Elimination of singularities of smooth mappings of 4-manifolds into 3-manifolds

The simplest singularities of smooth mappings are fold singularities. We say that a mapping f is a fold mapping if every singular point of f is of the fold type. We prove 1 that for a closed orientable 4-manifold M4 the following conditions are equivalent: (1) M4 admits a fold mapping into R3; (2) for every orientable 3-manifold N3, every homotopy class of mappings of M4 into N3 contains a fold mapping; (3) there exists a cohomology class x ∈ H2(M4;Z) such that x ` x is the first Pontryagin class of M4. For a simply connected manifold M4, we show that M4 admits no fold mappings into N3 if and only if M4 is homotopy equivalent to CP 2 or CP 2#CP 2.