Pushforward of the trivial bundle

Example 11.3.7. Consider the terminal map \(p\colon M \to \pt \), and consider the trivial vector bundle \(E = \R ^N \to \pt \). Then the pullback bundle \(p^*(\R ^N)\) is the trivial vector bundle \(\ul {\R ^N}\) over \(M\), and we obtain a map \[ p_*\colon \Sigma ^N(M_+) \overset {\text{Example 10.1.5}}{\cong } \Th (\ul {\R ^N}) \xrightarrow {p_*} \Th (\R ^N) \overset {\text{Example 10.1.5}}{\cong } S^N. \]