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 were homeomorphic but not diffeomorphic to the 7-sphere . 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 -dimensional manifold which is known to be homeomorphic to the sphere . There are a number of criteria for this: for instance, could admit a cover by two charts, or could admit a function with only two critical points. The goal is to prove then that is *not* diffeomorphic to . Obviously the standard invariants in topology see only homotopy type and are useless at telling apart and . One needs to define an invariant that relies on the smooth structure of in some way.

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