Pushforward along the diagonal

Example 11.3.8. Consider the diagonal map \(\Delta \colon M \to M \times M\), and let \(E = \nu \times \ul {0}\) be the product bundle over \(M \times M\), where \(\nu \) is the normal bundle of our fixed embedding \(\phi \colon M \hookrightarrow \R ^N\) and \(\ul {0}\) is the zero bundle. Observe that the pullback bundle \(\Delta ^*(E) = \Delta ^*(\nu \times \ul {0})\) is isomorphic to the direct sum \(\nu \oplus \ul {0}\) of these two bundles, and hence is isomorphic to \(\nu \) itself. The above construction thus produces a map \[ \Delta _*\colon \Th (\nu ) \to \Th (\nu \times \ul {0}) \overset {\text{Example 10.1.6}}{\cong } \Th (\nu ) \wedge \Th (\ul {0}) \cong \Th (\nu ) \wedge M_+. \]