The universal Pontryagin-Thom correspondence for unoriented and oriented bordism

The collapse map is a surjective homomorphism. By [F1] the collapse assignment gives well-defined homomorphisms ${\mathrm{\Omega }}_n^O\to {\pi }_n(M\mathrm{O})$ and ${\mathrm{\Omega }}_n^{SO}\to {\pi }_n(M\mathrm{S}\mathrm{O})$, additive under disjoint union by the pinch construction, and surjective in every degree. It remains to prove injectivity and identify the inverse.

Define the transverse-preimage assignment $\beta$. At a level $s\ge \max (n+1,2)$, let $g:S^{n+s}\to T_s$ be based, smooth and transverse near its zero preimage, with image in a finite Grassmannian Thom space. By [F3] its zero preimage $P$ is a compact closed $n$-manifold with the specified pulled-back normal bundle; in the oriented theory give $TP$ the orientation for which ${\nu }_P\oplus TP$ has the sphere orientation. Define $\beta ([g])=[P]$. Stabilization preserves this preimage and orientation: in the coordinate-first model of [F2], the new normal equation near $P$ is $(t,a(x))=0$, where $a$ is the old fibre coordinate, and the new normal bundle is ${\epsilon }^1\oplus {\nu }_P$. Use the disk/sphere suspension homeomorphism modified to be linear near zero by a radial homotopy; this preserves the basepoint and gives a representative smooth and transverse there. Such a modification exists fibrewise because the radial scale factor is positive and can be interpolated to $1$ on a smaller disk, with the boundary fixed. Adding the first ambient coordinate and the first normal coordinate therefore leaves the induced tangent orientation unchanged. Every stable class has such a representative: [F1] realizes it by a classified collapse, and [F2] permits stabilization to this range.

Independence of the transverse representative. If two such maps represent the same stable class, [F4] puts their stabilized representatives at a common level and gives a based homotopy between them. Their zero preimages remain the original manifolds by step 1.2. Its compact image lies in a finite Grassmannian Thom space by [F4]. Reparametrize the homotopy to be constant on endpoint collars. Apply the relative transverse-homotopy supplier [F3], protecting the closed basepoint track, to make it smooth and transverse near its zero preimage while fixing the endpoints. The zero preimage is a compact neat normal cobordism between the endpoint preimages. In the oriented theory orient $S^{n+s}\times I$ by $(-1{)}^n$ times sphere-then-time and orient the cobordism normal-first. At a constant collar, moving time past the $n$ endpoint tangent directions identifies this tangent orientation with time-then-endpoint; the boundary orientations are consequently negative at the incoming end and positive at the outgoing end. Thus the endpoint manifolds are bordant in the appropriate theory, proving that $\beta$ is well defined on the stable colimit.

The assignments are inverse. For a normally classified collapse of $M$, the zero preimage is exactly $M$. In compatible tubular coordinates its normal differential is the positive radius rescaling followed by the supplied normal classifying isomorphism; it is invertible, and preserves the normal orientation in the oriented case. Therefore the preimage orientation is the original tangent orientation, and $\beta \alpha ([M])=[M]$ by [F2, F3]. This makes $\alpha$ injective. It is surjective by step 1.1, so for any stable $x=\alpha ([M])$ we also have $\alpha \beta (x)=\alpha \beta \alpha ([M])=x$. Hence $\beta$ is its inverse, without an additional assertion that an arbitrary transverse map is homotopic to the collapse of its own preimage.

Compatibility and the oriented case. The pinch and external-sum constructions of [F2] show that the isomorphisms are compatible with disjoint union and with the product operations, and a null-cobordism gives the zero stable class by the cylinder-collapse argument of [F1]. The oriented signs were checked in steps 1.2–2.1 using the normal-first convention. Empty preimages and degree zero are included by the based quotient conventions of [F2], and all changes of level use the fixed structure maps and the stable colimit rather than any unstable dimension count.

Conclusion. Steps 1.1–4.1 give well-defined homomorphisms that are surjective and injective in every degree, hence isomorphisms ${\mathrm{\Omega }}_n^O\cong {\pi }_n(M\mathrm{O})$ and ${\mathrm{\Omega }}_n^{SO}\cong {\pi }_n(M\mathrm{S}\mathrm{O})$, with the inverse described by the transverse preimage; the same steps give the stated compatibility with disjoint union, products and null-cobordism.