Linking between singular locus and regular fibers

Given a null-cobordant oriented framed link $L$ in a closed oriented $3$–manifold $M$, we determine those links in $M\setminus L$ which can be realized as the singular point set of a generic map $M\to {\mathbb{R}}^2$ that has $L$ as an oriented framed regular fiber. Then, we study the linking behavior between the singular point set and regular fibers for generic maps of $M$ into ${\mathbb{R}}^2$.