Non-singular extensions of circle-valued Morse functions

TL;DR

This work addresses when a circle-valued Morse function on a closed surface can be extended to a non-singular submersion on a 3-manifold with boundary that surface. The authors develop a combinatorial criterion based on the labeled Reeb graph $W_{f}^{\pm}$, a target graph map $h:V\to S^{1}$, and an allowable collapse $C:W_{f}^{\pm}\to V$, giving necessary and sufficient conditions for extendability. The proof is constructive: starting from an allowable collapse, they assemble a 3-manifold and a submersion using a Curley-type construction to realize the extension; conversely, any extension yields such a collapse. This extends the classical non-singular extension problem to circle-valued Morse functions and provides a concrete framework for deciding extendability via combinatorial data on Reeb graphs, with potential implications for 3-manifold topology and Morse theory.

Abstract

In this paper, we consider the non-singular extension problem for circle-valued Morse functions on closed orientable surfaces. The problem asks, given a circle-valued Morse function $f\colon M\to S^{1}$ on a closed orientable surface $M$, under what condition there exist a compact orientable 3-dimensional manifold $N$ with $\partial N = M$ and a submersion $G\colon N \to S^{1}$ such that $G|_{\partial N}=f$. We provide necessary and sufficient conditions for the existence of a non-singular extension of a circle-valued Morse function as the main theorem when a submersion on a collar neighborhood is given.

Figures (10)

• Figure 1: List of local behaviors of a collapse around the vertices of $W_{f}^{\pm}$. The signs on the symbols $M$, $N$, $S$, $G$, and $J$ correspond to the signs assigned to the vertices of $W_{f}^{\pm}$ and those symbols are adopted from C.
• Figure 2:
• Figure 3:
• Figure 4:
• Figure 5: We depict the neighborhoods of the vertices of $W_{G}$, where the top and bottom portions of the dotted circles are aligned with $\overline{G}$ according to the orientation of $S^{1}$. The numbers on the edges indicate the genera of the level surfaces of $G$ corresponding to those edges. The symbols correspond to those in \ref{['fig:2']}.

Theorems & Definitions (14)

• Theorem 1.1
• Definition 1: Circle-valued Morse function
• Remark 1
• Lemma 2.1
• Lemma 2.2
• Definition 2: Reeb graph
• Definition 3: Non-singular extension
• Definition 4: Labeled Reeb graph
• Definition 5: Collapse
• Definition 6: Allowable