Twisted collapse for the diagonal

Example 11.3.11. Consider the diagonal embedding \(\Delta \colon M \hookrightarrow M \times M\). Its normal bundle is isomorphic to the tangent bundle \(T_M\) of \(M\). As a twist, we use the external direct sum \(E = \ul {0} \times \nu \) over \(M \times M\), where \(\nu = \nu (\phi )\) as before. The pullback of \(E\) along \(\Delta \) is isomorphic to \(\nu \), as in Example 11.3.8, and we see that there is a trivialization \(\nu (\Delta ) \oplus \Delta ^*(E) \cong T_M \oplus \nu \cong \ul {\R ^N}\). The twisted Pontryagin-Thom map in this case takes the form \[ \PT (\Delta ,\ul {0} \times \nu )\colon M_+ \wedge \Th (\nu ) \cong \Th (\ul {0}) \wedge \Th (\nu ) \cong \Th (\ul {0} \oplus \nu ) \to \Th (\ul {\R ^N}) \cong \Sigma ^N(M_+). \]