TITLE: How to Tackle the Smooth Poincare Conjecture
QUESTION [33 upvotes]: The last remaining problem in this whole "everything is a sphere" business, is the Smooth Poincare Conjecture in dimension 4: If $X\simeq_\text{homo.eq.} S^4$ then $X\approx_\text{diffeo} S^4$.  Freedman showed that this holds if we replace "diffeomorphism" by "homeomorphism", so another viewpoint would be that $S^4$ has no exotic smooth structures.
Restated: If $X$ is a connected, closed smooth 4-manifold with $\pi_1X=1$ and $H_*X\cong H_*S^4$, then $X\approx_\text{diffeo}S^4$.
This was told to me by Michael Hutchings, who motivationally remarked a way to solve it (Edit: not an original suggestion of his):
Find a symplectic structure on $X-\lbrace pt\rbrace$ which is standard near the puncture-point.
Then we're done by "Recognition of $\mathbb{R}^4$": Let $(M,\omega)$ be a noncompact symplectic 4-manifold such that $H_\ast(M)\cong H_\ast (pt)$. Suppose there exist compact sets $K_0\subset M$ and $K_1\subset\mathbb{R}^4$ and a symplectomorphism $\phi:(M-K_0,\omega)\to(\mathbb{R}^4-K_1,\omega_\text{std})$. Then $\phi$ extends to a symplectomorphism $(M,\omega)\to(\mathbb{R}^4,\omega_\text{std})$, after removing slightly bigger compact sets.
I am ignorant to the size of this wall (the conjecture) and the ability to make an indent in it. But the above idea is pretty cool, even if not anymore helpful than the original statement. The Poincare Conjecture (in dimension 3) was solved using Hamilton's idea of Ricci flow. This leads me to ask:
Is there another idea proposed to tackle this conjecture? Or a failed attempt?

REPLY [22 votes]: It seems to me your question implicitly assumes that the 4d smooth Poincaré conjecture (S4PC) is true.  But if you were to take a poll of experts  I think you would find that most of them suspect it is false.  (See also Ryan's comment to the original question.)  There are many large families of potential counterexamples to the S4PC.  These are smooth 4-manifolds which are homotopy equivalent (and hence homeomorphic) to the 4-sphere, but no one knows how to prove that they are diffeomorphic to the 4-sphere.  
(Akbulut and Gompf have made recent progress in showing that some, but not nearly all, of these potential counterexamples are standard 4-spheres, so I think people are more open to the idea that the S4PC might be true than they were four years ago.)
There are similarly many suspected counterexamples to the closely related Andrews-Curtis conjecture.  Andrew Casson has some interesting (and unpublished) numerical evidence that the AC conjecture is false.
So if you are interested in this problem, you might want to spend as much time trying to prove that one of the potential counterexamples is an actual counterexample as you do trying to prove the conjecture.