Topological and Homological Structure of Latent Manifolds in Geometric World Models: Volume IV — Persistent Homology, Cohomology, and Qualitative Invariants of Latent Dynamics

This fourth volume complements the Riemannian calculus developed in Volumes I–III by examining the topological and homological structure of latent manifolds in geometric world models. While the metric, connection, and curvature describe local geometric properties, topological invariants capture global qualitative features—connectedness, holes, voids, and higher-dimensional cycles—that persist under continuous deformations. The article develops the relevant tools of algebraic topology (simplicial homology, persistent homology, de Rham cohomology) and shows how they constrain and illuminate the organisation of latent dynamics, multi-stability, and generalisation in world models. Relations between the Hodge Laplacian and the graph Laplacians of Volume III are emphasised, together with practical computational pipelines.