Bordism groups of special generic mappings
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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 $S^r$.References
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Additional Information
- Rustam Sadykov
- Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
- MR Author ID: 687348
- Received by editor(s): August 14, 2003
- Received by editor(s) in revised form: November 10, 2003
- Published electronically: August 24, 2004
- Communicated by: Paul Goerss
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 133 (2005), 931-936
- MSC (2000): Primary 55N22; Secondary 55P42, 57R45
- DOI: https://doi.org/10.1090/S0002-9939-04-07586-0
- MathSciNet review: 2113946