Morse functions with regular level sets consisting of $2$-dimensional spheres, $2$-dimensional tori, or Klein Bottles

TL;DR

This paper characterizes 3-dimensional manifolds using specific Morse functions with regular level sets of spheres, tori, or Klein Bottles, extending previous classifications to non-orientable and more complex cases.

Contribution

It extends prior work by classifying 3-manifolds via Morse functions with particular regular level sets, including non-orientable and connected sum cases.

Findings

Characterization of 3-manifolds with Morse functions having sphere, torus, or Klein Bottle level sets.

Extension of Saeki's orientable case to non-orientable manifolds.

Classification of Morse functions with specified regular level sets, generalizing previous results.

Abstract

In this paper, we study Morse functions with regular level sets consisting of spheres, tori, or Klein Bottles on $3$-dimensional closed manifolds. We characterize $3$-dimensional manifolds represented by connected sums each of whose summands is the product $S^1 \times S^2$ of the circle $S^1$ and the sphere $S^2$, lens spaces, or non-orientable closed and connected manifolds of genus $1$ by a certain subclass of such Morse functions. This is a kind of extensions of the orientable case, by Saeki, in 2006. This is a variant of its extension by the author for $3$-dimensional orientable manifolds represented by connected sums each of whose summands is the product $S^1 \times S^2$, lens spaces, or torus bundles over $S^1$ by a certain class of Morse-Bott functions. We also classify Morse functions with given regular level sets consisting of $S^2$, $S^1 \times S^1$, or Klein Bottles in a certain sense, generalizing some previous work by the author.