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

TL;DR

This paper provides a detailed characterization of certain 3-manifolds constructed from Lens spaces, $S^2 imes S^1$, and torus bundles over the circle using Morse-Bott functions, extending previous classifications and strengthening known results.

Contribution

It introduces a new characterization of these 3-manifolds via Morse-Bott functions, building upon and explicitly strengthening Saeki's earlier work from 2006.

Findings

Characterization of 3-manifolds via Morse-Bott functions.

Extension of previous classification results.

Explicit description of manifolds with specific Morse-Bott functions.

Abstract

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.