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Bordism groups of special generic mappings

Author: Rustam Sadykov
Journal: Proc. Amer. Math. Soc. 133 (2005), 931-936
MSC (2000): Primary 55N22; Secondary 55P42, 57R45
Published electronically: August 24, 2004
MathSciNet review: 2113946
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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 $S^r$.

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Additional Information

Rustam Sadykov
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611

Keywords: Pontrjagin-Thom construction, special generic mappings, bordisms
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
Article copyright: © Copyright 2004 American Mathematical Society

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