Framed cobordant submanifolds have homotopic Pontryagin-Thom maps

(The collapse is a homotopy between the end maps.) Choose tube data for the normal datum of $W$ in $X\times I$ as in [F1] and form the collapse $c_W:X\times I\to S^k$. By [F1], $c_W$ is continuous and based, and its restrictions $c_W(\cdot ,0)$ and $c_W(\cdot ,1)$ are the Pontryagin-Thom maps of $(N_0,{\phi }_0)$ and $(N_1,{\phi }_1)$ computed with the induced boundary tube data. Reading $c_W$ as a based homotopy $X_{+}\times I\to S^k$ between those two end maps, [F4] turns it into a based homotopy between the end maps.

(Replacing the induced tube data.) The induced boundary tube data are compatible tube data for $N_i$ in $X$; by [F2] the Pontryagin-Thom map of $N_i$ computed with them is based homotopic to the Pontryagin-Thom map of $(N_i,{\phi }_i)$ computed with any other compatible tube data, in particular with the data used to define $f_{(N_i,{\phi }_i)}$. Concatenating these two based homotopies with the end maps of step 1.1 yields a based homotopy $X_{+}\times I\to S^k$ from $f_{(N_0,{\phi }_0)}$ to $f_{(N_1,{\phi }_1)}$, by [F4].

(Conclusion.) Step 2.1 exhibits the required based homotopy; ignoring basepoints gives the homotopy of maps $X\to S^k$, and the explicit homotopy is the collapse $c_W$ together with the two tube-comparison homotopies at the ends. For empty ends the maps are constant at the basepoint. For $k=0$, each path $t\mapsto c_W(x,t)$ in the discrete space $S^0$ is constant, so the end characteristic maps agree. No choice beyond the inherited ${\mathrm{A}\mathrm{C}}_{\omega }$ is used.