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…)