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 $f:N\to P$ is called an $\Omega^{I}$-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 $\Omega^{I}$-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.