three-manifolds and the global geometry of the universe: a mathematical approach to cosmic topology

This paper investigates the mathematical foundations of cosmic topology by studying three-manifolds and their possible geometries as spatial models of the universe. building upon thurston’s geometrization theorem and einstein’s field equations, we analyze how local curvature and global topology interact in shaping cosmological models. we emphasize the classification of 3-manifolds, the role of fundamental groups in identifying multiply connected spaces, and the mathematical implications for global geometry. the approach is mathematical in nature yet motivated by cosmological observations.