Stable homotopy groups of spheres and higher singularities

Abstract

We will construct an isomorphism of the group of all cobordism classes of fold-maps of degree 0 of $n$-dimensional closed oriented manifolds to the $n$-sphere to the $n$-th stable homotopy group $\pi _{n}^{s}$ of spheres. As an application we will show that elements of $\pi _{n}^{s}$ are detected by higher singularities of certain maps in dimensions $n < 8$.