Rigorous Proof of the Fractal Poincar´e Duality Theorem in Non-Manifolds

This paper rigorously proves the extension of the Poincar duality to noncompactfractal manifolds. By investigating fractal manifolds with Hausdorff dimension DHand satisfying the condition of Ahl regularity, we establish the following resultthrough comprehensive application of fractal measure theory, recursive constructionof chain complexes, and fractal differential structures:Hkc(X) ∼= HDH−kBM (X),where Hkc(X) represents compactly supported cohomology and HDH−kBM (X) denotesBorel-Moore homology. The proof incorporates the fractal Stokes theorem, recur?sive fractal chain construction, and spectral properties of the fractal Laplace oper?ator, providing a theoretical foundation for topology research in fractal geometry.This work is particularly relevant for studies on topology and fractals in quantumgravity and condensed matter physics.Keywords: Fractal Poincar´e Duality; Noncompact Fractal Manifolds; AhlforsRegular Measure; Fractal Chain Complex; Fractal Stokes Theorem; Hausdorff Di?mension; Borel-Moore Homology