On the stratification of a compact 3-manifold by the trajectory spaces of a Morse-Smale flow

Abstract

We consider a Morse function $f$ and a Morse-Smale gradient-like vector field $X$ on a compact connected oriented 3-manifold $M$ such that $f$ has only one critical point of index 3. Based on Laudenbach's ideas, we will show that the flow of $X$ can be isotoped into one so that the trajectory spaces of the new flow provide a stratification for $M$. We will construct "natural" tubular neighborhoods about each given trajectory space of the new flow such that these neighborhoods are stratified by open subsets of trajectory spaces that co-bound the given one. In connection with this we introduce the concept of {\it conic stratification} of a manifold and point out that this is the appropriate condition the stratification of $M$ by trajectory spaces should be required to satisfy.