Orbit spaces of gradient vector fields (Q2874003)

The authors study the orbit spaces of generalized gradient vector fields for Morse functions. The study of these vector fields was started by Pajitnov. The spaces of generalized gradient vector fields for Morse functions are non-Hausdorff. The authors show that these spaces are locally contractible. They also show that the quotient map associated to each such an orbit space is a weak homotopy equivalence and has the path lifting property. The work in this paper is motivated by the desire to show that a class of dynamically important flows (namely, the gradient flows for Morse functions) have semilocally simply-connected orbit spaces. Another motivation is an observation of Arnold saying that there exists an uncountable collection of smooth vector fields on \(\mathbb{R}^5\) which are pairwise topologically equivalent but not smoothly equivalent. A third motivation is a remark made independently by Sullivan and by Verjovsky saying that manifolds of dimension 3 and 4 might be studied by looking at their Morse-gradient orbit spaces, which have dimensions 2 and 3 respectively.