Tube independence of the Pontryagin-Thom map

(The two collapses and the metric comparison.) Let $c_i:X_{+}\to {\mathrm{Th}}_{h_i}(\nu (N\subseteq X))$ be the collapse built from the $i$-th tube data, $i=1,2$. By [F2] both are based continuous maps, and there is a based homotopy between $c_1$ and $r^{-1}\circ c_2$, where $r:{\mathrm{Th}}_{h_1}(\nu )\to {\mathrm{Th}}_{h_2}(\nu )$ is the canonical radial comparison of metrics.

(Composing with the framing identification.) Let ${\mathrm{\Phi }}_{\phi }^{(i)}:{\mathrm{Th}}_{h_i}(\nu )\to N_{+}\wedge S^k$ be the framing homeomorphisms, and $q_i:=p\circ {\mathrm{\Phi }}_{\phi }^{(i)}$. By [F3], ${\mathrm{\Phi }}_{\phi }^{(2)}={\mathrm{\Phi }}_{\phi }^{(1)}\circ r^{-1}$ as based maps, hence $q_2\circ r=q_1$: the metric comparisons and the framing homeomorphisms cancel exactly. Therefore $f_2=q_2\circ c_2=q_1\circ (r^{-1}{c}_2)$ and $f_1=q_1\circ c_1$, and composing the based homotopy of step 1.1 with the fixed based map $q_1$ gives a based homotopy $f_1\simeq f_2$ by [F4].

(Conclusion.) Steps 1.1-2.1 show that any two Pontryagin-Thom maps built from compatible charts, metrics and radii are based homotopic, the framing being held fixed. The argument used only continuity, the tube-independence of the collapse and the exact metric compatibility of the framing homeomorphism; no choice beyond the inherited ${\mathrm{A}\mathrm{C}}_{\omega }$ occurs, and for $N=\mathrm{\varnothing }$ both sphere-valued maps are constant at the basepoint. For $k=0$, $N$ is clopen in $X$, and both maps send $N$ to the nonbasepoint of $S^0$ and its complement to the basepoint.