Poincaré duality is a fundamental theorem in algebraic topology that establishes an isomorphism between the k-th homology group and the (n-k)-th cohomology group of a closed, oriented n-dimensional manifold, providing a deep connection between the topology of a manifold and its algebraic invariants. This duality reveals that the structure of a manifold is intricately linked to its dimensions, and it plays a crucial role in the classification and study of manifolds.