Poincaré duality

Poincaré duality is a fundamental concept in mathematics, specifically in topology, which studies shapes and spaces. It states that for a well-behaved, closed, oriented manifold (a kind of multi-dimensional shape), there is a deep relationship between its 'holes' of different dimensions. Essentially, the number of holes in one dimension (like tunnels or voids) corresponds to holes in the complementary dimension (like cavities), creating a balanced duality. This symmetry helps mathematicians understand the structure of complex shapes by linking their various topological features through algebraic tools.