Topological Methods for the Study of Complex Geometric Structures

Abstract

Complex geometric structures arise naturally in manifold theory, algebraic and differential geometry, singularity theory, geometric modelling, and data-driven representations of shape. Their local metric properties alone are often insufficient to determine their global organization, since spaces with similar local geometry may exhibit fundamentally different connectivity, loop structure, higher-dimensional cycles, bundle twisting, or multiscale topological behaviour. This paper develops a unified framework for the study of such structures through five complementary families of topological methods: homotopy theory and fundamental groups, homology and cohomology, fiber bundles and characteristic classes, Morse theory, and persistent homology. The emphasis is placed on the mathematical information encoded by each method, the relationships among the associated invariants, and the circumstances in which one topological descriptor reveals structural features that remain invisible to another. Standard benchmark spaces, including spheres, tori, orientable surfaces, and complex projective spaces, are used to illustrate the comparative strength of the proposed framework. In addition, a stability-aware computational pipeline is formulated for sampled, discretized, or noisy geometric data by connecting classical topological invariants with filtered simplicial complexes and persistence diagrams. The resulting synthesis provides a systematic local-to-global perspective for analysing complex geometric structures and offers practical criteria for selecting appropriate topological tools according to the geometric, algebraic, and computational characteristics of the problem.