A framing identifies the Thom target with a sphere smash product

(The framing carries one Thom model to the other.) Fix a metric $h$ on $\nu$ and let ${\phi }_{\ast }h$ be the metric on $N\times {\mathbb{R}}^k$ obtained by transporting $h$ across the bundle isomorphism $\phi$. Since $\phi$ is a fibrewise linear homeomorphism over ${\mathrm{i}\mathrm{d}}_N$, it maps $D_h(\nu )$ onto $D_{{\phi }_{\ast }h}(N\times {\mathbb{R}}^k)$ and $S_h(\nu )$ onto $S_{{\phi }_{\ast }h}(N\times {\mathbb{R}}^k)$; passing to the quotients it induces a homeomorphism ${\mathrm{Th}}_h(\nu )\to {\mathrm{Th}}_{{\phi }_{\ast }h}(N\times {\mathbb{R}}^k)$ carrying basepoint to basepoint. This map is based and functorial: for $N=\mathrm{\varnothing }$ it is the unique map of one-point spaces, while for $k=0$ it is the identity on $N_{+}$.

(The trivial model and metric independence.) By [F2] applied to the trivial bundle $N\times {\mathbb{R}}^k$ there is a canonical based homeomorphism ${\mathrm{Th}}_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}}(N\times {\mathbb{R}}^k)\cong N_{+}\wedge S^k$ for the product metric, natural in $N$. Composing with step 1.1 and the exact metric comparison $r:{\mathrm{Th}}_{{\phi }_{\ast }h}\cong {\mathrm{Th}}_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}}$ from [F4] gives a based homeomorphism ${\mathrm{\Phi }}_{\phi }:{\mathrm{Th}}_h(\nu )\to N_{+}\wedge S^k$. If $h,h^{\mathrm{\prime }}$ are two metrics on $\nu$, the two composites differ by the canonical radial homeomorphism $r^{\mathrm{\prime }}\circ r^{-1}$ of [F4], which is exactly the asserted independence: the construction is natural in the framing, because a bundle isomorphism intertwines the transported metrics and hence the two routes through the framing isomorphism and metric comparison. For $k=0$, $N\times {\mathbb{R}}^0=N$ and [F2] gives $\mathrm{Th}(N)=N_{+}\cong N_{+}\wedge S^0$; for $N=\mathrm{\varnothing }$, all four spaces are the one-point based space and all maps are the identity.

(The sphere-valued collapse.) Let $p:N_{+}\wedge S^k\to S^0\wedge S^k$ be the smash of the based collapse $N_{+}\to S^0$ (which sends $N$ to the nonbasepoint) with ${\mathrm{i}\mathrm{d}}_{S^k}$, followed by the canonical unitality homeomorphism $S^0\wedge S^k\cong S^k$ of [F5]; the composite $p$ is a based map. For any based $c:X_{+}\to \mathrm{Th}(\nu )$, the composite $p\circ {\mathrm{\Phi }}_{\phi }\circ c:X_{+}\to S^k$ is based because each factor is based, and it is independent of which unitality homeomorphism is used by the coherence clause of [F5].

(Conclusion.) Steps 1.1-3.1 construct the based homeomorphism ${\mathrm{\Phi }}_{\phi }$, prove its naturality in the framed data, its metric independence up to the canonical radial homeomorphism, and the based sphere-valued composite with any based map out of $X_{+}$. The degenerate cases $k=0$ and $N=\mathrm{\varnothing }$ were treated in steps 1.1-2.1. Nothing beyond the inherited ${\mathrm{A}\mathrm{C}}_{\omega }$ is used: all maps are the canonical ones induced by $\phi$ and the supplied metrics.