On the Topological Dynamics of Dynamical Manifolds and Their Fundamental Group

The paper aims to deduce the relation between the category of topology and algebra from viewpoint of geometry and dynamical system. We introduce and define a dynamical manifold as a manifold associated with a time parameter. We obtain the induced chain of topological dynamics on the fundamental group from the chain of dynamical maps on a dynamical manifold. For many adjunctions in this context, we deduce the limit topological dynamics and conditional topological dynamics on the fundamental group. We use the category of commutative diagrams as chains of dynamical manifolds to deduce the chains on fundamental groups. Also, we describe how the manifold changes in a dynamical system from the view of the fundamental group.