Topology Invariants and Dimensional Flow Constraints of Fractal Manifolds

This paper proposes a framework for topological invariants applicable to fractalmanifolds and explores the topological constraint mechanism of fractal dimensionalflow. By introducing concepts such as fractal homology groups, fractal cohomol?ogy, and fractal Euler characteristic, we redefine the topological invariants of frac?tal manifolds. Furthermore, by combining fractional calculus principles, we con?struct the equations of motion on fractal manifolds and discuss their applicationsin quantum gravity and condensed matter physics. The research indicates that theevolution of fractal dimensional flow must satisfy topological conservation condi?tions, and its path is subject to strict number-theoretic constraints. The theoreticalframework presented here offers a new perspective on the quantum nature of fractalspacetime and provides potential directions for future experimental validation.