Differentiable structure on $S^4$

Question

How to show that the claim that there is exactly one differentiable structure on $S^4$ implies the smooth four-dimensional Poincaré conjecture (homotopy equivalent to $S^4$ implies diffeomorphic to $S^4$)?

Answer 1

If $M$ is a closed smooth four-manifold homotopy equivalent to $S^4$, then $M$ is homeomorphic to $S^4$. This follows from Freedman's Theorem.

Theorem (Freedman): For any unimodular form $b$ over $\mathbb{Z}$, there is a closed simply connected topological four-manifold whose intersection form is isomorphic to $b$. If $b$ is even, the manifold is unique up to homeomorphism, while if $b$ is odd, there are two such manifolds (which are distinguished by their Kirby-Siebenmann invariant).

The Kirby-Siebenmann invariant is the obstruction to admitting a PL structure. As a smooth structure induces a PL structure, smooth manifolds have vanishing Kirby-Siebenmann invariant. Therefore, we obtain the following:

Corollary: Two closed smooth simply connected four-manifolds are homeomorphic if and only if they have isomorphic intersection forms.

If $M$ is a closed smooth simply connected four-manifold homotopy equivalent to $S^4$, then it has intersection form zero. But $S^4$ is another closed smooth simply connected four-manifold with intersection form zero, so by the above, $M$ is homeomorphic to $S^4$ (the topological four-dimensional Poincaré conjecture). That is, $M$ can be viewed as the topological manifold $S^4$ with a choice of smooth structure. If $S^4$ were to admit only one smooth structure up to diffeomorphism, then $M$ would be diffeomorphic to $S^4$ (the smooth four-dimensional Poincaré conjecture).