Dynamics, Cohomology and Topology

For a smooth Morse-Smale vector field with Lyapunov constraints (Lyapunov function) one shows how and why the non-triviality of the cohomology, as concluded from its additive structure, detects rest points and the multiplicative structure of the cohomology detects instantons (trajectories between rest points). The same remains true for Lyapunov closed one form, a more general Lyapunov constraint, but in this presentation this fact is discussed only informally. These observations are based on the smooth " manifold with corner structures" of the stable/unstable sets and of the set of trajectories of such vector fields. (This paper is a written version of two talks with the same title given at IMAR Bucharest in November 2023.)