Exotic smooth structure and exotic sphere

Let $X$ and $Y$ be smooth manifolds. Then $X$ is an exotic smooth structure on $Y$ when the underlying topological manifolds of $X$ and $Y$ are homeomorphic and $X$ is not diffeomorphic to $Y$. An exotic smooth $n$-sphere is a closed connected smooth $n$-manifold that is homeomorphic, but not diffeomorphic, to the standard smooth sphere $S^n$.

Two clarifications belong to the definition. First, an orientation is extra data when oriented classes are compared: a homeomorphism or diffeomorphism of the underlying manifolds need not preserve any chosen orientation, and an exotic sphere admits no diffeomorphism to the standard sphere in either orientation. Second, the definition concerns the smooth category: a homotopy equivalence alone does not assert a homeomorphism, so no exoticity statement in this library is derived from homotopy data by itself. The existence of exotic spheres is proved later on this page by exhibiting an explicit seven-dimensional example.

The homeomorphism and diffeomorphism predicates fix the two categories being compared, and the negation is the exoticness assertion. The reference smooth sphere is the standard round $S^n$ with its standard smooth structure; a manifold counted as an exotic $n$-sphere therefore carries a smooth structure that is not diffeomorphic to that one, while its underlying topological manifold is still homeomorphic to $S^n$. This is exactly the distinction Milnor's 1956 paper introduced, and it is the distinction that separates the exotic seven-spheres of this page from the standard seven-sphere.