Stabilized framed cobordism and the framed bordism group

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) is inherited from the framed-cobordism relation). For $d\ge 0$ and $k\ge 1$ let ${\mathrm{C}\mathrm{o}\mathrm{b}}_d(k)$ be the set of framed cobordism classes of closed framed $d$-submanifolds of $S^{d+k}$ (codimension $k$; the fixed-codimension theorem applies with $n=d+k\ge k$, Framed cobordism of framed submanifolds, Framed cobordism is an equivalence relation).

The equatorial stabilization $\sigma ={\sigma }_k:{\mathrm{C}\mathrm{o}\mathrm{b}}_d(k)\to {\mathrm{C}\mathrm{o}\mathrm{b}}_d(k+1)$ sends the class of $(N,\phi )$ to the class of $(i(N),{\nu }_{\mathrm{e}\mathrm{q}}\oplus \phi ),$ where $i:S^{d+k}\hookrightarrow S^{d+k+1}$ is the equatorial inclusion, with the new ambient coordinate placed first, and ${\nu }_{\mathrm{e}\mathrm{q}}$ is the outward unit normal of the chosen northern hemisphere along its equator (a trivialization of the normal line, equipped with the product metric). It is well defined: $i(N)$ is a closed embedded $d$-submanifold of $S^{d+k+1}$, the framing ${\nu }_{\mathrm{e}\mathrm{q}}\oplus \phi$, the equatorial normal prepended to $\phi$, is a trivialization of $\nu (i(N)\subseteq S^{d+k+1})\cong i_{\ast }\nu (N\subseteq S^{d+k})\oplus {\epsilon }^1$ over the equatorial collar, and if $W\subseteq S^{d+k}\times I$ is a framed cobordism from $(N_0,{\phi }_0)$ to $(N_1,{\phi }_1)$ then $(i\times {\mathrm{i}\mathrm{d}}_I)(W)\subseteq S^{d+k+1}\times I$ is a framed cobordism from $(i(N_0),{\nu }_{\mathrm{e}\mathrm{q}}\oplus {\phi }_0)$ to $(i(N_1),{\nu }_{\mathrm{e}\mathrm{q}}\oplus {\phi }_1)$: the equatorial product structure is preserved by $i$, the product ends map to product ends, and the framing maps to the framing with the constant normal field ${\nu }_{\mathrm{e}\mathrm{q}}$ prepended. Hence $\sigma$ carries classes to classes.

The framed bordism group ${\mathrm{\Omega }}_d^{\mathrm{f}\mathrm{r}}$ is the colimit of the sequence ${\mathrm{C}\mathrm{o}\mathrm{b}}_d(1)\stackrel{\sigma }{\to }{\mathrm{C}\mathrm{o}\mathrm{b}}_d(2)\stackrel{\sigma }{\to }{\mathrm{C}\mathrm{o}\mathrm{b}}_d(3)\stackrel{\sigma }{\to }\cdots ,$ realised as the quotient of the disjoint union ${⨆}_{k\ge 1}{\mathrm{C}\mathrm{o}\mathrm{b}}_d(k)$ by the equivalence relation generated by $s\sim {\sigma }_k(s)$ for all $s\in {\mathrm{C}\mathrm{o}\mathrm{b}}_d(k)$ and all $k$. Thus an element of ${\mathrm{\Omega }}_d^{\mathrm{f}\mathrm{r}}$ is represented by a framed $d$-submanifold of some sphere $S^{d+k}$, two representatives being equal exactly when they become framed cobordant after finitely many equatorial stabilizations; a stable framed bordism class is such a stabilization class of embedded framed manifolds. This is distinct from a stable framing on a fixed manifold, which is a trivialization after adding trivial summands, considered up to homotopy and further stabilization.

At a common level $k\ge 2$, define the proposed sum of two representatives by transporting addition in ${\pi }_{d+k}(S^k)$ through the fixed-codimension bijection ${\mathrm{\Phi }}_k$: $x{+}_{k}y={\mathrm{\Phi }}_k^{-1}({\mathrm{\Phi }}_k(x)+{\mathrm{\Phi }}_k(y))\mathrm{.}$ The stable theorem below proves compatibility with stabilization, so this operation descends to the colimit and is an abelian group law with the empty manifold as zero. It also proves its geometric interpretation: put the two framed representatives, with their transported framings, into separate affine half-space charts and take their disjoint union. The placement is part of this description; a union of two arbitrary intersecting or linked embeddings cannot be justified by the abstract unoriented bordism group law. No choice of representatives is built into the resulting operation, since independence is proved through the classifying bijection. The Pontryagin-Thom correspondence in fixed codimension supplies the levelwise bijections; The stable Pontryagin-Thom theorem identifies framed bordism with stable stems ↗ supplies the well-definedness of this colimit operation.