Donaldson's theorem

In mathematics, and especially differential topology and gauge theory, Donaldson's theorem states that a definite intersection form of a compact, oriented, smooth manifold of dimension 4 is diagonalisable. If the intersection form is positive (negative) definite, it can be diagonalized to the identity matrix (negative identity matrix) over the integers. The original version^[1] of the theorem required the manifold to be simply connected, but it was later improved to apply to 4-manifolds with any fundamental group.^[2]