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

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 [1], 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 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.