Twisted Pontryagin–Thom collapse map

Construction 11.3.9 (Twisted Pontryagin-Thom collapse map). The Pontryagin-Thom collapse map also has a ‘twisted’ version, where in addition to a smooth embedding \(i\colon N \hookrightarrow N'\) of compact manifolds without boundary we are given a vector bundle \(E\) over \(N'\). It takes the form \[ \PT (i,E)\colon \Th (E) \to \Th (\nu (i) \oplus i^*E). \] To define it, choose a metric on \(E\) and consider the composite embedding \(N \xhookrightarrow {i} N' \xhookrightarrow {s_0} E\), where the second map is the zero-section. Its normal bundle is the direct sum \(\nu (i) \oplus i^*E\), so the ordinary Pontryagin–Thom construction gives a collapse map \[ E_+ \longrightarrow \Th (\nu (i) \oplus i^*E). \] We may choose its tubular neighborhood inside the open unit disk bundle of \(E\). The collapse map then sends the complement of that disk bundle, and in particular the unit sphere bundle, to the basepoint. It therefore factors through the quotient \(E_+ \to E^+\cong \Th (E)\) of Remark 10.1.4, giving the desired twisted collapse map. Its based homotopy class is independent of the metric and tubular-neighborhood choices.