  We prove that a hyper-K\"ahler fourfold satisfying a mild topological
assumption is of K3$^{[2]}$ deformation type. This proves in particular a
conjecture of O'Grady stating that hyper-K\"ahler fourfolds of K3$^{[2]}$
numerical type are of K3$^{[2]}$ deformation type. Our topological assumption
concerns the existence of two integral degree-2 cohomology classes satisfying
certain numerical intersection conditions.
  There are two main ingredients in the proof. We first prove a topological
version of the statement, by showing that our topological assumption forces the
Betti numbers, the Fujiki constant, and the Huybrechts-Riemann-Roch polynomial
of the hyper-K\"ahler fourfold to be the same as those of K3$^{[2]}$
hyper-K\"ahler fourfolds. The key part of the article is then to prove the
hyper-K\"ahler SYZ conjecture for hyper-K\"ahler fourfolds for divisor classes
satisfying the numerical condition mentioned above.
