On the space of trajectories of a generic gradient like vector field

Abstract

In this paper the authors explain, in great detail, how to equip the compactified (un)stable sets and trajectory spaces of a gradient-like vector field with the structure of a smooth manifold with corners, in a canonical way. This is done for vector fields which are gradient-like with respect to a proper Morse function, satisfy the Smale transversality condition, and are of standard form, ∑i≤qxi∂∂xi−∑i>qxi∂∂xi, near the zeros (critical points). As an application, the authors discuss the integration homomorphism relating the de Rham complex and the Thom–Smale complex with coefficients in a representation of the fundamental group.