Stabilizing a framed submanifold suspends its Pontryagin-Thom map

Choose a point outside $N$, possible because a positive-codimension submanifold has empty interior. A plane-rotation path can move that point to the chosen sphere basepoint; transporting $N$ and its framing along this path gives a framed cobordism (flatten the time at its ends). Thus we may choose a representative avoiding the basepoint. Take its tube small enough to avoid that point too. Its collapse $f$ is then based on $S^n$ itself, $n=d+k$, and constant on a neighbourhood of its basepoint, with $f^{-1}(y_0)=N$ and centre differential $\phi$.

Write the sphere complements as ${\mathbb{R}}^n$ and ${\mathbb{R}}^k$, choosing an equatorial stereographic chart of the next sphere in which the equator is $\{0\}\times {\mathbb{R}}^n$ and the chosen new normal points in the positive first coordinate. The smash $S^1\wedge S^n$ is the one-point compactification of $\mathbb{R}\times {\mathbb{R}}^n$: the quotient of the compact product of the two one-point compactifications has exactly this open complement of its collapsed wedge, and its neighbourhoods at the collapsed point have compact complements. Under this identification ${\mathrm{i}\mathrm{d}}_{S^1}\wedge f$ is $F(t,x)=(t,z(f(x)))(f(x)\mathrm{\ne }\mathrm{\infty }),F(t,x)=\mathrm{\infty }(f(x)=\mathrm{\infty })\mathrm{.}$ It is continuous by the smash quotient, represents $E[f]$, and near its centre fibre it is smooth. Its centre preimage is exactly $\{0\}\times N$, with normal differential $(a,v)\mapsto (a,\phi (v))$, the prepended framing of [F1]. This is a statement about the class, not an assertion that a radial stabilized tube collapse equals a suspension pointwise.

By the continuous local-smooth version of [F3], $F$ is homotopic to the collapse of its framed centre preimage, namely $\sigma (N,\phi )$. Hence its free class is $\mathrm{P}\mathrm{T}(\sigma (N,\phi ))$, and [F4] identifies the corresponding based classes. Since $[F]=E[f]$, the desired identity follows. Both constructions respect classes, so it gives a map of directed systems.