Cobordisms of Maps without Prescribed Singularities

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 -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 -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.