Topological properties of moduli spaces of flow lines

A lot of information about a smooth manifold can be determined from the flow lines of a gradient field associated to a smooth(Morse-Smale) function on the manifold. But the set of flow lines with an appropriate topology is also an interesting object of study in itself. In many cases, perturbations of the gradient field (and of the associated flow lines) will give rise to local partial orders on the manifold. Flow lines can then be interpreted as directed paths with respect to this order information.The project seeks to achieve information on the topological properties of such a space of flow lines between given critical points (e.g.,connected components, homotopy/homology groups) and to compare the results with concurrency set-ups.