Lower Bounds for Equilibria of Covering Flows on Manifolds

Let X be a vector field on a compact manifold M, let ρ: N → M be a universal covering map, and let X(ρ) be the induced vector field on N. In this article, we obtain lower bounds on the number of bounding surfaces for the regions of stability in N with respect to X(ρ). The results are particularly useful because they apply to Euclidean spaces in which most practical applications occur. We develop homology and homotopy methods which allow us to obtain results for noncompact manifolds in cases where Morse–Smale approach does not give adequate lower bounds.