Construction of Exotic Spheres

Milnor was constructing exotic spheres (at least in dimension 7) by bundle theory. Having proven the existence of such an exotic beast, I wonder if something as this is possible:

Let $\mathbb{S}^n$ be the n-sphere with a standard structure, $\Sigma^n$ be one of the exotic spheres. Topologically, we have $\mathbb{S}^n = \mathbb{D}^n \cup_{\partial\mathbb{D}^n} \mathbb{D}^n$, where boundary identification is understood.

Question: does there exist a $g:\partial\mathbb{D}^n \rightarrow \partial\mathbb{D}^n$, where I don't specify (on purpose) what kind of a map $g$ is, such that $\mathbb{D}^n \cup_g \mathbb{D}^n = \Sigma^n$ ?

Solution 1:

Yes. This procedure is called clutching (and the resulting spheres are clutched spheres or twisted spheres In this procedure $g$ is a diffeomorphism.

If $g$ extends to a diffeomorphism of the whole ball, it is easy to see that $\Sigma$ is diffeomorphic to the standard sphere. On the other hand, as $g$ does extend to a homeomorphism of the whole ball (by Alexander's trick), $\Sigma$ is homeomorphc to the standard ball.

A theorem of Smale guarantees that in dimension $>4$, every exotic sphere can be constructed this way. In dimension $\leq 3$, we know there is no exotic spheres. And in dimension $4$, no exotic clutched sphere exists (because every diffeomorphism of $S^3$ extends to $D^4$, a theorem due to Cerf), but we still don't know whether exotic spheres exist.