The diagonalizability theorem (Donaldson 1983a, 1983b, 1987a): If the intersection form of a smooth, closed, simply connected 4-manifold is positive- or negative-definite then it is diagonalizable over the integers.
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For compact simply-connected manifolds of dimension 4 Simon Donaldson found examples with an infinite number of inequivalent PL structures, and Michael Freedman found the E8 manifold which not only has no PL structure, but is not even homeomorphic to a simplicial complex.