Construction 11.3.6. Let \(f\colon X \to Y\) be a continuous map of topological spaces. Let \(p\colon E \to Y\) be a vector bundle over \(Y\), and let \(f^*(E) \to X\) be its pullback bundle. By applying functoriality of the Thom space construction (Construction 10.1.7) to the commutative diagram
we obtain a continuous map \[ f_*\colon \Th (f^*E) \to \Th (E). \]
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