Morse-Smale flows and homotopy theory

Abstract

THIS ARTICLE investigates the relationship between the types of Morse-Smale flows which exist on a given manifold and the topology of that manifold. It is a now classical result that a Morse function f on a manifold M gives rise to a CW decomposition of that manifold with one cell for each critical point of f. In this article we formalize this correspondence and show it holds in the more general context of Morse-Smale flows with closed orbits corresponding to a pair of cells attached in one of two ways. More importantly, we use these ideas to investigate the set of orbits connecting two rest points (or closed orbits) p and 4; that is, the set of orbits asymptotic to p in forward time and 9 in negative time. One of the main results is that if there are no “intermediate” rest points or closed orbits (i.e. W’(q) fI W”(p) contains no basic sets other than p and 9) then set of orbits connecting p to q admits a cross section by a framed submanifold N of W”(p) called the connecting manifold between p and q. Moreover in $3 we show the following relationship with the CW decomposition of A4 associated to the flow.