Donaldson's Theorem

Donaldson's Theorem, in differential topology and gauge theory, states that if a closed, oriented, smooth four-dimensional manifold has a definite intersection form, that form must be diagonalizable, reducing to the identity matrix over the integers in the positive definite case. Proved by Simon Donaldson using instanton gauge theory, the theorem was originally stated for simply connected manifolds and was later extended to four-manifolds with any fundamental group. It showed that many topological four-manifolds admit no smooth structure at all, revealing a sharp divide between topology and smooth geometry unique to dimension four.