Exotic spheres and homotopy groups: sanity check

Question

There is a homomorphism $\Theta_n \to \Pi_n/J_n$ where $\Theta_n$ is used to denote the group of diffeomorphism classes of $n$-spheres (with connected sum), $\Pi_n$ denote the $n$-th stable homotopy group and $J_n$ the image of the Whitehead homomorphism. For certain $n$ it is an isomorphism. Until recently I thought we use knowledge of this map, $\Theta_n$ and $J_n$ to compute $\Pi_n$. But it seems to go the other way around. Is it true that spectral sequences and other tools are used to determine $\Pi_n$ and from there one may determine $\Theta_n$? Furthermore, is it true that we can only do this for $n\geq 5$ although the most interesting dimension would be $4$?

Answer 1

As far as I know, we use our knowledge of $\Pi_n$, $\mathrm{Im}(J_n)$, and $bP_{n+1}$ to compute $\Theta_n$. $bP_{n+1}$ is fully understood (except when $n = 126$), while $\mathrm{Im}(J_n)$ is difficult to compute and no general procedure exists. So determining $\Theta_n$ is more of a matter of determining $\Pi_n$ and $\mathrm{Im}(J_n)$.

The methods of Kervaire-Milnor work for all dimensions, with the caveat that we must remember the actual definition of $\Theta_n$. $\Theta_n$ is actually the group of $h$-cobordism classes of oriented homotopy $n$-spheres. By the work of Smale on the $h$-cobordism theorem, this means that $\Theta_n$ is the group of exotic spheres for $n \geq 5$, which is the way many people think of $\Theta_n$. $\Theta_1$ and $\Theta_2$ are trivial for well-known reasons. $\Theta_3$ is trivial because the Poincaré conjecture is true.

Now the story for $\Theta_4$ is as follows. Freedman proved that the $h$-cobordism theorem holds for $n = 4$ topologically, i.e. $M^4$ and $N^4$ are $h$-cobordant if and only if they are homeomorphic. On the other hand, Freedman's topological classification of $4$-manifolds shows that $S^4$ is the only homotopy $4$-sphere up to homeomorphism. Hence $\Theta_4$ is trivial. On the other hand, Donaldson showed that the smooth $h$-cobordism theorem does not hold: if $M^4$ and $N^4$ are $h$-cobordant, then they are not necessarily diffeomorphic. So despite the fact that $\Theta_4$ is trivial, there may still exist exotic $4$-spheres, since $4$-manifolds can be $h$-cobordant without being diffeomorphic.