Functoriality of Thom spaces

Construction 10.1.7 (Functoriality of Thom spaces). Consider a fiberwise injective morphism of rank \(n\) vector bundles, i.e. a commutative diagram

in which \(\widetilde {f}\) induces injective \(\R \)-linear maps \(E_x \hookrightarrow E'_{f(x)}\) on fibers. Then the map \(\widetilde {f}\) induces a map \(S^{E} \to S^{E'}\) on fiberwise one-point compactifications, which preserves the sections at infinity and hence induces a continuous map \[ \Th (\widetilde {f}) \colon \Th (E) \to \Th (E'). \]