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. Milnor's 7-spheres give nice counterexamples to this result in dimension 7, and exotic $\mathbb{R}^4$'s give nice counterexamples in dimension 4. But I don't know about dimensions 5 and 6. Is the result true or false in dimensions 5 and 6? And, if false, what are some classic counterexamples, and do stronger constraints -- say compactness or closedness -- happen to make it true?



exotic differentiable structures on manifolds in dimensions 5 and 6
It is false in dimension 5 and 6. Spheres happen to be standard, but some other (compact and closed) manifolds happen to admit different smooth (and PL) structures.


Simple example are tori. For example, $\mathbb T^5$ admits 3 different PL structures that give rise to 3 different differentiable structures. See, e.g., Hsiang, Shaneson "Fake tori" or Wall's book on surgery.


