Fold-maps and the space of base point preserving maps of spheres

Let f : N --> P be a smooth map between n-dimensional oriented manifolds which has only fold singularities. Such a map is called a fold-map. For a connected closed oriented manifold P, we shall define a fold-cobordism class of a fold-map into P of degree m, under a certain cobordism equivalence. Let Omega(fold,m)(P) denote the set of all fold-cobordism classes of fold-maps into P of degree in. Let F-m denote the space. lim(k --> infinity) where F-k(m) denotes the space of all base point preserving maps of degree 72 of Sk-1. In this paper we shall prove that there exists a surjection of Omega(fold,m)(P) to the set of homotopy classes [p, F-m], which induces many fold-cobordism invariants.