We give two types of singularities of maps between 4q-manifolds whose Thom polynomials with integer coefficients have nonvanishing coefficient of Pontrjagin class Pq . 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 singilarities of an extended map between D~4q of the fold map.
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