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

Abstract

Let $f : N \to P$ 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 foldcobordism classes of fold-maps into $P$ of degree $m$. Let $F^{m}$ denote the space $\lim _{k\to \infty}F_{k}^{m}$, where $F_{k}^{m}$ denotes the space of all base point preserving maps of degree $m$ of $S^{k-1}$. 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.