It has been known since Milnor’s famous paper that two smooth manifolds can be homeomorphic without being diffeomorphic. Milnor showed that certain sphere bundles over {S^4} were homeomorphic but not diffeomorphic to the 7-sphere {S^7}. In later papers, Milnor constructed a number of additional examples of exotic spheres.

In this post, I’d like to give a detailed presentation of the argument in Milnor’s first paper.

1. Distinguishing homeomorphic manifolds

Suppose you have a {4k-1}-dimensional manifold {M} which is known to be homeomorphic to the sphere {S^{4k-1}}. There are a number of criteria for this: for instance, {M} could admit a cover by two charts, or {M} could admit a function with only two critical points. The goal is to prove then that {M} is not diffeomorphic to {S^{4k-1}}. Obviously the standard invariants in topology see only homotopy type and are useless at telling apart {M} and {S^{4k-1}}. One needs to define an invariant that relies on the smooth structure of {M} in some way.

It can be shown that any such {M} is an oriented boundary, {M = \partial B}, for a {4k}-manifold {B}. This is a deep fact, but in practice, the manifolds {M} come with explicit {B}‘s already, and one might as well define the invariant below on boundaries. Milnor’s strategy is to define an invariant in terms of {B} (which will depend very much on the smooth structure on {M}), in such a way that it will turn out to not depend on the choice of {B}. (more…)