Pontryagin–Thom collapse map

Construction 10.1.8 (Pontryagin–Thom collapse map). Let \(i\colon N \hookrightarrow N'\) be a smooth embedding of smooth manifolds, with \(N\) compact, and let \(\nu (i)\) denote its normal bundle. Choose a tubular neighborhood, given by an open neighborhood \(U \subseteq N'\) of \(N\) and a diffeomorphism from a neighborhood of the zero section in \(\nu (i)\) onto \(U\). Collapsing the complement of a smaller tubular neighborhood to the basepoint gives a pointed map \[ \PT (i)\colon N'_+ \longrightarrow \Th (\nu (i)), \] called the Pontryagin–Thom collapse map. Its homotopy class is independent of the auxiliary choices. If \(N'\) is equipped with a basepoint away from \(N\), we may choose the tubular neighborhood of \(N\) to avoid this basepoint, giving a pointed map \[ \PT (i)\colon N' \longrightarrow \Th (\nu (i)). \]