Characterizing $3$-dimensional manifolds represented as connected sums of Lens spaces, $S^2\times S^1$, and torus bundles over the circle by certain Morse-Bott functions

We characterize $3$-dimensional manifolds represented as connected sums of Lens spaces, copies of $S^2\times S^1$, and torus bundles over the circle by certain Morse-Bott functions. This adds to our previous result around 2024, classifying Morse functions whose preimages containing no singular points are disjoint unions of spheres and tori on $3$-dimensional manifolds represented as connected sums of connected sums of Lens spaces and copies of $S^2\times S^1$: we have strengthened and explicitized Saeki’s result, characterizing the manifolds via such functions, in 2006. We apply similar arguments. However we discuss in a self-contained way essentially.