Bordism groups of special generic mappings

Abstract:

The Pontrjagin-Thom construction expresses a relation between the oriented bordism groups of framed immersions $M^m\looparrowright \mathbb {R}^n, m<n$, and the stable homotopy groups of spheres. We apply the Pontrjagin-Thom construction to the oriented bordism groups $\mathcal M_{m,n}$ of mappings $M^m\to \mathbb {R}^n, m>n$, with mildest singularities. Recently, O. Saeki showed that for $m\ge 6$, the group $\mathcal M_{m,1}$ is isomorphic to the group of smooth structures on the sphere of dimension $m$. Generalizing, we prove that $\mathcal M_{m,n}$ is isomorphic to the $n$-th stable homotopy group $\pi ^{st}_n( \mathrm {BSDiff}_r,\mathrm {BSO}_{r+1})$, $r=m-n$, where $\mathrm {SDiff}_r$ is the group of oriented auto-diffeomorphisms of the sphere $S^{r}$ and $\mathrm {SO}_{r+1}$ is the group of rotations of $S^r$.