Regular level sets of Lyapunov graphs of nonsingular Smale flows on 3-manifolds

TL;DR

This paper characterizes 3-manifolds admitting nonsingular Smale flows with specific regular level sets and explores how to realize templates as basic sets of such flows, linking template genus to manifold topology.

Contribution

It provides a complete characterization of 3-manifolds with certain regular level sets in NSF and establishes a connection between template genus and the topology of the realizing manifold.

Findings

A 3-manifold admits an NSF with a regular level set homeomorphic to (n+1)T^2 iff it is a connected sum of a manifold M' and n S^1×S^2.

Conditions for realizing a template as a basic set of an NSF on a 3-manifold.

Relationship between the genus of the template and the topological structure of the 3-manifold.

Abstract

In this paper, we first discuss the regular level set of a nonsingular Smale flow (NSF) on a 3-manifold. The main result about this topic is that a 3-manifold $M$ admits an NSF flow which has a regular level set homeomorphic to $(n+1)T^{2}$ $(n\in \mathbb{Z}, n\geq 0)$ if and only if $M=M'\sharp n S^{1}\times S^{2}$. Then we discuss how to realize a template as a basic set of an NSF on a 3-manifold. We focus on the connection between the genus of the template $T$ and the topological structure of the realizing 3-manifold $M$.