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

TL;DR

This work establishes a necessary and sufficient condition for reconstructing a Morse function on a 3-manifold with prescribed regular level sets $F_a$ and $F_b$, strengthening prior sufficiency results by introducing cobordism-type invariants $P(S)$ and $P_o(S)$. The main result, Theorem $\{'thm:1'\}$, links the reconstruction problem to parity and inequality constraints between $F_a$ and $F_b$ and to the existence of a Morse function whose Reeb digraph matches a given acyclic digraph, with the regular level sets corresponding to $F_a$ and $F_b$. The approach combines Reeb-graph theory, explicit handle-attachment constructions, and a careful necessity argument via Morse theory to derive the cobordism conditions $P_o(F_b) \le P(F_a)$ and $P_o(F_a) \le P(F_b)$. This completes the 3D case for the reconstruction problem and clarifies orientability considerations, while situating the result relative to prior work on sufficiency and to higher-dimensional extensions. The findings have implications for designing Morse functions with prescribed level-set topology on 3-manifolds.

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.

Theorems & Definitions (5)

• Theorem 1: An extension of kitazawa2
• Theorem 2
• Theorem 3
• proof
• proof : A proof of Theorem \ref{['thm:1']}