Proposition 10.2.5. Let \(p\colon E\to X\) be a vector bundle over a paracompact Hausdorff space of the homotopy type of a cell complex. Then the underlying pointed anima of its Thom space is isomorphic to the Thom anima of \(\xi _E\): \[ \Pi _{\infty }(\Th (E)) \cong \Th (\xi _E). \]
Proof. We first identify the colimit of any pointed family with the cofiber of its section. We then compare this homotopy cofiber with the point-set quotient defining the classical Thom space.
Set \(B:=\Pi _{\infty }(X)\). Under the straightening-unstraightening equivalence \(\Fun (B,\An _*) \iso (\An _{/B})_*\), the spherical fibration \(\xi _E\) corresponds to the pointed object of \(\An _{/B}\) given by \[ \Pi _{\infty }(X) \xrightarrow {\Pi _{\infty }(s_{\infty })} \Pi _{\infty }(S^E) \longrightarrow \Pi _{\infty }(X). \] More generally, suppose that a functor \(\xi \colon B\to \An _*\) corresponds to a pointed object \(B\xrightarrow {s}A\to B\) of \(\An _{/B}\). By Proposition 23.5.2, the colimit of the underlying functor \(B\to \An \) is \(A\), while the colimit of the constant basepoint diagram is \(B\). Since colimits in pointed objects are obtained by forming a pushout over the basepoint, it follows that \[ \colim _B\xi \cong \cofib (s\colon B\to A). \] Applied to \(\xi _E\), this gives an isomorphism \[ \Th (\xi _E) \cong \cofib \big ( \Pi _{\infty }(X) \xrightarrow {\Pi _{\infty }(s_{\infty })} \Pi _{\infty }(S^E) \big ). \]
It remains to identify this cofiber with the classical Thom space. Choose a fiberwise metric on \(E\), which exists by paracompactness. It identifies the fiberwise one-point compactification \(S^E\) with the unit sphere bundle of \(E\oplus \ul {\R }\). Under this identification, the section \(s_{\infty }\) is the section given by the north pole in each fiber. The pair \((S^n,\infty )\) admits an NDR structure (cf. Corollary 2.3.11), which may be chosen \(O(n)\)-equivariantly and therefore induces an NDR structure on \((S^E,s_{\infty }(X))\). Moreover, \(S^E\) has the homotopy type of a cell complex, since it is the total space of a fibration over a space of the homotopy type of a cell complex with fiber \(S^n\); see Milnor (1959), Theorem 3. Thus \(X\) and \(S^E\) satisfy the hypotheses of Corollary 2.3.11, giving the isomorphism \[ \cofib \big ( \Pi _{\infty }(X) \xrightarrow {\Pi _{\infty }(s_{\infty })} \Pi _{\infty }(S^E) \big ) \simeq \Pi _{\infty }\big (S^E/s_{\infty }(X)\big ) = \Pi _{\infty }(\Th (E)), \] as desired. □
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