Proposition 11.3.12 (Pontryagin–Thom calculus). Pontryagin–Thom collapse maps have the following properties, up to based homotopy.
- (1)
-
Isotopy invariance. Isotopic embeddings induce homotopic collapse maps. The same holds for twisted collapse maps, provided the twisting bundle is fixed throughout the isotopy.
- (2)
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Compatibility with composition. Let \(N \xhookrightarrow {i} P \xhookrightarrow {j} Q\) be smooth embeddings of compact manifolds, and let \(E\) be a vector bundle over \(Q\). After choosing a splitting of the normal-bundle sequence \[ 0 \to \nu (i) \to \nu (ji) \to i^*\nu (j) \to 0, \] the composite \[ \Th (E) \xrightarrow {\PT (j,E)} \Th (\nu (j) \oplus j^*E) \xrightarrow {\PT (i,\nu (j)\oplus j^*E)} \Th (\nu (i) \oplus i^*\nu (j) \oplus (ji)^*E) \] agrees up to homotopy with \(\PT (ji,E)\) under the resulting identification of target Thom spaces.
- (3)
-
Transverse base change. Let \(f\colon M_1 \to M_2\) be a smooth map between compact smooth manifolds without boundary and let \(i_2\colon N_2 \hookrightarrow M_2\) be a closed smooth submanifold. Assume that \(f\) is transverse to \(N_2\), and set \(N_1 := f^{-1}(N_2)\), with inclusion \(i_1\colon N_1 \hookrightarrow M_1\) and induced map \(\bar f\colon N_1 \to N_2\). Then \(N_1\) is a smooth submanifold and there is a natural isomorphism \(\nu (i_1) \cong \bar f^*\nu (i_2)\). Under this identification, the square
commutes up to based homotopy. More generally, for a vector bundle \(E\) over \(M_2\), the square
commutes up to based homotopy.
Proof sketch. Choose Riemannian metrics and tubular neighborhoods. An isotopy of embeddings has a tubular neighborhood over the parameter interval, which gives the homotopy in (1). For (2), choose nested tubular neighborhoods for \(N \subseteq P \subseteq Q\); collapsing successively then agrees with collapsing the resulting tubular neighborhood of \(N\) in \(Q\). For (3), transversality identifies the normal bundle of \(N_1\) with \(\bar f^*\nu (i_2)\). Compactness allows the tubular neighborhoods to be chosen compatibly over the whole square. Comparing the corresponding collapse quotients gives the displayed homotopies. The same choices made in the total space of \(E\) prove the twisted statements. □
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