On reconstructing Morse functions with prescribed level sets on $3$-dimensional manifolds and a necessary and sufficient condition for the reconstruction

TL;DR

This paper establishes a necessary and sufficient condition for reconstructing Morse functions with prescribed regular level sets on 3D manifolds, advancing previous work that only provided sufficient conditions.

Contribution

It provides a complete characterization for the reconstruction problem, strengthening earlier results by including both necessary and sufficient conditions.

Findings

Derived a necessary and sufficient condition for Morse function reconstruction

Extended previous results from only sufficient conditions to a full characterization

Applicable to 3-dimensional manifolds with prescribed regular level sets

Abstract

We discuss a necessary and sufficient condition for reconstruction of Morse functions with prescribed (regular) level sets on $3$-dimensional manifolds. The present work strengthens a previous result of the author where only sufficient conditions are studied. Our new work is also regarded as a kind of addenda.