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 prove111After the paper was written, O. Saeki informed the author that he obtained similar results using a different approach [O. Saeki, Comment. Math. Helv. 78 (2003) 627]. for a closed oriented 4-manifold M4 the following conditions are equivalent: admits a fold mapping into R3; r every orientable 3-manifold N3, every homotopy class of mappings of M4 into N3 contains a fold mapping; ere exists a cohomology class x∈H2(M4;Z) such that x⌣x is the first Pontrjagin class of M4. simply connected manifold M4, we show that M4 admits no fold mappings into N3 if and only if M4 is homotopy equivalent to CP2 or CP2 # CP2.