Characteristic Numbers and Cobordism Obstructions

By [F2] the class $\alpha \times \beta$ is represented by the cocycle $J(\phi ,\psi )T$, and by [F4] the class $c\times d$ is represented by the cycle $S(z\otimes w)$, which is a cycle because $S(z\otimes w)$ has boundary $\mathrm{\partial }z\times w+(-1{)}^{m}z\times \mathrm{\partial }w=0$ by [F3]. The shuffle equivalence [F5] supplies the inverse $T$ and a homotopy $K$ with $dK+Kd=TS-1$ on the tensor complex, so the pairing can be computed on these representatives.

Evaluation gives $(JT)(S(z\otimes w))=J(TS(z\otimes w))$. Since $z\otimes w$ is a cycle, $TS(z\otimes w)-z\otimes w=dK(z\otimes w)$. The cocycle $J=J(\phi ,\psi )$ annihilates this boundary by [F2], so the value is $J(z\otimes w)$. This equals $\phi (z)\psi (w)$ when $(m,n)=(p,q)$, and is zero for every other bidegree with $m+n=p+q$, by the defining bidegree support of $J$.

For $(m,n)=(p,q)$, [F1] identifies these values with the two Kronecker pairings, proving the identity. Representative independence follows from [F1, F2]. The front/back formula in [F6] identifies $JA$ with ${\mathrm{pr}}_X^{\ast }\phi ⌣{\mathrm{pr}}_Y^{\ast }\psi$: only the $(p,q)$ cut survives. Since $Jd=0$, its difference from $JT$ is $J(A-T)=JHd=\delta (JH)$, so these cochains represent the same class. This comparison uses no additive Kunneth bijectivity. Extension of integral cycles to $R$ commutes with shuffle and evaluation, proving the integral-input version. This includes degree zero, empty spaces and the zero ring.

Since $M$ and $N$ are closed, $M\times N$ is a closed smooth $(m+n)$-manifold with the product smooth structure and empty boundary [F8], and the product orientation of [F2] is an $R$-orientation of it. By [F1] the fundamental class $[M\times N]$ is the unique class in $H_{m+n}(M\times N;R)$ whose restriction at each point is the local generator attached to the ray of the product orientation, and [F5] makes the class $[M]\times [N]$ well defined. It therefore suffices to prove that for every $(x,y)$ the restriction of $[M]\times [N]$ to $H_{m+n}(M\times N,M\times N\setminus \{(x,y)\};R)$ is the local generator attached to the tensor ray of ${\mu }_x$ and ${\nu }_y$.

Model computation. In ${\mathbb{R}}^m$ let $c$ be an affine $m$-simplex whose image contains $0$ in its interior and whose affine parametrization is orientation-preserving, and in ${\mathbb{R}}^n$ let $d$ be such an $n$-simplex; write ${\rho }_m,{\rho }_n,{\rho }_{m+n}$ for the positive local generators of the standard orientations. For $m,n>0$ choose the two simplices so that $(0,0)$ avoids the internal faces of their finite shuffle triangulation; this is possible by varying the two interior barycentric coordinates to avoid finitely many proper affine hyperplanes. The boundaries of $c$ and $d$ avoid the origin, so $c$ and $d$ are relative cycles representing ${\rho }_m$ and ${\rho }_n$, and by [F3] the boundary $\mathrm{\partial }(c\times d)=\mathrm{\partial }c\times d+(-1{)}^{m}c\times \mathrm{\partial }d$ lies in $C(({\mathbb{R}}^m\setminus \{0\})\times {\mathbb{R}}^n)+C({\mathbb{R}}^m\times ({\mathbb{R}}^n\setminus \{0\}))\subseteq C({\mathbb{R}}^{m+n}\setminus \{0\})$, so $c\times d$ is a relative cycle for the pair $({\mathbb{R}}^{m+n},{\mathbb{R}}^{m+n}\setminus \{0\})$. Its shuffle expansion is the sum of the terms $\mathrm{sgn}(\theta )(c\times d)\circ {\lambda }_{\theta }$ over all $(m,n)$-shuffles $\theta$ [F3]: the vertices of ${\lambda }_{\theta }$ are the successive vertices of the lattice path of $\theta$, so consecutive differences of its edge columns give the ordered coordinate increments of the path, with determinant $\mathrm{sgn}(\theta )$; subtracting successive columns does not change the determinant, so the coefficient $\mathrm{sgn}(\theta )$ makes every term positively oriented, and the images of these simplices have pairwise disjoint interiors and cover the product of the two simplex images, the standard lattice-path triangulation of a product simplex. Exactly one shuffle simplex contains the origin in its interior, and all other shuffle simplices avoid it. Its oriented class is therefore the positive local generator; equivalently $c\times d$ covers a neighbourhood of the origin exactly once, positively oriented, so by the chart convention of [F1] its class is the positive generator ${\rho }_{m+n}$ of the local group of ${\mathbb{R}}^{m+n}$ at the origin; that is, ${\rho }_m\times {\rho }_n={\rho }_{m+n}$ for the standard, hence product, orientation.

For relative cycles $c\in C_m(M,M\setminus \{x\};R)$ and $d\in C_n(N,N\setminus \{y\};R)$, the boundary formula makes $c\times d$ a relative cycle off $(x,y)$: its boundary terms are supported in $(M\setminus \{x\})\times N$ or $M\times (N\setminus \{y\})$. Changing $c$ by $\mathrm{\partial }e$ changes the product by $\mathrm{\partial }(e\times d)-(-1{)}^{m+1}e\times \mathrm{\partial }d$, a relative boundary because the second term misses $y$; changing $d$ by $\mathrm{\partial }f$ has the analogous effect, with the remaining term missing $x$. Chains already supported off $x$ or $y$ also produce chains off $(x,y)$. Thus the cross product descends to the local relative groups, and quotienting absolute cycles shows that the restriction of $[M]\times [N]$ is the cross product of their local restrictions.

First suppose $m,n>0$. Choose charts $\phi :(U,x)\to ({\mathbb{R}}^m,0)$ and $\psi :(V,y)\to ({\mathbb{R}}^n,0)$, replacing either chart by its composition with a reflection of the corresponding Euclidean space if necessary, in the integral case so that $d{\phi }_x$ and $d{\psi }_y$ carry the orientation rays of $M$ at $x$ and of $N$ at $y$ to the standard rays. Over ${\mathbb{F}}_2$ take any charts, since either sign gives the canonical local generator. In the integral case this also makes $d(\phi \times \psi {)}_{(x,y)}=d{\phi }_x\oplus d{\psi }_y$ carry the product ray to the standard ray of ${\mathbb{R}}^{m+n}={\mathbb{R}}^m\times {\mathbb{R}}^n$ [F2]. The chart maps induce isomorphisms of the local pairs and, by naturality [F4, F5], carry the relative cross product of step 1.3 to the relative cross product in the models, while by the chart convention of [F1] the local generators ${\mu }_x$, ${\nu }_y$ correspond to the positive generators ${\rho }_m$, ${\rho }_n$ of the model local groups and the product-orientation generator to ${\rho }_{m+n}$. The required pointwise identity at $(x,y)$ is therefore exactly the model identity ${\rho }_m\times {\rho }_n={\rho }_{m+n}$ proved in step 1.2. If a factor is zero-dimensional, its local fundamental class is its supplied sign times the point cycle (or the unique nonzero point cycle over ${\mathbb{F}}_2$). The point-factor shuffle has a single term; bilinearity carries that sign into the product local generator, so no nonexistent orientation-reversing zero-dimensional chart is needed.

Steps 1.2, 1.3 and 2.1 show that the restriction of $[M]\times [N]$ at every point $(x,y)$ of $M\times N$ is the local generator of the product orientation, so the characterizing property of the fundamental class [F1] gives $[M\times N]=[M]\times [N]$ in $H_{m+n}(M\times N;R)$; this is the integral identity for $R=\mathbb{Z}$. For $R={\mathbb{F}}_2$ the same shuffle formula has all signs equal to $1$, the local groups are ${\mathbb{F}}_2$ with the canonical orientation of [F7], and the computation of step 1.2 is unchanged, so the identity holds in $H_{m+n}(M\times N;{\mathbb{F}}_2)$ as well. For a disjoint union $M={⨆}_i{M}_i$ the fundamental class is the sum of the component fundamental classes and the cross product is bilinear [F1, F5], so applying the identity to each component pair gives $[M\times N]={\sum }_{i,j}[M_i\times N_j]=({\sum }_i[M_i])\times ({\sum }_j[N_j])=[M]\times [N]$; this is the asserted compatibility with the componentwise definition, and by [F6] the same reduction would already follow from checking one point in each connected component. If $M$ or $N$ is empty then $M\times N$ is empty and both sides are the zero class in the zero group; if $m=0$ or $n=0$ the corresponding shuffle set has one element of sign $1$ and the computation of step 1.2 covers the point factors.

By [F1, F3], $w(T(M\times N))={\pi }_1^{\ast }w(TM){\pi }_2^{\ast }w(TN)=w(TM)\times w(TN)$. Thus $w_{i_l}(T(M\times N))={\sum }_{a_l+b_l=i_l}{w}_{a_l}(TM)\times w_{b_l}(TN)$. Multiplying these finite sums gives the displayed indexed expansion; Koszul signs disappear over ${\mathbb{F}}_2$. By [F5] each term whose factor degrees are $m,n$ evaluates to the product of its factor numbers. A factor class above its manifold dimension is zero by top-homology vanishing and field duality; since the two degrees sum to $m+n$, every term with unequal factor degrees has such an over-dimension factor. Hence precisely the wrong-degree terms contribute zero, as stipulated in [F2].

Pontryagin defect. Complexifying the splitting [F1] and using naturality and the Whitney formula for Chern classes [F4] gives $c((T(M\times N){)}_{\mathbb{C}})={\pi }_1^{\ast }c(T{M}_{\mathbb{C}})\cdot {\pi }_2^{\ast }c(T{N}_{\mathbb{C}})$. Writing $c(T{M}_{\mathbb{C}})={\sum }_i{c}_i$ and $c(T{N}_{\mathbb{C}})={\sum }_j{c}_j$ and comparing even parts with the definition $p_k=(-1{)}^k{c}_{2k}$ [F4] gives $p(T(M\times N))-p(TM)\times p(TN)={\sum }_{a,b\ge 0}(-1{)}^{a+b+1}{c}_{2a+1}(T{M}_{\mathbb{C}})\times c_{2b+1}(T{N}_{\mathbb{C}}),$ because the even-even terms reassemble to the cross product of the two total Pontryagin classes and each odd-odd term appears with the sign recorded. Each odd Chern class of a complexified real bundle is two-torsion by [F4], so the right-hand side, a sum of cross products of two-torsion classes, is two-torsion; hence it vanishes in $H^{\ast }(M\times N;\mathbb{Z}[1\mathrm{/}2])$, giving the stated identity over $\mathbb{Z}[1\mathrm{/}2]$, which also follows directly from the away-from-two multiplicativity in [F4]. If the odd Chern classes of $T{M}_{\mathbb{C}}$ and $T{N}_{\mathbb{C}}$ all vanish the correction is zero, so the identity is integral.

The complex-manifold case. If $M$ and $N$ are complex manifolds, then $TM$ and $TN$ are complex vector bundles, and the complexification of an underlying real complex bundle is $V\oplus \bar{V}$: the map $v\otimes z\mapsto (zv,z\bar{v})$ is complex linear, with inverse $(a,\bar{b})\mapsto \frac{a+b}{2}\otimes 1+\frac{a-b}{2i}\otimes i$, where $v\mapsto \bar{v}$ is the canonical antilinear copy. The formulas respect local frames. Thus $(T{M}_{\mathbb{R}}{)}_{\mathbb{C}}\cong TM\oplus \overline{TM}$; by the conjugation formula of [F4], $c(\overline{TM})$ is obtained from $c(TM)$ by $c_i\mapsto (-1{)}^i{c}_i$, so the odd part of $c(TM\oplus \overline{TM})$ cancels in pairs and the correction of step 2.1 vanishes. Hence the integral class identity $p(T(M\times N))=p(TM)\times p(TN)$ holds for complex-manifold factors, and in particular for products of complex projective spaces.

For $J=(j_1,\dots ,j_t)$, write $p_{j_l}(T(M\times N))={\sum }_{a_l+b_l=j_l}{p}_{a_l}(TM)\times p_{b_l}(TN)+{\delta }_{j_l}$, where each ${\delta }_{j_l}$ is two-torsion by step 2.1. In the product every term containing a correction remains two-torsion and pairs to zero with the integral fundamental class by [F6]. The remaining product is the product of these individually prescribed homogeneous sums, and expands over all tuples $(a_l,b_l)$ in the statement, with positive Koszul signs. Terms of bidegree $(4a,4b)$ evaluate by [F5] to the product of the two factor numbers. For any other bidegree of the same total degree one factor exceeds its dimension; over $\mathbb{Q}$ it vanishes by top-homology vanishing and field duality [F7], so its integral product term has zero integral evaluation by coefficient naturality and injectivity of $\mathbb{Z}\to \mathbb{Q}$ [F6]. This proves the integer formula, including the wrong-degree convention.

The evaluation of a specified cross product of top-degree monomials is the single product of evaluations by [F5]. When one factor is zero-dimensional, the formulas reduce componentwise to the sum of signed point contributions over $\mathbb{Z}$ or point parity over ${\mathbb{F}}_2$; these need not equal $1$. Empty factors give zero. No choice beyond the stated supplier assumptions is used.

Unoriented case. Let $(W,{\theta }_0,{\theta }_1)$ be a bordism from $M_0$ to $M_1$ [F1]. The collars identify $[0,1)\times M_0$ and $(-1,0]\times M_1$ with collar neighbourhoods, and their zero-slice restrictions identify the boundary parts with $M_0$ and $M_1$, so $\mathrm{\partial }W$ with its canonical mod-two fundamental class is, up to the diffeomorphisms ${\theta }_i{\mathrm{\mid }}_{\{0\}\times M_i}$, the disjoint union $M_0\bigsqcup M_1$; in particular $\mathrm{\partial }W$ is a closed smooth $n$-manifold of the boundary type covered by [F2]. Applying the boundary-vanishing proposition [F2] to the boundary $\mathrm{\partial }W$ of $W$ gives $w^I[\mathrm{\partial }W]=0$ for every monomial of total degree $n$, while the componentwise definition of the Stiefel-Whitney number [F3] gives $w^I[\mathrm{\partial }W]=w^I[M_0]+w^I[M_1]$ in ${\mathbb{F}}_2$, the identification of the two boundary parts with $M_0$ and $M_1$ being a diffeomorphism and the mod-two numbers carrying no orientation sign. Hence $w^I[M_0]=w^I[M_1]$ for every monomial of total degree $n$.

Oriented case. Let $(W,{\theta }_0,{\theta }_1)$ now be an oriented bordism from $(M_0,o_0)$ to $(M_1,o_1)$ [F1]. The induced boundary orientation of $\mathrm{\partial }W$ restricts to $-o_0$ on $(\mathrm{\partial }W{)}_0$ and to $o_1$ on $(\mathrm{\partial }W{)}_1$, so by [F2] applied to the oriented boundary $\mathrm{\partial }W$ of $W$ we have $p_J[\mathrm{\partial }W]=0$ for every partition $J$ of $k$. The componentwise additivity [F3] and the collar identifications give $p_J[\mathrm{\partial }W]=p_J[(\mathrm{\partial }W{)}_0]+p_J[(\mathrm{\partial }W{)}_1]=-p_J[M_0]+p_J[M_1]$, where the sign uses the orientation-reversal rule of [F3] and the fact that ${\theta }_0$ identifies $(\mathrm{\partial }W{)}_0$ with $M_0$ carrying the negative orientation. Hence $p_J[M_0]=p_J[M_1]$. For the Stiefel-Whitney numbers of the oriented pair, the same bordism is in particular an unoriented bordism, so step 1.1 applies and gives $w^I[M_0]=w^I[M_1]$ for every monomial of total degree $4k$.

The numbers therefore descend to the cobordism-class groups of [F4]. This includes dimension zero: the empty monomial is point parity in the unoriented theory and signed count in the oriented theory, and the same boundary-vanishing argument proves invariance without assuming a classification of compact one-manifolds. Empty manifolds have zero numbers. AC is inherited from [F2, F3]; the comparison itself uses only the supplied bordism and finite component sums.

Unoriented case. Suppose $M$ is null-cobordant, so by [F1] there is a bordism from $M$ to the empty $n$-manifold. The empty manifold is a closed smooth $n$-manifold, and by [F3] each of its Stiefel-Whitney numbers is the empty componentwise sum $0$. Applying the invariance theorem [F2] to the pair $(M,\mathrm{\varnothing })$ gives $w^I[M]=w^I[\mathrm{\varnothing }]=0\in {\mathbb{F}}_2$ for every monomial $w^I$ of total degree $n$.

Oriented case. Suppose now that $(M,o)$ is a closed oriented $4k$-manifold that is null-cobordant in the oriented theory. By [F1] there is an oriented bordism from $(M,o)$ to the empty oriented $4k$-manifold, whose Pontryagin numbers are the empty sums $0$ by [F3]. Applying the oriented half of [F2] gives $p_J[M]=p_J[\mathrm{\varnothing }]=0\in \mathbb{Z}$ for every partition $J$ of $k$; the same comparison gives the vanishing of the Stiefel-Whitney numbers of $M$ as well.

Equivalence and conventions. Taking contrapositives, a nonzero $w^I[M]$ obstructs unoriented null-cobordism of $M$, and a nonzero $p_J[M]$ obstructs oriented null-cobordism; this is the stated equivalence, since a manifold is null-cobordant precisely when its class is zero in the corresponding bordism group [F1]. If $M$ is presented as the boundary of a compact $W$, the collar data of the null-cobordism definition give a bordism from $M$ to $\mathrm{\varnothing }$, so steps 1.1 and 1.2 re-derive the published boundary-vanishing propositions [F4]. In degree zero, a closed $0$-manifold is a finite set of signed points and its only Stiefel-Whitney number is the cardinality mod $2$; null-cobordism forces an even cardinality by [F4], matching step 1.1, and in the oriented theory the signed count is zero, matching step 1.2. For $n=0$ the oriented case $k=0$ and the empty manifold are both covered by the empty-sum convention; no choice beyond the cited suppliers is used, since only bordism data and the componentwise sums are compared.

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.