Poincaré Duality is a fundamental concept in algebraic topology that establishes a relationship between the homology and cohomology groups of a manifold. Specifically, it states that for a closed oriented manifold, there is an isomorphism between the k-th homology group and the (n-k)-th cohomology group, where n is the dimension of the manifold. This duality connects various aspects of topology, emphasizing the interplay between singular homology and cohomology, the cup product structure, and the algebraic organization of cohomology rings.
congrats on reading the definition of Poincaré Duality. now let's actually learn it.