It's a result of low-dimensional topology that in dimensions 3 and lower, two manifolds are homeomorphic if and only if they are diffeomorphic. The 28 differentiable structures on Milnor's 7-sphere give a nice counterexample to this result in dimension 7, and I have heard the result is also false in dimension 4. But I don't know about dimensions 5 and 6. So my question comes in two parts:
-- is there a relatively famous or classic counterexample to the result in dimension 4?
-- is the result true or false in dimensions 5 and 6, and if false, what are some classic counterexamples?