Exotic 7-Spheres

In this talk, we will go over Milnor's famous paper showing the existence of exotic spheres in dimension 7; that is, closed smooth manifolds which are homeomorphic to the 7-dimensional sphere, but not diffeomorphic to it. We will construct such exotic spheres as generalizations of the quaternionic Hopf fibration, and then distinguish them via a mod-7 invariant for smooth 7-dimensional homology spheres, which is constructed in a manner analogous to the Rokhlin invariant.