Fractal Cohomology, Duality, and Renormalization Group Flow in Quantum Gravity

This paper establishes a rigorous theory of homology for fractal manifolds with non-integer Hausdorff dimensions DH, and it derives the duality between fractal and classical homology through a generalized Poincar´e duality. By integrating category theory and geometric measure theory, we define fractal chain complexes, prove the Fractal Stokes’ Theorem, and construct weighted homology groups H(α) n (X). Further, we obtain the beta function β(DH) governing dimensional renormalization f low in the fractal AdS/CFT correspondence and validate it via holographic stress tensor anomalies. High-dimensional fractal numerical algorithms (including cellular automata and Monte Carlo methods) are developed, alongside explicit conformal block expansions in fractal conformal field theory (CFT). This work bridges fractal geometry, quantum gravity, and condensed matter physics, providing testable predictions for cross-dimensional phenomena.