The Templates of Nonsingular Smale Flows on Three Manifolds

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 $n$.