \widetilde R^n(\Th (E))\) whose restriction to every fiber is a unit in \(\pi _0(R)\).
Proof. A class \(\theta \in \widetilde R^n(\Th (E))\) is a morphism \(\th ([E])[-n]\otimes R\to R\) of \(R\)-modules. Since the source is the colimit of \(\xi _{[E],R}\), adjunction identifies this with a morphism \(\xi _{[E],R}\to R_{\Pi _\infty (X)}\) of local systems. It is an isomorphism precisely when each restriction \(\theta _x\in \pi _0(R)\) is a unit. □
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