The Topological H-Theorem: Topological Obstructions to Global Entropy Dissipation

We investigate the relationship between the topology of a system’s state space and thepossibility of defining a global arrow of time in entropy-driven dynamics, at a conceptualand structural level. In classical thermodynamics and dynamical systems, irreversibility isoften associated with the existence of a monotone Lyapunov or entropy functional governingthe evolution toward equilibrium. In this work, we propose a conceptual framework suggestingthat the existence of such a global, nonsingular entropy structure imposes nontrivialconstraints on the topology of the underlying manifold.We introduce the notion of an Entropy-Admissible Topology, defined as a manifold thatsupports a globally defined, bounded, and monotone functional compatible with a smoothirreversible evolution. Within this framework, we formulate what we refer to as the TopologicalH-Theorem as an obstruction principle: topological complexity may prevent the existenceof any global entropy structure, independently of the detailed form of the dynamics.Focusing on three-dimensional manifolds, we argue that the presence of hyperbolic componentsor nontrivial JSJ decompositions leads to the coexistence of dynamically incompatibleasymptotic regimes, obstructing global bounded entropy dissipation. The Ricciflow, equipped with Perelman’s entropy functionals, is discussed as a canonical and wellunderstoodwitness illustrating how such topological obstructions manifest through singularityformation and topology change.The results presented here should be understood as a foundational and programmaticcontribution rather than a complete theorem-proof development. The primary aim of thiswork is to articulate a structural connection between topology and irreversibility and tooutline a research framework for making this connection mathematically precise in futurestudies.