Pontryagin–Thom transformation

Construction 10.3.5 (Pontryagin–Thom transformation). Let \((M,f,\sigma )\) be a \(B\)-bordism cycle of degree \(n\) over \(X\). The closed smooth manifold \(M\) has the homotopy type of a finite cell complex by [Milnor (1963), Theorem 3.5 and Corollary 6.7], so the Thom-spectrum constructions above apply to its vector bundles. Choose a smooth embedding \(i\colon M \hookrightarrow \R ^{n+k}\) for some \(k\), compactify the ambient space to \(S^{n+k}\) with basepoint at infinity, and let \(\nu \) be the rank \(k\) normal bundle. Applying \(\Sigma ^{\infty }\) to the pointed collapse map of Construction 10.1.8 and shifting gives a map of spectra \[ \S [n] \longrightarrow \th (\nu )[-k]. \] By Lemma 10.2.21, the target is the Thom spectrum of the stable normal bundle. The map \((\sigma ,f)\colon M \to B\times X\) therefore induces a map \[ \th (\nu )[-k] \longrightarrow MB \otimes \S [X]. \] The composite determines an element of \(\pi _n(MB \otimes \S [X])\). The standard arguments proving independence of the embedding and bordism invariance are included in the Pontryagin–Thom theorem quoted below. They show that the construction defines a natural homomorphism \[ \PT \colon \Omega _n^B(X) \longrightarrow \pi _n(MB\otimes \S [X]). \]