Existence Theorems of Fold-maps

Let N and P be smooth manifolds of dimensions n and p (n ≥ p ≥ 2) respectively. A smooth map having only fold singularities is called a fold-map. We will study conditions for a continuous map f : N → P to be homotopic to a fold-map from the viewpoint of the homotopy principle. By certain homotopy principles for fold-maps, we prove that if there exists a fiberwise epimorphism TN ⊕ θN → TP covering f , then there exists a fold-map homotopic to f , where θN is the trivial line bundle. We also give an additional condition for finding a fold-map which folds only on a finite number of spheres of dimension p− 1.