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? 



How to Tackle the Smooth Poincare Conjecture
In principle the Ricci flow (with surgery) could also be used to prove the smooth Poincare' conjecture in dimension $4$. 


There are some major problems to be overcome in this approach (problems which did not arise in dimension $3$, such as the absence of Hamilton-Ivey pinching estimates, and the new "hole-punch" singularities, as opposed to the "neck-pinch") and I know that some people are indeed working on these issues.


If you are interested, there are quite a few papers on Ricci flow on $4$-manifolds, starting with the work of Hamilton ([here](http://www.ams.org/mathscinet-getitem?mr=0862046) and [here](http://www.ams.org/mathscinet-getitem?mr=1456308)) and more recent work of Chen-Zhu ([here](http://www.ams.org/mathscinet-getitem?mr=2258799) and [here](http://arxiv.org/abs/0810.1999)) and many others.


