Thom's theorem: Stiefel-Whitney numbers detect unoriented bordism

The only-if direction. If $M$ and $N$ are unoriented-cobordant then all their Stiefel-Whitney numbers agree by [F1], and a null-cobordant manifold has all numbers zero by [F1]; this is the easy half of both formulations.

The converse: vanishing numbers force the Thom class to be zero. Suppose every Stiefel-Whitney number of $M$ vanishes. By [F2] the Pontryagin-Thom class $\alpha (M)\in {\pi }_n(M\mathrm{O})$ determines null-cobordism, and for a large representative rank $r$ the numbers are the evaluations $w^I[M]=⟨u_r⌣\overline{w^I}({\gamma }_r),\alpha (M)⟩$. By [F3] every detector coordinate $D_{r,n}$ in degree $n$ is a finite linear combination of normal-class monomials $u_r⌣a({\gamma }_r)$ of degree $n$, and each such normal monomial is, by the degreewise inverse substitution of [F2] (which is involutive), a finite linear combination of the tangent-number functionals $u_r⌣\overline{w^I}({\gamma }_r)$. Hence every detector coordinate evaluates to zero on $\alpha (M)$, since each of those functionals evaluates to a Stiefel-Whitney number and all of them vanish by hypothesis.

Zero detector implies null-cobordism. The detector coordinates are compatible with the fixed-coordinate structure maps and injective on the cofinal tail $r\ge n+2$ by [F4]. Since step 1.2 shows that all coordinates of $\alpha (M)$ vanish at a representative rank in that tail, injectivity gives $\alpha (M)=0$; by [F2] the manifold $M$ is null-cobordant. The argument uses the batch-30 Steenrod/Thom module-coalgebra and free-generator construction, the strict mod-two comparison, the finite-generation comparison through the universal-coefficient cone, the relative Hurewicz range, and the proved suspension compatibility through their supplier chain; no unstable dimension count or integral/mod-two identification replaces those inputs.

Equality of numbers and injectivity. For closed $n$-manifolds $M,N$, the numbers of the disjoint union satisfy $w^I[M\bigsqcup N]=w^I[M]+w^I[N]$ in ${\mathbb{F}}_2$ by the componentwise definition [F1], and $[N]=-[N]$ in the unoriented bordism group [F4]. Hence all numbers of $M$ and $N$ agree if and only if all numbers of $M\bigsqcup N$ vanish, if and only if $M\bigsqcup N$ is null-cobordant by step 2.1, if and only if $[M]+[N]=0$, if and only if $[M]=[N]$. Therefore the coordinate map ${\mathrm{\Omega }}_n^O\to {\prod }_I{\mathbb{F}}_2$ is well defined and injective in every degree, so the unoriented bordism ring is detected by the Stiefel-Whitney numbers; no polynomial presentation of that ring is asserted.

Degree zero and empty case. For $n=0$ the sole monomial is the empty product, whose number is the parity of the cardinality by [F1], the stable group is detected in degree zero through ${\widehat{H}}^0={\mathbb{F}}_{2}U$ and ranks $r\ge 2$ by [F3] and [F4], and the same argument gives that a finite set is null-cobordant exactly when its cardinality is even. The empty manifold has the zero class in ${\mathrm{\Omega }}_0^O$ and all numbers zero, consistent with the injectivity of step 3.1. All rank thresholds use the cofinal tail $r\ge n+2$ of [F4] and the fixed-coordinate structure maps; no further choice is made.