Homotopic maps with a common regular value have framed-cobordant preimages

Reparametrize $F$ by a smooth $\rho :I\to I$ equal to zero on $[0,\delta ]$ and one on $[1-\delta ,1]$, where $0<\delta <1\mathrm{/}2$. Extend the resulting homotopy to a smooth map $\tilde{F}:X\times \mathbb{R}\to S^k$ by $f_0$ for $t<0$ and $f_1$ for $t>1$. Smoothness across the ends follows from the constant collars. On a neighbourhood of the closed set $A=X\times ((-\mathrm{\infty },0]\cup [1,\mathrm{\infty }))$ this map is transverse to $\{y\}$, since its spatial derivatives there are those of $f_i$, surjective at their $y$-preimages.

Apply [F2] in the boundaryless manifold $X\times \mathbb{R}$, with $Z=\{y\}$ and closed set $A$. Obtain a transverse smooth map $G$ equal to $\tilde{F}$ near $A$. Compactness of $X$ supplies $0<\epsilon <\delta$ with $G(x,t)=f_0(x)$ for $0\le t<\epsilon$ and $G(x,t)=f_1(x)$ for $1-\epsilon <t\le 1$: a finite cover of each compact end slice by product neighbourhoods gives a positive minimum time width.

Set $W=(G{\mathrm{\mid }}_{X\times I}{)}^{-1}(y)$. In a local chart at $y$ with differential $\beta$, [F3] makes $W$ a closed, hence compact, neat codimension-$k$ submanifold, with normal framing $\mathrm{\Psi }=\beta \circ \overline{dG}$. The end restriction is transverse because it equals $f_i$. Step 2.1 gives the literal product ends $f_i^{-1}(y)\times {\mathrm{\Theta }}_i$, and there $dG$ is the pullback of $d{f}_i$ and annihilates the time direction. Thus $\mathrm{\Psi }$ is the pullback of $f_{i\ast }b$ throughout each collar. No global diffeomorphism extending the chosen target chart is required.

By [F4], $(W,\epsilon ,\mathrm{\Psi })$ is the required framed cobordism. Empty preimages cause no exception; for $k=0$ the preimages are clopen and the normal framings are the unique rank-zero maps. Countable choice is inherited from [F2] and [F3].