A note on nonremovable cusp singularities (Q1348171)

Let \(f:M^4\to N^3\) be a smooth map. The author proves that if \(N^3\) is an orientable 3-manifold and \(M^4\) is a closed orientable 4-manifold such that \(\text{rank}_\mathbb{Z} H_2(M^4, \mathbb{Z}_2)=1\), then there does not exist a smooth map \(f:M^4\to N^3\) with only fold singularities. As a consequence he proves that every stable map \(f:M^4\to N^3\) always has cusp singularities, although the Thom polynomial of cusp singularities of \(f\) always vanishes.