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?

non-smoothis somewhat misleading let me clarify this point: Perelman used the Ricci flow (with surgery) to prove the 3-dimensional Poincare conjecture. In dimension three it doesn't matter whether you sayhomeomorphicordiffeomorphic. $\endgroup$ – Robert Haslhofer May 20 '12 at 6:44deeprationale -- it's just that for "sufficiently large" 4-manifolds, there appears to be always arguments to this effect. And "sufficiently large" manifolds have been getting smaller and smaller recently. See Stern's notes from this talk: math.uci.edu/~rstern/Cornell_2012.pdf $\endgroup$ – Ryan Budney May 20 '12 at 7:38