Generic smooth maps with sphere fibers

Abstract

In this paper, we study various topological properties of generic smooth maps between manifolds whose regular fibers are disjoint unions of homotopy spheres. In particular, we show that if a closed $4$-manifold admits such a generic map into a surface,then it bounds a $5$-manifold with nice properties. As a corollary, we 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.