Generic smooth maps with sphere fibers (Q2569286)

The authors study various topological properties of generic smooth maps between manifolds whose fibers are disjoint unions of homotopy spheres. This is a generalization of the class of special generic maps. They have fold points and cusp points as their singularities. In particular it is shown that if a closed 4-manifold admits such a generic map into a surface, then it bounds a 5-manifold with nice properties using the method of Stein factorization. As a corollary, the authors show that each regular fiber of such a generic map of the 4-sphere into the plane is a homotopy ribbon 2-link and that any spun 2-knot of a classical knot can be realized as a component of a regular fiber of such a map. They also get a characterization of the standard 4-sphere in terms of sphere maps and derive that such 4-manifolds which admit the spherical maps should be null-cobordant.