Torus fibrations on $3$-dimensional manifolds which are smooth functions with finitely many critical values and stable fold maps

We present smooth functions which are torus fibrations with finitely many critical values on $3$-dimensional closed manifolds, from (stable) fold maps into the plane. Fold maps are higher dimensional versions of Morse functions and introduced and studied first in the 1950s by Thom and Whitney. Saeki, Sakuma, and so on, are important researchers, studying related singularity theory and differential topology mainly, since the 1990s. $3$-dimensional closed manifolds always admit ones into the plane. Surface fibrations are important tools and objects in complex geometry, symplectic geometry, and $4$-dimensional geometry. It is important to construct important examples of the fibrations. In our paper, we present a $3$-dimensional related new study. Related to this, the author has succeeded in explicit construction of explicit functions with prescribed level sets and fold maps whose singular values sets are concentric on manifolds such as connected sums of sphere bundles over spheres.