The Templates of Nonsingular Smale Flows on Three Manifolds

TL;DR

This paper explores the relationship between template theory and Smale flows on 3-manifolds, using symbolic dynamics and surgeries to understand basic sets and manifold structures.

Contribution

It establishes connections between template moves, surgeries, and the topology of 3-manifolds with nonsingular Smale flows, including modeling basic sets on connected sums of $S^1 imes S^2$.

Findings

Template theory relates to basic sets of Smale flows.

Surgeries influence the number of $S^1 imes S^2$ factors.

Any template can model a basic set on some connected sum of $S^1 imes S^2$.

Abstract

In this paper, we first discuss some connections between template theory and the description of basic sets of Smale flows on 3-manifolds due to F. B\'eguin and C. Bonatti. The main tools we use are symbolic dynamics, template moves and some combinatorial surgeries. Second, we obtain some relationship between the surgeries and the number of $S^1 \times S^2$ factors of $M$ for a nonsingular Smale flow on a given closed orientable 3-manifold $M$. Besides these, we also prove that any template $T$ can model a basic set $\Lambda$ of a nonsingular Smale flow on $nS^1 \times S^2$ for some positive integer $n$.