On the cobordism group of Morin maps (Q1288901)

A smooth map \(f\: M^n \to {\mathbb R}^{n+k}\) is called a Morin map if \(\operatorname{rank}(df) \geq n-1\) at each point of \(M\). The cobordism group of Morin maps, \(\Sigma^1(n,k)\), may be defined in the standard way and it is a result of \textit{U. Koschorke} [Vector fields and other vector bundle morphisms -- a singularity approach, Lect. Notes Math. 847 (1981; Zbl 0459.57016)] that \(\Sigma^1(n,k) \otimes {\mathbb Q} = 0\). The author proves here that \(\Sigma^1(n,k)\) is finite and without odd torsion if \(k\) is odd or if \(k\) is even and \(n \leq 4k+2\). The result is proved by showing that a particular homotopy group, \(\pi_{n+k}(I(k))\), of a classifying space \(I(k)\) for Morin maps is the cobordism group desired and then showing, by an analysis of associated Thom spaces and the generalized Whitehead theorem, that \(\pi_{n+k}(I(k))\otimes {\mathbb Z}/p = 0\) for \(p\) odd.