Georgia Institute of Technology College of Sciences

In his 1956 paper "On manifolds homeomorphic to the 7-sphere'', John Milnor constructed some examples of manifolds that are homeomorphic, but not diffeomorphic, to the standard unit sphere. They are now called exotic 7-spheres. This example established that the differential structure of a manifold can carry information not given by its topological structure. Thus, Milnor founded differential topology as a stand-alone field. On my first reading of the paper, I thought that many of the choices Milnor made on his road to constructing the first exotic spheres seemed rather strange and arbitrary. Why 7-spheres? Why look for them among $S^3$-bundles over $S^4$ with structure group $SO(4)$? And what's the motivation behind his complicated mod 7-valued lambda invariant that detects exotica in these examples? Fortunately, Milnor answered some of these questions in his essay "Classification of $(n-1)$-connected $2n$-dimensional manifolds and the discovery of exotic spheres''. This talk is an attempt to understand this essay.