Let N and P be smooth closed manifolds of dimensions n and p respectively. Given a Thom-Boardman symbol I, a smooth map f : N → P is called an Ω I -regular map if and only if the Thom-Boardman symbol of each singular point of f is not greater than I in the lexicographic order. We will represent the group of all cobordism classes of Ω I -regular maps of n-dimensional closed manifolds into P in terms of certain stable homotopy groups. As an application we will study the relationship among the stable homotopy groups of spheres, the above cobordism group and higher singularities.

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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 T N ⊕ θN → T P 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.

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In the 2-jet space J 2 (n, p) of smooth map germs (R n , 0) → (R p , 0) with n ≥ p ≥ 2, we consider the subspace Ω n−p+1,0 (n, p) consisting of all 2-jets of regular germs and map germs with fold singularities. In this paper we determine the homotopy type of the space Ω n−p+1,0 (n, p). Let N and P be smooth (C ∞ ) manifolds of dimensions n and p. A smooth map f : N → P is called a fold-map if f has only fold singularities. We will prove that this homotopy type is very useful in finding invariants of fold-maps. For instance, by applying the homotopy principle for fold-maps in [An3] and [An4] we prove that if n − p + 1 is odd and P is connected, then there exists a surjection of the set of cobordism classes of fold-maps into P to the stable homotopy groupHere, ν k P is the normal bundle of P in R p+k and γ G n−p+1, denote the canonical vector bundles of dimension over the grassman manifold G n−p+1, . We also prove the oriented version.