Leading terms of Thom polynomials and J- images

Abstract

We give two types of singularities of maps between $4q$-manifolds whose Thom polynomials with integer coefficients have nonvanishing coefficient of Pontrjagin class $P_q$. We show that an element of the $J$-image of dimension $4q-1$ has a fold map between $S^{4q-1}$ and can be detected by the leading terms of Thom polynomials of those singularities of an extended map between $D^{4q}$ of the fold map.