Pontryagin–Thom collapse for an embedding in a sphere

Example 11.3.10. Consider our fixed embedding \(\phi \colon M \hookrightarrow \R ^N\). Since \(M\) is compact, its image is closed in \(S^N\) and misses the point at infinity. Thus \(M\hookrightarrow S^N\) is a pointed smooth embedding, and the Pontryagin-Thom construction provides a map \[ \PT (\phi )\colon S^N \to \Th (\nu ). \]