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.
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An obstruction to the manifold version was formulated by Andrew Casson and Dennis Sullivan in 1967–9 (originally in the simply-connected case), using the Rochlin invariant and the cohomology group H3(M;Z/2Z).