Unlinking singular loci from regular fibers and its application to submersions

Given a null-cobordant oriented framed link L in a closed oriented 3–manifold M , we study the condition for the existence of a generic smooth map of M to the plane that has L as an oriented framed regular fiber such that the singular point set is unlinked with L. As an application, we give a singularity theoretical proof to the theorem, originally proved by Hector, Peralta-Salas and Miyoshi, about the realization of a link in an open oriented 3–manifold as a regular fiber of a submersion to the plane.