# Enter Simon Donaldson

Almost immediately after the first one, a second thunderbolt struck.

Theorem (Donaldson 1982). The diagonal quadratic forms $m\langle 1\rangle$ and $m\langle -1\rangle$ are the only definite forms that can be realized as intersection forms of a smooth 4-manifold.

Hence, out of the vast realm of definite quadratic forms, only a single one is realized by a smooth manifold for a given signature. In particular, combining Freedman's and Donaldson's results and the classification of quadratic forms, we get a succinct statement.

Corollary. Smooth, simply connected 4-manifolds are homeomorphic if and only if their intersection forms have the same rank, signature and parity.

The proof of this theorem uses differential geometry, specifically gauge theory, marking the start of a conquest of low-dimensional geometry by physics.