What is the relationship between diffeomorphisms of the sphere modulo isotopy and exotic spheres?

Question

In his "Classification of (n-1)-connected 2n-dimensional manifolds and the discovery of exotic spheres", Milnor observes that since his exotic 7-spheres admit a Morse function with only two critical points, they are diffeomorphic to two 7-disks glued along a diffeomorphism $g : S^6 \to S^6$, which can't be isotopic to the identity due to the fact the result can't be diffeomorphic to $S^7$ (here I suppose Milnor forgot to comment that $g$ preserves orientation).

How far does this relationship go in general? If I take an orientation-preserving diffeomorphism $g : S^n \to S^n$ which is not isotopic to the identity, and construct a topological sphere by gluing two copies of $D^{n + 1}$ along $g$, will that sphere be exotic? Can every exotic sphere be obtained in this way?

Answer 1

Look under "Twisted Spheres" in the Wikipedia article: https://en.wikipedia.org/wiki/Exotic_sphere. For n>6, all diffeomorphisms not isotopic to the identity give an exotic sphere. Edit: As Mike Miller points out and the article as well, all exotic spheres are thus obtainable.

Answer 2

Grumpy Parsnip answered the question, but let me summarize the situation and add some other aspects of the story.

Denote by

• $Diff^\partial(D^n)$ the group of diffeomorphisms of the $n$-disc that are the identity on a neighborhood of the boundary, by
• $Diff^+(S^n)$ the group of orientation preserving diffeomorphisms and by
• $\Theta_n$ the group of exotic spheres (which coincides with the group of h-cobordism classes of homotopy spheres provided $n\neq 4$ by the h-cobordism theorem).

We have maps $$\pi_0(Diff^\partial(D^n))\rightarrow \pi_0(Diff^+(S^n))\rightarrow\mu(Diff^+(S^n))\rightarrow \Theta_{n+1},$$

where

• $\mu(Diff(M))$ denotes the group of pseudoisotopy classes of diffeomorphisms of a manifold $M$,
• the first map is induced by the map $Diff^\partial(D^n)\rightarrow Diff^+(S^n)$ that crushes the boundary of the $n$-disc,
• the second map is provided as isotopy implies pseudoisotopy and
• the last map is given by the twisted spheres construction (gluing two $n+1$ discs via the given diffeomorphism to obtain a, possibly exotic, spheres).

What can we say about the quality of these maps?

• The last map is an isomorphism for $n\neq 4 $ by Smale's h-cobordism theorem.
• The middle map is an isomorphism for $n\ge 5$ by the pseudoistopy theorem of Cerf which says that $\mu(Diff(M))\cong \pi_0(Diff(M))$ for simply connected manifolds of dimension $n\ge 5$.

So we are left with the question in which dimensions the first map is an isomorphism as well. We have a map $Diff^+(S^n)\rightarrow SO(n+1)$ given by taking the derivative at a fixed point $x$ and identifying the frame bundle of $S^n$ with $SO(n+1)$. This is a fibration with fiber the group of diffeomorphism that fix $x$ and the tangent space of $x$, which is homotopy equivalent to the group of diffeomorphisms fixing $x$ and a neighborhood thereof, which is $Diff^\partial(D^n)$. We hence get a fibration sequence $$Diff^\partial(D^n)\rightarrow Diff^+(S^n)\rightarrow SO(n+1),$$ which splits up to homotopy (a splitting is given by the inclusion $SO(n+1)\subseteq Diff^+(S^n)$), so we get $$Diff^+(S^n)\simeq SO(n+1)\times Diff^\partial(D^n).$$ As $\pi_0(SO(n+1))$ is trivial, the first map in the above chain is always an isomorphism.

Now the question arises whether actually $SO(n+1)$ is equivalent to the path component of the identity of $Diff^+(S^n)$. This is

• true in dimensions 1-3,
• unknown in dimension 4 and
• wrong in dimensions 5 and above.