It is known that $\mathbb{R}^4$ has exotic smooth structures, and there are many such examples in higher dimensions, such as the famous 7-sphere. My (probably very naive) question is, for every $n\geq4$, does there exist an $n$-manifold with exotic smooth structures?


In other words, for every $n\geq4$, does there exist topological $n$-manifolds which admit more than one diffeomorphism class of smooth structures?



Does every dimension $n\geq4$ admit a manifold with an exotic smooth structure?
Yes. For every $n\ge 5$ there are exotic tori. 


In fact, the PL-structures on $T^n$ are in one-to-one correspondence with $H^3(T^n;\mathbb{Z}/2)$, and every one of these is smoothable (Reference: "Surgery on Compact Manifolds" by C. T. C. Wall, Chapter 15A). Since any smooth manifold admits a unique PL-structure up to PL-isomorphism, it follows that there are many manifolds homeomorphic but not diffeomorphic to the standard torus.


