On the number of periodic orbits of Morse-Smale flows on graph manifolds

TL;DR

This paper establishes an upper bound on the minimal number of periodic orbits in Morse-Smale flows on certain 3-manifolds, using handle decompositions and flow constructions.

Contribution

It introduces a method to bound the minimal number of periodic orbits for Morse-Smale flows on Seifert and graph manifolds, extending previous flow construction techniques.

Findings

Provides an explicit upper bound for n(Y) on Seifert and graph manifolds

Combines handle decompositions with Morse-Smale flow constructions

Extends understanding of flow dynamics on 3-manifolds

Abstract

For a closed oriented 3-manifold $Y$ we define $n(Y)$ to be the minimal non-negative number such that in each homotopy class of non-singular vector fields of $Y$ there is a Morse-Smale vector field with less or equal to $n(Y)$ periodic orbits. We combine the construction process of Morse-Smale flows given in [2] with handle decompositions of compact orientable surfaces to provide an upper bound to the number $n(Y)$ for oriented Seifert manifolds and oriented graph manifolds prime to $\stwo\times\sone$.