Morse Theory on Fractal Manifolds with Boundary

This paper systematically develops Morse theory on fractal mani?folds with boundary. By introducing fractal differential structures andfractal chain complexes, we define fractal Morse functions and estab?lish Morse inequalities adapted to fractal manifolds. Using the fractalStokes theorem and the spectral properties of the fractal Laplace oper?ator, we rigorously analyze the relationship between critical points andthe homology groups of the manifold, proving the deep connection be?tween critical points and fractal topological properties. Furthermore,numerical simulations of fractal manifolds are conducted to analyzethe behavior of gradient flows and their impact on the distributionof critical points, verifying the accuracy of fractal Morse theory. Theresults demonstrate the potential of this theory in addressing complexgeometric structures and broad topological applications, providing anovel pathway for the integration of fractal geometry and topology.