Normal framings, stable normal framings and tangential framings are distinct data

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$, inherited from the normal-bundle suppliers. Three different data types occur in this pair and are not interchangeable. An actual normal framing of a closed embedded submanifold $N\subseteq S^m$ is a smooth bundle isomorphism $\phi :\nu (N\subseteq S^m)\to N\times {\mathbb{R}}^k$ for the honest normal bundle, with $k$ the codimension of $N$ in $S^m$ (Framings of a normal bundle); it is part of the data of a framed submanifold and the Pontryagin-Thom map depends on it. A stable normal framing is a trivialization of $\nu (N)\oplus {\epsilon }^j$ for some $j$, considered up to homotopy and adding further trivial summands; this is the sense in which the stable normal bundle of a compact manifold is independent of the chosen embedding (Stable normal bundle of a compact smooth manifold, Stable normal bundle is independent of the embedding). A stable tangential framing is a trivialization of $TN\oplus {\epsilon }^{j^{\mathrm{\prime }}}$ considered up to homotopy and further stabilization.

The passage between the actual and the stable notions loses information. An actual framing $\phi$ of $\nu (N)$ determines the stable normal framing of $\phi \oplus \mathrm{i}\mathrm{d}$ on $\nu (N)\oplus {\epsilon }^j$ for every $j$, and, once the splitting $TN\oplus \nu (N)=T{S}^m{\mathrm{\mid }}_N$ of the restricted ambient tangent bundle and the stable trivialization $T{S}^m\oplus {\epsilon }^1\cong {\epsilon }^{m+1}$ of the sphere are fixed, it also determines a stable tangential framing of $N$, by adding trivial summands to both sides; conversely a stable normal framing and the same ambient data determine a stable tangential framing up to homotopy. What is not available is a converse at the level of actual data: a stable framing is an equivalence class under adding trivial summands, it does not single out a codimension $k$, a level sphere $S^m$ or an embedding, and no actual framing of $\nu (N)$ is recovered from it without choosing a level and splitting off the added summands.

Consequently the left-hand side of the stable Pontryagin-Thom theorem is a colimit. The elements of ${\mathrm{\Omega }}_d^{\mathrm{f}\mathrm{r}}$ are stabilization classes of actual normal framings, not framed submanifolds of a fixed sphere (Stabilized framed cobordism and the framed bordism group), and only at a sufficiently large level does a representative of a stable class become an actual framed submanifold, where the levelwise correspondence of The stable Pontryagin-Thom theorem identifies framed bordism with stable stems becomes available. Identifying the fixed-codimension and the stable statements without keeping track of that stabilization is precisely the error this remark rules out.