Bordism groups of special generic mappings

Abstract

The Pontrjagin-Thom construction expresses a relation between the oriented bordism groups of framed immersions M m → R n , m n, with mildest singularities. Recently, O. Saeki showed that for m > 6, the group M m,1 is isomorphic to the group of smooth structures on the sphere of dimension m. Generalizing, we prove that M m,n is isomorphic to the n-th stable homotopy group π st n (BSDiff r , BSO r+1 ), r = m - n, where SDiff r is the group of oriented auto-diffeomorphisms of the sphere S r and SO r+1 is the group of rotations of S r .