This paper develops DCE Topology, a systematic framework that simultaneously extends the dimension parameter α and the curvature parameter K to complex numbers within the context of topology. Building upon the foundational work on DCE Geometry, we introduce the notion of a (−1)-dimensional topological cell N(−1) = 1 and establish the generating function FX(t) = (1 + t) dim X+1 as the universal encoding of all topological information. We rigorously construct negativedimensional simplicial complexes K−n and prove that their homology groups satisfy H0(K−n) ∼= Q and Hq(K−n) = 0 for q ≥ 1, with extended Euler characteristic ˜χ(K−n) = 0 under zeta regularization. The curvature extension principle is implemented through the curvature-parametrized trigonometric functions CosK(x) = cos(√Kx) and SinK(x) = √1Ksin(√Kx), which satisfy the unified law of cosines and are entire functions of K. We extend the Gauss-Bonnet theorem, the Riemann-Roch theorem, and the Atiyah-Singer index theorem to the DCE framework, proving the unified form Te±p,q(MαK) = p(p ± q)α+1T (MαK). The duality principles D(MαK) = M−α−21/K and the spectral properties of the Laplacian on MαK are rigorously established. All open problems and conjectures are transformed into proven theorems through complete derivations and rigorous proofs. Physical applications include negative-dimensional topological quantum field theories, fractional quantum Hall effects with effective dimension deff = ν(d−1), and complex-curvature-induced topological phase transitions.
shifa liu (Wed,) studied this question.