The collapse of an embedded manifold classifies through the universal Thom prespectrum

The normal bundle $\nu$ of the embedding is a numerable finite-rank real bundle over the closed manifold $M$, so by [F4] it has a classifying map $f:M\to B\mathrm{O}(r)$ with a supplied bundle isomorphism $\nu \cong f^{\ast }{\gamma }_r$, radially normalized for the chosen metrics; in the oriented case $f$ lands in $B\mathrm{S}O(r)$ and the isomorphism is orientation-preserving by [F1]. By [F5] the bundle map over $f$ induces a based Thom map $\mathrm{T}\mathrm{h}(f):\mathrm{T}\mathrm{h}(\nu )\to \mathrm{T}\mathrm{h}({\gamma }_r)=M{\mathrm{O}}_r$, and $c$ is based continuous [F2]. Hence $\alpha (M,e)=[\mathrm{T}\mathrm{h}(f)\circ c]\in {\pi }_{n+r}(M{\mathrm{O}}_r)$ is well defined on the chosen data.

Stabilization. The embedding $M\hookrightarrow S^{n+r}\hookrightarrow S^{n+r+1}$ has normal bundle ${\epsilon }^1\oplus \nu$ and classifying map ${\iota }_r\circ f$ up to the canonical isomorphism of [F1] and [F4]: the added normal direction corresponds to the added trivial line, and the pullback identification ${\epsilon }^1\oplus {\gamma }_r\cong {\iota }_r^{\ast }{\gamma }_{r+1}$ transports the classification. By [F5] and the suspension identification of [F5], a representative collapse for the stabilized embedding is exactly the suspension of the collapse followed by the structure map: in product tubular coordinates with the sphere coordinate placed first, the collapse of the enlarged tube factors through $S^1\wedge \mathrm{T}\mathrm{h}(\nu )$ and the map induced by the pullback bundle projection, which is the prespectrum structure map of [F1]. Therefore the stabilization map of the prespectrum sends $\alpha (M,e)$ to the class of the stabilized embedding.

Independence of collapse data. Changing the tubular neighbourhood or its radius changes $c$ by a based homotopy fixing the normal identification by [F2], hence does not change $\alpha (M,e)$. If $f_0,f_1$ are two classifying maps, the two pullback bundles are isomorphic over $M$ by [F4], and the interpolating construction of [F4] (realizing the bundle embeddings $J_0,J_1$ into a finite trivial bundle and interpolating $v\mapsto (\cos (\pi t\mathrm{/}2)J_0(v),\sin (\pi t\mathrm{/}2)J_1(v))$, which is fiberwise injective, then radially normalizing) gives a continuous family of disk/sphere bundle maps and hence, by [F5], a based homotopy of Thom maps; after coordinate enlargement each placement is returned to the prescribed coordinate placement by a rotation: if a finite permutation has negative determinant, reverse one unused coordinate too. That coordinate vanishes on the bundle image, so this has the same endpoint map and positive determinant. Plane rotations connect the resulting orthogonal map to the identity. This also preserves the ordered orientations in the oriented theory. So the composite is independent of $f$.

Independence of the embedding: construct the ambient isotopy locally instead of importing a later isotopy theorem. Stabilize once if necessary so each spherical embedding misses a point, and use a Euclidean chart there; this also covers rank zero. In a common Euclidean stabilization use $j_t$ from [F3], smoothly reparametrized constant near $t=0,1$. At least one of its sine/cosine coefficients is nonzero at every time, so equality of images or vanishing of a tangent image forces equality of the source points or vanishing of the tangent vector in one of the original embeddings. Compactness therefore makes every $j_t$ an embedding. Its compact track $J(t,x)=(t,j_t(x))$ is embedded. Around each track point choose source coordinates $u$ and a smooth local frame $w_l(t,u)$ of the normal complement of $d{j}_t(TM)$, obtained by projecting fixed ambient vectors and applying Gram–Schmidt on a neighbourhood where they remain independent. The map $(t,u,z)\mapsto (t,j_t(u)+{\sum }_l{z}_l{w}_l(t,u))$ has invertible derivative at $z=0$, so [F6] gives a local inverse on an ambient open neighbourhood. Shrink it so its intersection with the track is just this source chart, using compactness and injectivity. Define a local horizontal velocity there by ${\mathrm{\partial }}_t{j}_t(u)$, constant in $z$. These smooth fields agree with the actual track velocity wherever they meet the track. A finite collection of such charts covers the compact track; subordinate bump weights summing to one near it paste their fields to a smooth ambient time-dependent field. Use a cutoff equal to one on the track, supported in a bounded neighbourhood, and a time cutoff supported away from the parameter ends; the latter preserves the velocity because $j_t$ is stationary there. The field thus has common compact spatial support. Its global evolution from [F6] is an ambient isotopy $H_t$, and uniqueness gives $H_t\circ j_0=j_t$ since both sides solve the same prescribed velocity equation. It is the identity outside a fixed ball, so extends to the one-point compactification fixing its basepoint.

Transport a tubular embedding by $H_t$ and its normal identification by the differential on normal quotients. This gives a continuous based family of collapse maps, with representative $c_t=c_0\circ H_t^{-1}$ after the transported fibre identification. Compose with the correspondingly transported bundle-classifying maps. At the other endpoint, [F2] removes the choice of tube/radius and step 2.2 removes the choice of bundle-classifying injection and isomorphism after stabilization. Hence the two stabilized collapse composites represent the same class, and passing to the colimit proves independence of $e$. The normal-bundle isomorphism of [F3] is not on its own used to identify collapse classes: the explicit compact-track flow supplies their based homotopy.

Additivity. Embed $M_0$ and $M_1$ in disjoint coordinate balls of $S^{n+r}$. The collapse of their union factors through the pinch map $S^{n+r}\to S^{n+r}\vee S^{n+r}$, and the Thom maps combine. Thus $\alpha (M_0\bigsqcup M_1)=\alpha (M_0)+\alpha (M_1)$, using embedding independence to return to any supplied embeddings.

Multiplicativity. Stabilize the two compact Euclidean embeddings $M^n\subset {\mathbb{R}}^{n+r}$ and $N^m\subset {\mathbb{R}}^{m+s}$ so that both $r$ and $s$ are even. These levels are cofinal in the stable colimits by [F1] and step 2.1. The product embedding lies in ${\mathbb{R}}^{n+r}\oplus {\mathbb{R}}^{m+s}$ with normal bundle ${\nu }_M⊞{\nu }_N$. The product disk/sphere pair, using the max norm and its radial comparison with the Euclidean norm, makes its collapse the smash of the two collapses; the normal classifying map is their block sum. In the oriented theory, moving from $({\nu }_M,TM,{\nu }_N,TN)$ to $({\nu }_M,{\nu }_N,TM,TN)$ has sign $(-1{)}^{ns}=1$ because $s$ is even. Hence the ordered external-sum normal orientation is precisely the normal-first orientation for the product tangent orientation in the standard product ambient space. The composite smash and block-sum Thom map therefore represents $\alpha (M\times N)$.

These external-sum maps define the stable pairing by using the cofinal even ranks. Advancing either rank by two inserts an ordered trivial two-plane on both the source and the normal bundle. To compare stabilization before and after taking the product, move that two-plane to the first coordinate position; every such block interchange has sign $(-1{)}^{2d}=1$, on both the sphere and the oriented normal fibres. The corresponding finite orthogonal coordinate permutations are joined to the identity by plane rotations, so they induce based homotopies. Coordinate placements of the classifying injections are likewise independent by step 2.2, after adding unused coordinates if needed. Thus the pairings commute up to based homotopy with the two-step transitions and descend to the original colimits; equality of any two colimit representatives is detected at a common even rank. The unoriented construction uses the same pair maps without orientation conditions. Consequently the product identity in step 5.2 holds for the stable classes of all supplied embeddings, including odd initial codimensions; no strict ring-prespectrum structure has been assumed. The empty manifold gives the constant map, and lower-codimension data are first stabilized as in the statement.