Dynamics, Cohomology and Topology

TL;DR

This paper explores how cohomology and topology reveal the structure of Morse-Smale vector fields, specifically how cohomology detects rest points and instantons through the manifold's corner structures.

Contribution

It demonstrates the relationship between cohomology structures and the dynamics of Morse-Smale vector fields using manifold with corner structures.

Findings

Cohomology detects rest points via additive structure.

Cohomology detects instantons via multiplicative structure.

Manifold with corner structures underlie the analysis.

Abstract

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.)