Lemma 10.2.19. Let \(X\) be a paracompact Hausdorff space and let \(f\colon X \to \Gr _n(\R ^{\infty })\) be a continuous map. Let \(p\colon E \to X\) be the rank \(n\) vector bundle classified by \(f\). Then the map \(\xi _E\colon \Pi _{\infty }(X) \to \An _*\) is given by the composite \[ \Pi _{\infty }(X) \xrightarrow {\Pi _{\infty }(f)} \Pi _{\infty }(\Gr _n(\R ^{\infty })) \iso \bB O(n) \xrightarrow {J_n^{\mathrm {un}}} \An _*, \] where the middle isomorphism is the one from Proposition 9.2.11.
Proof. By definition, \(p\colon E \to X\) is the pullback of the universal bundle \(p_{\univ }\colon E_{\univ } \to \Gr _n(\R ^{\infty })\). The induced square of sphere bundles is carried to a pullback of animae by Proposition 2.3.19. Under straightening-unstraightening, pullback corresponds to precomposition, so it suffices to prove the claim for \(p_{\univ }\). The Stiefel manifold \(V_n(\R ^\infty )\) of orthonormal \(n\)-frames is contractible, and its projection to \(\Gr _n(\R ^\infty )\) is a principal \(O(n)\)-bundle. It therefore identifies \(\Pi _{\infty }(\Gr _n(\R ^\infty ))\) with \(\bB O(n)\) by Proposition 9.2.11. Under this identification, the fiberwise one-point compactification of the universal bundle is the associated pointed sphere bundle \[ V_n(\R ^\infty ) \times _{O(n)} S^n \longrightarrow \Gr _n(\R ^\infty ). \] Its straightening is precisely \(J_n^{\mathrm {un}}\). □
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