Donaldson's theorem

Donaldson's theorem is a result in mathematics that helps classify certain types of smooth, four-dimensional shapes called "4-manifolds." It states that for specific symmetrical, "positive-definite" 4-manifolds, their intersection forms (which encode how surfaces inside these shapes intersect) must be equivalent to a standard diagonal form, like a grid of points. This theorem provides critical constraints on how these 4-manifolds can be smoothly structured, playing a key role in understanding the possible shapes and properties of four-dimensional spaces in topology.