Exotic sphere

Description

An exotic sphere is a smooth manifold $\Sigma$ that is homeomorphic to the standard sphere $S^n$ but not diffeomorphic to it. Equivalently, it is $S^n$ with a smooth structure that is not diffeomorphic to the standard one.

Milnor found the first examples in dimension $7$. He took total spaces of bundles over $S^4$ with fibre $S^3$. For suitable bundles he built a Morse function with exactly two critical points, so the total space is homeomorphic to $S^7$. He then defined an invariant of such a manifold from a compact $8$-manifold that it bounds, using the Hirzebruch signature theorem. This invariant is zero for the standard $S^7$ and nonzero for some of the bundles, so these are not diffeomorphic to $S^7$.