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

AbstractThe 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]. that for a closed oriented 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 Pontrjagin 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 CP2 or CP2#CP2